path: root/src/hyper/pages/ZDSOLVE.pht

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch1}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull1}{ZeroDimensionalSolvePackageXmpPageEmpty1}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull1}{\hidepaste}
\tab{5}\spadcommand{R := Integer\bound{R }}
\indentrel{3}\begin{verbatim}
   (1)  Integer
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty1}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty1}{ZeroDimensionalSolvePackageXmpPagePatch1}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty1}{\showpaste}
\tab{5}\spadcommand{R := Integer\bound{R }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch2}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull2}{ZeroDimensionalSolvePackageXmpPageEmpty2}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull2}{\hidepaste}
\tab{5}\spadcommand{ls : List Symbol := [x,y,z,t]\bound{ls }}
\indentrel{3}\begin{verbatim}
   (2)  [x,y,z,t]
                                      Type: List Symbol
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty2}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty2}{ZeroDimensionalSolvePackageXmpPagePatch2}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty2}{\showpaste}
\tab{5}\spadcommand{ls : List Symbol := [x,y,z,t]\bound{ls }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch3}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull3}{ZeroDimensionalSolvePackageXmpPageEmpty3}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull3}{\hidepaste}
\tab{5}\spadcommand{ls2 : List Symbol := [x,y,z,t,new()$Symbol]\bound{ls2 }}
\indentrel{3}\begin{verbatim}
   (3)  [x,y,z,t,%A]
                                      Type: List Symbol
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty3}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty3}{ZeroDimensionalSolvePackageXmpPagePatch3}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty3}{\showpaste}
\tab{5}\spadcommand{ls2 : List Symbol := [x,y,z,t,new()$Symbol]\bound{ls2 }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch4}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull4}{ZeroDimensionalSolvePackageXmpPageEmpty4}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull4}{\hidepaste}
\tab{5}\spadcommand{pack := ZDSOLVE(R,ls,ls2)\free{ls }\free{ls2 }\free{R }\bound{pack }}
\indentrel{3}\begin{verbatim}
   (4)
  ZeroDimensionalSolvePackage(Integer,[x,y,z,t],[x,y,z,t,
  %A])
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty4}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty4}{ZeroDimensionalSolvePackageXmpPagePatch4}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty4}{\showpaste}
\tab{5}\spadcommand{pack := ZDSOLVE(R,ls,ls2)\free{ls }\free{ls2 }\free{R }\bound{pack }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch5}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull5}{ZeroDimensionalSolvePackageXmpPageEmpty5}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull5}{\hidepaste}
\tab{5}\spadcommand{p1 := x**2*y*z + x*y**2*z + x*y*z**2 + x*y*z + x*y + x*z + y*z\bound{p1 }}
\indentrel{3}\begin{verbatim}
             2       2     2
   (5)  x y z  + (x y  + (x  + x + 1)y + x)z + x y
                               Type: Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty5}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty5}{ZeroDimensionalSolvePackageXmpPagePatch5}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty5}{\showpaste}
\tab{5}\spadcommand{p1 := x**2*y*z + x*y**2*z + x*y*z**2 + x*y*z + x*y + x*z + y*z\bound{p1 }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch6}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull6}{ZeroDimensionalSolvePackageXmpPageEmpty6}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull6}{\hidepaste}
\tab{5}\spadcommand{p2 := x**2*y**2*z + x*y**2*z**2 + x**2*y*z + x*y*z + y*z + x + z\bound{p2 }}
\indentrel{3}\begin{verbatim}
           2 2     2 2     2
   (6)  x y z  + (x y  + (x  + x + 1)y + 1)z + x
                               Type: Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty6}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty6}{ZeroDimensionalSolvePackageXmpPagePatch6}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty6}{\showpaste}
\tab{5}\spadcommand{p2 := x**2*y**2*z + x*y**2*z**2 + x**2*y*z + x*y*z + y*z + x + z\bound{p2 }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch7}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull7}{ZeroDimensionalSolvePackageXmpPageEmpty7}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull7}{\hidepaste}
\tab{5}\spadcommand{p3 := x**2*y**2*z**2 + x**2*y**2*z + x*y**2*z + x*y*z + x*z + z + 1\bound{p3 }}
\indentrel{3}\begin{verbatim}
         2 2 2      2      2
   (7)  x y z  + ((x  + x)y  + x y + x + 1)z + 1
                               Type: Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty7}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty7}{ZeroDimensionalSolvePackageXmpPagePatch7}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty7}{\showpaste}
\tab{5}\spadcommand{p3 := x**2*y**2*z**2 + x**2*y**2*z + x*y**2*z + x*y*z + x*z + z + 1\bound{p3 }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch8}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull8}{ZeroDimensionalSolvePackageXmpPageEmpty8}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull8}{\hidepaste}
\tab{5}\spadcommand{lp := [p1, p2, p3]\free{p1 }\free{p2 }\free{p3 }\bound{lp }}
\indentrel{3}\begin{verbatim}
   (8)
         2       2     2
   [x y z  + (x y  + (x  + x + 1)y + x)z + x y,
       2 2     2 2     2
    x y z  + (x y  + (x  + x + 1)y + 1)z + x,
     2 2 2      2      2
    x y z  + ((x  + x)y  + x y + x + 1)z + 1]
                          Type: List Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty8}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty8}{ZeroDimensionalSolvePackageXmpPagePatch8}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty8}{\showpaste}
\tab{5}\spadcommand{lp := [p1, p2, p3]\free{p1 }\free{p2 }\free{p3 }\bound{lp }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch9}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull9}{ZeroDimensionalSolvePackageXmpPageEmpty9}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull9}{\hidepaste}
\tab{5}\spadcommand{triangSolve(lp)$pack\free{lp }\free{pack }}
\indentrel{3}\begin{verbatim}
   (9)
   [
     {
          20     19      18      17       16      15
         z   - 6z   - 41z   + 71z   + 106z   + 92z
       + 
             14       13       12       11       10
         197z   + 145z   + 257z   + 278z   + 201z
       + 
             9       8       7       6      5       4
         278z  + 257z  + 145z  + 197z  + 92z  + 106z
       + 
            3      2
         71z  - 41z  - 6z + 1
       ,

                      19            18             17
             14745844z   + 50357474z   - 130948857z
           + 
                         16             15             14
             - 185261586z   - 180077775z   - 338007307z
           + 
                         13             12             11
             - 275379623z   - 453190404z   - 474597456z
           + 
                         10             9             8
             - 366147695z   - 481433567z  - 430613166z
           + 
                         7             6             5
             - 261878358z  - 326073537z  - 163008796z
           + 
                         4             3            2
             - 177213227z  - 104356755z  + 65241699z
           + 
             9237732z - 1567348
        *
           y
       + 
                 19           18            17
         1917314z   + 6508991z   - 16973165z
       + 
                    16            15            14
         - 24000259z   - 23349192z   - 43786426z
       + 
                    13            12            11
         - 35696474z   - 58724172z   - 61480792z
       + 
                    10            9            8
         - 47452440z   - 62378085z  - 55776527z
       + 
                    7            6            5
         - 33940618z  - 42233406z  - 21122875z
       + 
                    4            3           2
         - 22958177z  - 13504569z  + 8448317z  + 1195888z
       + 
         - 202934
       ,

               3       2       3    2               2
             (z  - 2z)y  + (- z  - z  - 2z - 1)y - z  - z
           + 
             1
        *
           x
       + 
          2
         z  - 1
       }
     ]
             Type: List RegularChain(Integer,[x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty9}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty9}{ZeroDimensionalSolvePackageXmpPagePatch9}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty9}{\showpaste}
\tab{5}\spadcommand{triangSolve(lp)$pack\free{lp }\free{pack }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch10}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull10}{ZeroDimensionalSolvePackageXmpPageEmpty10}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull10}{\hidepaste}
\tab{5}\spadcommand{univariateSolve(lp)$pack\free{lp }\free{pack }}
\indentrel{3}\begin{verbatim}
   (10)
   [
     [
       complexRoots =
            12      11      10     9     8      7      6
           ?   - 12?   + 24?   + 4?  - 9?  + 27?  - 21?
         + 
              5     4     3      2
           27?  - 9?  + 4?  + 24?  - 12? + 1
       ,

       coordinates =
         [
                       11        10         9        8
             63x + 62%A   - 721%A   + 1220%A  + 705%A
           + 
                    7         6        5         4       3
             - 285%A  + 1512%A  - 735%A  + 1401%A  - 21%A
           + 
                  2
             215%A  + 1577%A - 142
           ,

                       11        10         9        8
             63y - 75%A   + 890%A   - 1682%A  - 516%A
           + 
                  7         6         5         4        3
             588%A  - 1953%A  + 1323%A  - 1815%A  + 426%A
           + 
                    2
             - 243%A  - 1801%A + 679
           ,
          z - %A]
       ]
     ,

                     6    5    4    3    2
     [complexRoots= ?  + ?  + ?  + ?  + ?  + ? + 1,
                          5       3
      coordinates= [x - %A ,y - %A ,z - %A]]
     ,

                     2
     [complexRoots= ?  + 5? + 1,
      coordinates= [x - 1,y - 1,z - %A]]
     ]
Type: List Record(complexRoots: SparseUnivariatePolynomial Integer,coordinates: List Polynomial Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty10}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty10}{ZeroDimensionalSolvePackageXmpPagePatch10}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty10}{\showpaste}
\tab{5}\spadcommand{univariateSolve(lp)$pack\free{lp }\free{pack }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch11}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull11}{ZeroDimensionalSolvePackageXmpPageEmpty11}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull11}{\hidepaste}
\tab{5}\spadcommand{lr := realSolve(lp)$pack\free{lp }\free{pack }\bound{lr }}
\indentrel{3}\begin{verbatim}
   (11)
   [
     [%R1,

         1184459    19   2335702    18   5460230    17
         ������� %R1   - ������� %R1   - ������� %R1
         1645371          548457          182819
       + 
         79900378    16   43953929    15   13420192    14
         �������� %R1   + �������� %R1   + �������� %R1
          1645371          548457           182819
       + 
         553986    13   193381378    12   35978916    11
         ������ %R1   + ��������� %R1   + �������� %R1
          3731           1645371           182819
       + 
         358660781    10   271667666    9   118784873    8
         ��������� %R1   + ��������� %R1  + ��������� %R1
          1645371           1645371           548457
       + 
         337505020    7   1389370    6   688291    5
         ��������� %R1  + ������� %R1  + ������ %R1
          1645371          11193          4459
       + 
         3378002    4   140671876    3   32325724    2
         ������� %R1  + ��������� %R1  + �������� %R1
          42189          1645371          548457
       + 
           8270       9741532
         - ���� %R1 - �������
            343       1645371
       ,

            91729    19   487915    18   4114333    17
         - ������ %R1   + ������ %R1   + ������� %R1
           705159         705159          705159
       + 
           1276987    16   13243117    15   16292173    14
         - ������� %R1   - �������� %R1   - �������� %R1
            235053          705159           705159
       + 
           26536060    13   722714    12   5382578    11
         - �������� %R1   - ������ %R1   - ������� %R1
            705159           18081          100737
       + 
           15449995    10   14279770    9   6603890    8
         - �������� %R1   - �������� %R1  - ������� %R1
            235053           235053          100737
       + 
           409930    7   37340389    6   34893715    5
         - ������ %R1  - �������� %R1  - �������� %R1
            6027          705159          705159
       + 
           26686318    4   801511    3   17206178    2
         - �������� %R1  - ������ %R1  - �������� %R1
            705159          26117         705159
       + 
           4406102       377534
         - ������� %R1 + ������
            705159       705159
       ]
     ,

     [%R2,

         1184459    19   2335702    18   5460230    17
         ������� %R2   - ������� %R2   - ������� %R2
         1645371          548457          182819
       + 
         79900378    16   43953929    15   13420192    14
         �������� %R2   + �������� %R2   + �������� %R2
          1645371          548457           182819
       + 
         553986    13   193381378    12   35978916    11
         ������ %R2   + ��������� %R2   + �������� %R2
          3731           1645371           182819
       + 
         358660781    10   271667666    9   118784873    8
         ��������� %R2   + ��������� %R2  + ��������� %R2
          1645371           1645371           548457
       + 
         337505020    7   1389370    6   688291    5
         ��������� %R2  + ������� %R2  + ������ %R2
          1645371          11193          4459
       + 
         3378002    4   140671876    3   32325724    2
         ������� %R2  + ��������� %R2  + �������� %R2
          42189          1645371          548457
       + 
           8270       9741532
         - ���� %R2 - �������
            343       1645371
       ,

            91729    19   487915    18   4114333    17
         - ������ %R2   + ������ %R2   + ������� %R2
           705159         705159          705159
       + 
           1276987    16   13243117    15   16292173    14
         - ������� %R2   - �������� %R2   - �������� %R2
            235053          705159           705159
       + 
           26536060    13   722714    12   5382578    11
         - �������� %R2   - ������ %R2   - ������� %R2
            705159           18081          100737
       + 
           15449995    10   14279770    9   6603890    8
         - �������� %R2   - �������� %R2  - ������� %R2
            235053           235053          100737
       + 
           409930    7   37340389    6   34893715    5
         - ������ %R2  - �������� %R2  - �������� %R2
            6027          705159          705159
       + 
           26686318    4   801511    3   17206178    2
         - �������� %R2  - ������ %R2  - �������� %R2
            705159          26117         705159
       + 
           4406102       377534
         - ������� %R2 + ������
            705159       705159
       ]
     ,

     [%R3,

         1184459    19   2335702    18   5460230    17
         ������� %R3   - ������� %R3   - ������� %R3
         1645371          548457          182819
       + 
         79900378    16   43953929    15   13420192    14
         �������� %R3   + �������� %R3   + �������� %R3
          1645371          548457           182819
       + 
         553986    13   193381378    12   35978916    11
         ������ %R3   + ��������� %R3   + �������� %R3
          3731           1645371           182819
       + 
         358660781    10   271667666    9   118784873    8
         ��������� %R3   + ��������� %R3  + ��������� %R3
          1645371           1645371           548457
       + 
         337505020    7   1389370    6   688291    5
         ��������� %R3  + ������� %R3  + ������ %R3
          1645371          11193          4459
       + 
         3378002    4   140671876    3   32325724    2
         ������� %R3  + ��������� %R3  + �������� %R3
          42189          1645371          548457
       + 
           8270       9741532
         - ���� %R3 - �������
            343       1645371
       ,

            91729    19   487915    18   4114333    17
         - ������ %R3   + ������ %R3   + ������� %R3
           705159         705159          705159
       + 
           1276987    16   13243117    15   16292173    14
         - ������� %R3   - �������� %R3   - �������� %R3
            235053          705159           705159
       + 
           26536060    13   722714    12   5382578    11
         - �������� %R3   - ������ %R3   - ������� %R3
            705159           18081          100737
       + 
           15449995    10   14279770    9   6603890    8
         - �������� %R3   - �������� %R3  - ������� %R3
            235053           235053          100737
       + 
           409930    7   37340389    6   34893715    5
         - ������ %R3  - �������� %R3  - �������� %R3
            6027          705159          705159
       + 
           26686318    4   801511    3   17206178    2
         - �������� %R3  - ������ %R3  - �������� %R3
            705159          26117         705159
       + 
           4406102       377534
         - ������� %R3 + ������
            705159       705159
       ]
     ,

     [%R4,

         1184459    19   2335702    18   5460230    17
         ������� %R4   - ������� %R4   - ������� %R4
         1645371          548457          182819
       + 
         79900378    16   43953929    15   13420192    14
         �������� %R4   + �������� %R4   + �������� %R4
          1645371          548457           182819
       + 
         553986    13   193381378    12   35978916    11
         ������ %R4   + ��������� %R4   + �������� %R4
          3731           1645371           182819
       + 
         358660781    10   271667666    9   118784873    8
         ��������� %R4   + ��������� %R4  + ��������� %R4
          1645371           1645371           548457
       + 
         337505020    7   1389370    6   688291    5
         ��������� %R4  + ������� %R4  + ������ %R4
          1645371          11193          4459
       + 
         3378002    4   140671876    3   32325724    2
         ������� %R4  + ��������� %R4  + �������� %R4
          42189          1645371          548457
       + 
           8270       9741532
         - ���� %R4 - �������
            343       1645371
       ,

            91729    19   487915    18   4114333    17
         - ������ %R4   + ������ %R4   + ������� %R4
           705159         705159          705159
       + 
           1276987    16   13243117    15   16292173    14
         - ������� %R4   - �������� %R4   - �������� %R4
            235053          705159           705159
       + 
           26536060    13   722714    12   5382578    11
         - �������� %R4   - ������ %R4   - ������� %R4
            705159           18081          100737
       + 
           15449995    10   14279770    9   6603890    8
         - �������� %R4   - �������� %R4  - ������� %R4
            235053           235053          100737
       + 
           409930    7   37340389    6   34893715    5
         - ������ %R4  - �������� %R4  - �������� %R4
            6027          705159          705159
       + 
           26686318    4   801511    3   17206178    2
         - �������� %R4  - ������ %R4  - �������� %R4
            705159          26117         705159
       + 
           4406102       377534
         - ������� %R4 + ������
            705159       705159
       ]
     ,

     [%R5,

         1184459    19   2335702    18   5460230    17
         ������� %R5   - ������� %R5   - ������� %R5
         1645371          548457          182819
       + 
         79900378    16   43953929    15   13420192    14
         �������� %R5   + �������� %R5   + �������� %R5
          1645371          548457           182819
       + 
         553986    13   193381378    12   35978916    11
         ������ %R5   + ��������� %R5   + �������� %R5
          3731           1645371           182819
       + 
         358660781    10   271667666    9   118784873    8
         ��������� %R5   + ��������� %R5  + ��������� %R5
          1645371           1645371           548457
       + 
         337505020    7   1389370    6   688291    5
         ��������� %R5  + ������� %R5  + ������ %R5
          1645371          11193          4459
       + 
         3378002    4   140671876    3   32325724    2
         ������� %R5  + ��������� %R5  + �������� %R5
          42189          1645371          548457
       + 
           8270       9741532
         - ���� %R5 - �������
            343       1645371
       ,

            91729    19   487915    18   4114333    17
         - ������ %R5   + ������ %R5   + ������� %R5
           705159         705159          705159
       + 
           1276987    16   13243117    15   16292173    14
         - ������� %R5   - �������� %R5   - �������� %R5
            235053          705159           705159
       + 
           26536060    13   722714    12   5382578    11
         - �������� %R5   - ������ %R5   - ������� %R5
            705159           18081          100737
       + 
           15449995    10   14279770    9   6603890    8
         - �������� %R5   - �������� %R5  - ������� %R5
            235053           235053          100737
       + 
           409930    7   37340389    6   34893715    5
         - ������ %R5  - �������� %R5  - �������� %R5
            6027          705159          705159
       + 
           26686318    4   801511    3   17206178    2
         - �������� %R5  - ������ %R5  - �������� %R5
            705159          26117         705159
       + 
           4406102       377534
         - ������� %R5 + ������
            705159       705159
       ]
     ,

     [%R6,

         1184459    19   2335702    18   5460230    17
         ������� %R6   - ������� %R6   - ������� %R6
         1645371          548457          182819
       + 
         79900378    16   43953929    15   13420192    14
         �������� %R6   + �������� %R6   + �������� %R6
          1645371          548457           182819
       + 
         553986    13   193381378    12   35978916    11
         ������ %R6   + ��������� %R6   + �������� %R6
          3731           1645371           182819
       + 
         358660781    10   271667666    9   118784873    8
         ��������� %R6   + ��������� %R6  + ��������� %R6
          1645371           1645371           548457
       + 
         337505020    7   1389370    6   688291    5
         ��������� %R6  + ������� %R6  + ������ %R6
          1645371          11193          4459
       + 
         3378002    4   140671876    3   32325724    2
         ������� %R6  + ��������� %R6  + �������� %R6
          42189          1645371          548457
       + 
           8270       9741532
         - ���� %R6 - �������
            343       1645371
       ,

            91729    19   487915    18   4114333    17
         - ������ %R6   + ������ %R6   + ������� %R6
           705159         705159          705159
       + 
           1276987    16   13243117    15   16292173    14
         - ������� %R6   - �������� %R6   - �������� %R6
            235053          705159           705159
       + 
           26536060    13   722714    12   5382578    11
         - �������� %R6   - ������ %R6   - ������� %R6
            705159           18081          100737
       + 
           15449995    10   14279770    9   6603890    8
         - �������� %R6   - �������� %R6  - ������� %R6
            235053           235053          100737
       + 
           409930    7   37340389    6   34893715    5
         - ������ %R6  - �������� %R6  - �������� %R6
            6027          705159          705159
       + 
           26686318    4   801511    3   17206178    2
         - �������� %R6  - ������ %R6  - �������� %R6
            705159          26117         705159
       + 
           4406102       377534
         - ������� %R6 + ������
            705159       705159
       ]
     ,

     [%R7,

         1184459    19   2335702    18   5460230    17
         ������� %R7   - ������� %R7   - ������� %R7
         1645371          548457          182819
       + 
         79900378    16   43953929    15   13420192    14
         �������� %R7   + �������� %R7   + �������� %R7
          1645371          548457           182819
       + 
         553986    13   193381378    12   35978916    11
         ������ %R7   + ��������� %R7   + �������� %R7
          3731           1645371           182819
       + 
         358660781    10   271667666    9   118784873    8
         ��������� %R7   + ��������� %R7  + ��������� %R7
          1645371           1645371           548457
       + 
         337505020    7   1389370    6   688291    5
         ��������� %R7  + ������� %R7  + ������ %R7
          1645371          11193          4459
       + 
         3378002    4   140671876    3   32325724    2
         ������� %R7  + ��������� %R7  + �������� %R7
          42189          1645371          548457
       + 
           8270       9741532
         - ���� %R7 - �������
            343       1645371
       ,

            91729    19   487915    18   4114333    17
         - ������ %R7   + ������ %R7   + ������� %R7
           705159         705159          705159
       + 
           1276987    16   13243117    15   16292173    14
         - ������� %R7   - �������� %R7   - �������� %R7
            235053          705159           705159
       + 
           26536060    13   722714    12   5382578    11
         - �������� %R7   - ������ %R7   - ������� %R7
            705159           18081          100737
       + 
           15449995    10   14279770    9   6603890    8
         - �������� %R7   - �������� %R7  - ������� %R7
            235053           235053          100737
       + 
           409930    7   37340389    6   34893715    5
         - ������ %R7  - �������� %R7  - �������� %R7
            6027          705159          705159
       + 
           26686318    4   801511    3   17206178    2
         - �������� %R7  - ������ %R7  - �������� %R7
            705159          26117         705159
       + 
           4406102       377534
         - ������� %R7 + ������
            705159       705159
       ]
     ,

     [%R8,

         1184459    19   2335702    18   5460230    17
         ������� %R8   - ������� %R8   - ������� %R8
         1645371          548457          182819
       + 
         79900378    16   43953929    15   13420192    14
         �������� %R8   + �������� %R8   + �������� %R8
          1645371          548457           182819
       + 
         553986    13   193381378    12   35978916    11
         ������ %R8   + ��������� %R8   + �������� %R8
          3731           1645371           182819
       + 
         358660781    10   271667666    9   118784873    8
         ��������� %R8   + ��������� %R8  + ��������� %R8
          1645371           1645371           548457
       + 
         337505020    7   1389370    6   688291    5
         ��������� %R8  + ������� %R8  + ������ %R8
          1645371          11193          4459
       + 
         3378002    4   140671876    3   32325724    2
         ������� %R8  + ��������� %R8  + �������� %R8
          42189          1645371          548457
       + 
           8270       9741532
         - ���� %R8 - �������
            343       1645371
       ,

            91729    19   487915    18   4114333    17
         - ������ %R8   + ������ %R8   + ������� %R8
           705159         705159          705159
       + 
           1276987    16   13243117    15   16292173    14
         - ������� %R8   - �������� %R8   - �������� %R8
            235053          705159           705159
       + 
           26536060    13   722714    12   5382578    11
         - �������� %R8   - ������ %R8   - ������� %R8
            705159           18081          100737
       + 
           15449995    10   14279770    9   6603890    8
         - �������� %R8   - �������� %R8  - ������� %R8
            235053           235053          100737
       + 
           409930    7   37340389    6   34893715    5
         - ������ %R8  - �������� %R8  - �������� %R8
            6027          705159          705159
       + 
           26686318    4   801511    3   17206178    2
         - �������� %R8  - ������ %R8  - �������� %R8
            705159          26117         705159
       + 
           4406102       377534
         - ������� %R8 + ������
            705159       705159
       ]
     ]
           Type: List List RealClosure Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty11}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty11}{ZeroDimensionalSolvePackageXmpPagePatch11}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty11}{\showpaste}
\tab{5}\spadcommand{lr := realSolve(lp)$pack\free{lp }\free{pack }\bound{lr }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch12}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull12}{ZeroDimensionalSolvePackageXmpPageEmpty12}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull12}{\hidepaste}
\tab{5}\spadcommand{\# lr\free{lr }}
\indentrel{3}\begin{verbatim}
   (12)  8
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty12}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty12}{ZeroDimensionalSolvePackageXmpPagePatch12}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty12}{\showpaste}
\tab{5}\spadcommand{\# lr\free{lr }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch13}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull13}{ZeroDimensionalSolvePackageXmpPageEmpty13}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull13}{\hidepaste}
\tab{5}\spadcommand{[[approximate(r,1/1000000) for r in point] for point in lr]\free{lr }}
\indentrel{3}\begin{verbatim}
   (13)
   [
        10048059
     [- ��������,
         2097152

        450305731698538794352439791383896641459673197621_
         17682193358812083855163140589245671760914236296_
         95777403099833360761048898228916578137094309838_
         59733113720258484693913237615701950676035760116_
         59174549868153820987890948515234203928112931261_
         41329856546977145464661495487825919941188447041_
         72244049192156726354215802806143775884436463441_
         0045253024786561923163288214175
      /
        450305728302524548851651180698582663508310069375_
         73204652805547068656449495775099168672018894380_
         90408354817931718593862797624551518983570793048_
         77442429148870882984032418920030143612331486020_
         08214437337907553112436329198648954217042289495_
         71290016119498807957023663865443069392027148979_
         68826671232335604349152343406892427528041733857_
         4817381189277066143312396681216
       ,

        210626076882347507389479868048601659624960714869_
         06855387636837150206396808586496507900558895056_
         46893309447097099937802187329095325898785247249_
         02071750498366048207515661873872451468533306001_
         12029646351663813515432559822002503052839810868_
         37110614842307026091211297929876896285681830479_
         05476005638076266490561846205530604781619178201_
         15887037891389881895
      /
        210626060949846419247211380481647417534196295329_
         64341024139031423687579676852738885855909759652_
         11778862189872881953943640246297357061959812326_
         10365979902512686325867656720234210687703171018_
         42474841814232889218376812370627084702957062184_
         85928867400771937828499200923760593314168901000_
         66637389634759811822855673103707202647449677622_
         83837629939232800768
       ]
     ,

        2563013
     [- �������,
        2097152

       -
           261134617679192778969861769323775771923825996_
            30635417819227523304401898996680729283384907_
            68623593207442125925986733815932243504809294_
            83752303023733723680666816744617300172727135_
            3311571242897
         /
           116522540050522253058398191600458914375722661_
            02768589900087901348199149409224137539839713_
            94019523433320408139928153188829495755455163_
            96341761930839597754479714023146923426903492_
            1938055593984
       ,

        357259455027591722109658872961578827299851705467_
         56032395781981410060340917352828265906219023044_
         66963941971038923304526273329316373757450061978_
         9892286110976997087250466235373
      /
        103954826934559893687707124483402605580081455112_
         01705922005223665917594096594864423391410294529_
         50265179989960104811875822530205346505131581243_
         9017247289173865014702966308864
       ]
     ,

        1715967
     [- �������,
        2097152

       -
           421309353378430352108483951797708239037726150_
            39695862248289984366060306560763593745648137_
            73498376603121267822565801436206939519951465_
            18222580524697287410022543952491
         /
           944181414418537445864969203434922405243659747_
            09662536639306419607958058825854931998401916_
            99917659443264824641135187383583888147867340_
            19307857605820364195856822304768
       ,

        763583334711264422251562542441083122534747566900_
         85893388341621725019049943763467308768090428452_
         08919919925302105720971453918982731389072591403_
         5
      /
        262418876408609719978429761047806663393423046789_
         58516022785809785037845492057884990196406022669_
         66026891580103543567625039018629887141284916756_
         48
       ]
     ,

         437701
     [- �������,
        2097152

        168310690863834958832217233265422591356298631318_
         19510314527501614414974734553281507213648683555_
         79646781603507777199075077835213366484533654913_
         83623741304759
      /
        168310686809521338900170998270591363896307766873_
         12261111677851880049074252262986803258878109626_
         14140298597366984264887998908377068799998454233_
         81649008099328
       ,

        496155010983501018642268101342210873595871480100_
         37606397079680966469128267084728344431172391721_
         9104249213450966312411133
      /
        496154987275773831550919207821020902985289711861_
         10971262363840408293765926191431317025486746479_
         2718363492160482442215424
       ]
     ,

       222801
     [�������,
      2097152

       -
           899488488040242826510759512197069142713604569_
            25419782755730018652137599215881377166961263_
            49101655220195142994932299137183241705867672_
            383477
         /
           116788999866502637217776510069188858270896996_
            02299347696908357524570777794164352094737678_
            66507769405888942764587718542434255625992456_
            372224
       ,

       -
           238970488813315687832080154437380839561277150_
            92084910198474529918855095465195254678390166_
            13593999693886640036283570552321155037871291_
            458703265
         /
           535548727364509632609040328668993190598822544_
            46854114332215938336811929575628336714686542_
            90340746993656285925599117602120446183443145_
            479421952
       ]
     ,

       765693
     [�������,
      2097152

        855896921981671626787324476117819808872469895861_
         66701402137657543220023032516857861186783308402_
         03328837654339523418704917749518340772512899000_
         391009630373148561
      /
        294144244553301079097642841137639349981558021594_
         58569179064525354957230138568189417023302287798_
         90141296236721138154231997238917322156711965244_
         4639331719460159488
       ,

       -
           205761823058257210124765032486024256111130258_
            15435888088439236627675493822416593627122907_
            77612800192921420574408948085193743688582762_
            2246433251878894899015
         /
           267159820332573553809795235350145022057631375_
            98908350970917225206427101987719026671839489_
            06289863714759678360292483949204616471537777_
            775324180661095366656
       ]
     ,

      5743879
     [�������,
      2097152

        107628881696890684795554639477357020817145672494_
         26186140236631235747689608504342639713980725465_
         92772662158833449797698617455397887562900072984_
         76800060834355318980169340872720504761255988923_
         27575638305286889535354218094827710589175426028_
         90060941949620874083007858366669453501766248414_
         88732463225
      /
        313176895708031794664846194002355204419037661345_
         85849862285496319161966016162197817656155325322_
         94746529648276430583810894079374566460757823146_
         88858119555602920851521883888320031865840746939_
         94260632605898286123092315966691297079864813198_
         51571942927230340622934023923486703042068153044_
         0845099008
       ,

       -
           211328669918575091836412047556545843787017248_
            98654859943898281353352644446652845575264927_
            34931691731407872701432935503473348172076098_
            72054584900878007756416053431789468836611952_
            97399805029441626685500981279619504962102219_
            42878089359674925850594427768502251789758706_
            752831632503615
         /
           162761558493798758024290662434710458088914446_
            61684597180431538394083725255333098080703636_
            99585502216011211087103263609551026027769414_
            08739114812622116813978168258743807532259146_
            61319399754572005223498385689642856344480185_
            62038272378787354460106106141518010935617205_
            1706396253618176
       ]
     ,

      19739877
     [��������,
       2097152

       -
           299724993683270330379901580486152094921504038_
            75007071777012857667201925305794224789535660_
            24359860143101547801638082771611160372212874_
            84777803580987284314922548423836585801362934_
            17053217025823333509180096017899370239859353_
            04900460493389873837030853410347089908880814_
            85398113201846458245880061539477074169948729_
            58759602107502158919488144768548710315309312_
            95467332190133702671098200902282300510751860_
            71859284570302778073977965258138627622392869_
            96106809728023675
         /
           230843327485227859072891008119181102390650414_
            13214326461239367948739333192706089607021381_
            93417647898360620229519176632937631786851455_
            01476602720625902225250555174182368889688380_
            66366025744317604722402920931967294751602472_
            68834121141893318848728661844434927287285112_
            89708076755286489505658586403317856591038706_
            50061128015164035227410373609905560544769495_
            27059227070809593049491257519554708879259595_
            52929920110858560812556635485429471554031675_
            979542656381353984
       ,

       -
           512818926354822848909627639786894008060093841_
            06630804594079663358450092641094905204598253_
            16250084723010047035024497436523038925818959_
            28931293158470135392762143543439867426304729_
            39091228501338519906964902315660943719943337_
            95070782624011727587749989296611277318372294_
            62420711653791043655457414608288470130554391_
            26204193548854107359401577758966028223645758_
            64611831512943973974715166920465061850603762_
            87516256195847052412587282839139194642913955
         /
           228828193977843933053120879318129047118363109_
            24553689903863908242435094636442362497730806_
            47438987739144921607794682653851741189091711_
            74186814511497833728419182249767586835872948_
            66447308566225526872092037244118004814057028_
            37198310642291275676195774614443815996713502_
            62939174978359004147086012775237299648862774_
            26724876224800632688088893248918508424949343_
            47337603075939980268208482904859678177751444_
            65749979827872616963053217673201717237252096
       ]
     ]
                       Type: List List Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty13}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty13}{ZeroDimensionalSolvePackageXmpPagePatch13}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty13}{\showpaste}
\tab{5}\spadcommand{[[approximate(r,1/1000000) for r in point] for point in lr]\free{lr }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch14}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull14}{ZeroDimensionalSolvePackageXmpPageEmpty14}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull14}{\hidepaste}
\tab{5}\spadcommand{lpr := positiveSolve(lp)$pack\free{lp }\free{pack }\bound{lpr }}
\indentrel{3}\begin{verbatim}
   (14)  []
           Type: List List RealClosure Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty14}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty14}{ZeroDimensionalSolvePackageXmpPagePatch14}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty14}{\showpaste}
\tab{5}\spadcommand{lpr := positiveSolve(lp)$pack\free{lp }\free{pack }\bound{lpr }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch15}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull15}{ZeroDimensionalSolvePackageXmpPageEmpty15}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull15}{\hidepaste}
\tab{5}\spadcommand{f0 := x**3 + y + z + t- 1\bound{f0 }}
\indentrel{3}\begin{verbatim}
                  3
   (15)  z + y + x  + t - 1
                               Type: Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty15}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty15}{ZeroDimensionalSolvePackageXmpPagePatch15}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty15}{\showpaste}
\tab{5}\spadcommand{f0 := x**3 + y + z + t- 1\bound{f0 }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch16}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull16}{ZeroDimensionalSolvePackageXmpPageEmpty16}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull16}{\hidepaste}
\tab{5}\spadcommand{f1 := x + y**3 + z + t -1\bound{f1 }}
\indentrel{3}\begin{verbatim}
              3
   (16)  z + y  + x + t - 1
                               Type: Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty16}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty16}{ZeroDimensionalSolvePackageXmpPagePatch16}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty16}{\showpaste}
\tab{5}\spadcommand{f1 := x + y**3 + z + t -1\bound{f1 }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch17}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull17}{ZeroDimensionalSolvePackageXmpPageEmpty17}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull17}{\hidepaste}
\tab{5}\spadcommand{f2 := x + y + z**3 + t-1\bound{f2 }}
\indentrel{3}\begin{verbatim}
          3
   (17)  z  + y + x + t - 1
                               Type: Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty17}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty17}{ZeroDimensionalSolvePackageXmpPagePatch17}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty17}{\showpaste}
\tab{5}\spadcommand{f2 := x + y + z**3 + t-1\bound{f2 }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch18}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull18}{ZeroDimensionalSolvePackageXmpPageEmpty18}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull18}{\hidepaste}
\tab{5}\spadcommand{f3 := x + y + z + t**3 -1\bound{f3 }}
\indentrel{3}\begin{verbatim}
                      3
   (18)  z + y + x + t  - 1
                               Type: Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty18}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty18}{ZeroDimensionalSolvePackageXmpPagePatch18}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty18}{\showpaste}
\tab{5}\spadcommand{f3 := x + y + z + t**3 -1\bound{f3 }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch19}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull19}{ZeroDimensionalSolvePackageXmpPageEmpty19}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull19}{\hidepaste}
\tab{5}\spadcommand{lf := [f0, f1, f2, f3]\free{f0 }\free{f1 }\free{f2 }\free{f3 }\bound{lf }}
\indentrel{3}\begin{verbatim}
   (19)
             3               3
   [z + y + x  + t - 1, z + y  + x + t - 1,
     3                               3
    z  + y + x + t - 1, z + y + x + t  - 1]
                          Type: List Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty19}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty19}{ZeroDimensionalSolvePackageXmpPagePatch19}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty19}{\showpaste}
\tab{5}\spadcommand{lf := [f0, f1, f2, f3]\free{f0 }\free{f1 }\free{f2 }\free{f3 }\bound{lf }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch20}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull20}{ZeroDimensionalSolvePackageXmpPageEmpty20}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull20}{\hidepaste}
\tab{5}\spadcommand{lts := triangSolve(lf)$pack\free{lf }\free{pack }\bound{lts }}
\indentrel{3}\begin{verbatim}
   (20)
   [
       2           3        3
     {t  + t + 1, z  - z - t  + t,

                 3      2
         (3z + 3t  - 3)y
       + 
            2      3           6     3            3      2
         (3z  + (6t  - 6)z + 3t  - 6t  + 3)y + (3t  - 3)z
       + 
            6     3          9     6     3
         (3t  - 6t  + 3)z + t  - 3t  + 5t  - 3t
       ,
      x + y + z}
     ,

       16     13     10     7      4      2
     {t   - 6t   + 9t   + 4t  + 15t  - 54t  + 27,

                     15            14             13
             4907232t   + 40893984t   - 115013088t
           + 
                      12            11             10
             22805712t   + 36330336t   + 162959040t
           + 
                         9             8             7
             - 159859440t  - 156802608t  + 117168768t
           + 
                       6             5             4
             126282384t  - 129351600t  + 306646992t
           + 
                       3              2
             475302816t  - 1006837776t  - 237269088t
           + 
             480716208
        *
           z
       + 
            54       51        48      46         45
         48t   - 912t   + 8232t   - 72t   - 46848t
       + 
              43          42        40          39
         1152t   + 186324t   - 3780t   - 543144t
       + 
                38         37           36         35
         - 3168t   - 21384t   + 1175251t   + 41184t
       + 
                34           33          32           31
         278003t   - 1843242t   - 301815t   - 1440726t
       + 
                 30           29           28          27
         1912012t   + 1442826t   + 4696262t   - 922481t
       + 
                   26            25          24
         - 4816188t   - 10583524t   - 208751t
       + 
                  23            22          21
         11472138t   + 16762859t   - 857663t
       + 
                    20            19           18
         - 19328175t   - 18270421t   + 4914903t
       + 
                  17            16           15
         22483044t   + 12926517t   - 8605511t
       + 
                    14           13           12
         - 17455518t   - 5014597t   + 8108814t
       + 
                 11          10           9           8
         8465535t   + 190542t   - 4305624t  - 2226123t
       + 
                7           6          5          4
         661905t  + 1169775t  + 226260t  - 209952t
       + 
                  3
         - 141183t  + 27216t
       ,

                 3      2
         (3z + 3t  - 3)y
       + 
            2      3           6     3            3      2
         (3z  + (6t  - 6)z + 3t  - 6t  + 3)y + (3t  - 3)z
       + 
            6     3          9     6     3
         (3t  - 6t  + 3)z + t  - 3t  + 5t  - 3t
       ,
                   3
      x + y + z + t  - 1}
     ,
              2                       2
    {t,z - 1,y  - 1,x + y}, {t - 1,z,y  - 1,x + y},
            2
    {t - 1,z  - 1,z y + 1,x},

       16     13     10     7      4      2
     {t   - 6t   + 9t   + 4t  + 15t  - 54t  + 27,

                     29            28             27
             4907232t   + 40893984t   - 115013088t
           + 
                       26             25             24
             - 1730448t   - 168139584t   + 738024480t
           + 
                         23             22              21
             - 195372288t   + 315849456t   - 2567279232t
           + 
                       20              19              18
             937147968t   + 1026357696t   + 4780488240t
           + 
                          17              16
             - 2893767696t   - 5617160352t
           + 
                          15              14
             - 3427651728t   + 5001100848t
           + 
                        13              12             11
             8720098416t   + 2331732960t   - 499046544t
           + 
                           10              9
             - 16243306272t   - 9748123200t
           + 
                        8               7               6
             3927244320t  + 25257280896t  + 10348032096t
           + 
                           5               4             3
             - 17128672128t  - 14755488768t  + 544086720t
           + 
                         2
             10848188736t  + 1423614528t - 2884297248
        *
           z
       + 
              68        65         62       60          59
         - 48t   + 1152t   - 13560t   + 360t   + 103656t
       + 
                57          56         54           53
         - 7560t   - 572820t   + 71316t   + 2414556t
       + 
              52          51           50         49
         2736t   - 402876t   - 7985131t   - 49248t
       + 
                 48            47          46           45
         1431133t   + 20977409t   + 521487t   - 2697635t
       + 
                    44           43           42
         - 43763654t   - 3756573t   - 2093410t
       + 
                  41            40            39
         71546495t   + 19699032t   + 35025028t
       + 
                    38            37             36
         - 89623786t   - 77798760t   - 138654191t
       + 
                  35             34             33
         87596128t   + 235642497t   + 349607642t
       + 
                    32             31             30
         - 93299834t   - 551563167t   - 630995176t
       + 
                   29             28             27
         186818962t   + 995427468t   + 828416204t
       + 
                     26              25              24
         - 393919231t   - 1076617485t   - 1609479791t
       + 
                   23              22              21
         595738126t   + 1198787136t   + 4342832069t
       + 
                      20              19              18
         - 2075938757t   - 4390835799t   - 4822843033t
       + 
                    17              16              15
         6932747678t   + 6172196808t   + 1141517740t
       + 
                      14              13              12
         - 4981677585t   - 9819815280t   - 7404299976t
       + 
                     11               10               9
         - 157295760t   + 29124027630t   + 14856038208t
       + 
                       8               7              6
         - 16184101410t  - 26935440354t  - 3574164258t
       + 
                     5               4              3
         10271338974t  + 11191425264t  + 6869861262t
       + 
                      2
         - 9780477840t  - 3586674168t + 2884297248
       ,

               3      3      2      6      3           9
             3z  + (6t  - 6)z  + (6t  - 12t  + 3)z + 2t
           + 
                 6    3
             - 6t  + t  + 3t
        *
           y
       + 
            3      3      6      3      2
         (3t  - 3)z  + (6t  - 12t  + 6)z
       + 
            9      6      3          12     9     6     3
         (4t  - 12t  + 11t  - 3)z + t   - 4t  + 5t  - 2t
       ,
                   3
      x + y + z + t  - 1}
     ,
            2
    {t - 1,z  - 1,y,x + z},

       8    7    6     5     4     3      2
     {t  + t  + t  - 2t  - 2t  - 2t  + 19t  + 19t - 8,

                     7           6           5
             2395770t  + 3934440t  - 3902067t
           + 
                        4           3            2
             - 10084164t  - 1010448t  + 32386932t
           + 
             22413225t - 10432368
        *
           z
       + 
                  7           6           5           4
         - 463519t  + 3586833t  + 9494955t  - 8539305t
       + 
                    3            2
         - 33283098t  + 35479377t  + 46263256t - 17419896
       ,

               4      3      3       6      3      2
             3z  + (9t  - 9)z  + (12t  - 24t  + 9)z
           + 
                    3                    6      4      3
             (- 152t  + 219t - 67)z - 41t  + 57t  + 25t
           + 
             - 57t + 16
        *
           y
       + 
            3      4      6      3      3
         (3t  - 3)z  + (9t  - 18t  + 9)z
       + 
                3              2
         (- 181t  + 270t - 89)z
       + 
               6       4      3                    7
         (- 92t  + 135t  + 49t  - 135t + 43)z + 27t
       + 
              6      4       3
         - 27t  - 54t  + 396t  - 486t + 144
       ,
                   3
      x + y + z + t  - 1}
     ,
            3
    {t,z - t  + 1,y - 1,x - 1}, {t - 1,z,y,x},
    {t,z - 1,y,x}, {t,z,y - 1,x}, {t,z,y,x - 1}]
             Type: List RegularChain(Integer,[x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty20}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty20}{ZeroDimensionalSolvePackageXmpPagePatch20}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty20}{\showpaste}
\tab{5}\spadcommand{lts := triangSolve(lf)$pack\free{lf }\free{pack }\bound{lts }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch21}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull21}{ZeroDimensionalSolvePackageXmpPageEmpty21}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull21}{\hidepaste}
\tab{5}\spadcommand{univariateSolve(lf)$pack\free{lf }\free{pack }}
\indentrel{3}\begin{verbatim}
   (21)
   [
     [complexRoots= ?,
      coordinates= [x - 1,y - 1,z + 1,t - %A]]
     ,
    [complexRoots= ?,coordinates= [x,y - 1,z,t - %A]],
    [complexRoots= ? - 1,coordinates= [x,y,z,t - %A]],
    [complexRoots= ?,coordinates= [x - 1,y,z,t - %A]],
    [complexRoots= ?,coordinates= [x,y,z - 1,t - %A]],

     [complexRoots= ? - 2,
      coordinates= [x - 1,y + 1,z,t - 1]]
     ,
    [complexRoots= ?,coordinates= [x + 1,y - 1,z,t - 1]],

     [complexRoots= ? - 1,
      coordinates= [x - 1,y + 1,z - 1,t]]
     ,

     [complexRoots= ? + 1,
      coordinates= [x + 1,y - 1,z - 1,t]]
     ,

                     6     3     2
     [complexRoots= ?  - 2?  + 3?  - 3,

       coordinates =
                 3                  3
         [2x + %A  + %A - 1, 2y + %A  + %A - 1, z - %A,
          t - %A]
       ]
     ,

                     5     3     2
     [complexRoots= ?  + 3?  - 2?  + 3? - 3,

       coordinates =
                              3
         [x - %A,y - %A,z + %A  + 2%A - 1,t - %A]
       ]
     ,

                     4    3     2
     [complexRoots= ?  - ?  - 2?  + 3,

       coordinates =
                3                 3
         [x + %A  - %A - 1, y + %A  - %A - 1,
                3
          z - %A  + 2%A + 1, t - %A]
       ]
     ,

     [complexRoots= ? + 1,
      coordinates= [x - 1,y - 1,z,t - %A]]
     ,

                     6     3     2
     [complexRoots= ?  + 2?  + 3?  - 3,

       coordinates =
                 3                          3
         [2x - %A  - %A - 1, y + %A, 2z - %A  - %A - 1,
          t + %A]
       ]
     ,

                     6      4      3      2
     [complexRoots= ?  + 12?  + 20?  - 45?  - 42? - 953,

       coordinates =
         [
                          5       4       3        2
             12609x + 23%A  + 49%A  - 46%A  + 362%A
           + 
             - 5015%A - 8239
           ,

                          5       4       3        2
             25218y + 23%A  + 49%A  - 46%A  + 362%A
           + 
             7594%A - 8239
           ,

                          5       4       3        2
             25218z + 23%A  + 49%A  - 46%A  + 362%A
           + 
             7594%A - 8239
           ,

                          5       4       3        2
             12609t + 23%A  + 49%A  - 46%A  + 362%A
           + 
             - 5015%A - 8239
           ]
       ]
     ,

                     5      3      2
     [complexRoots= ?  + 12?  - 16?  + 48? - 96,

       coordinates =
                 3
         [8x + %A  + 8%A - 8,2y - %A,2z - %A,2t - %A]
       ]
     ,

                     5    4     3     2
     [complexRoots= ?  + ?  - 5?  - 3?  + 9? + 3,

       coordinates =
                 3                   3
         [2x - %A  + 2%A - 1, 2y + %A  - 4%A + 1,
                 3                   3
          2z - %A  + 2%A - 1, 2t - %A  + 2%A - 1]
       ]
     ,

                     4     3     2
     [complexRoots= ?  - 3?  + 4?  - 6? + 13,

       coordinates =
                  3      2
         [9x - 2%A  + 4%A  - %A + 2,
                 3      2
          9y + %A  - 2%A  + 5%A - 1,
                 3      2
          9z + %A  - 2%A  + 5%A - 1,
                 3      2
          9t + %A  - 2%A  - 4%A - 1]
       ]
     ,

                     4      2
     [complexRoots= ?  - 11?  + 37,

       coordinates =
                 2             2                   2
         [3x - %A  + 7, 6y + %A  + 3%A - 7, 3z - %A  + 7,
                 2
          6t + %A  - 3%A - 7]
       ]
     ,

     [complexRoots= ? + 1,
      coordinates= [x - 1,y,z - 1,t + 1]]
     ,

     [complexRoots= ? + 2,
      coordinates= [x,y - 1,z - 1,t + 1]]
     ,

     [complexRoots= ? - 2,
      coordinates= [x,y - 1,z + 1,t - 1]]
     ,
    [complexRoots= ?,coordinates= [x,y + 1,z - 1,t - 1]],

     [complexRoots= ? - 2,
      coordinates= [x - 1,y,z + 1,t - 1]]
     ,
    [complexRoots= ?,coordinates= [x + 1,y,z - 1,t - 1]],

                     4     3      2
     [complexRoots= ?  + 5?  + 16?  + 30? + 57,

       coordinates =
                     3       2
         [151x + 15%A  + 54%A  + 104%A + 93,
                     3       2
          151y - 10%A  - 36%A  - 19%A - 62,
                    3       2
          151z - 5%A  - 18%A  - 85%A - 31,
                    3       2
          151t - 5%A  - 18%A  - 85%A - 31]
       ]
     ,

                     4    3     2
     [complexRoots= ?  - ?  - 2?  + 3,

       coordinates =
                3                  3
         [x - %A  + 2%A + 1, y + %A  - %A - 1, z - %A,
                3
          t + %A  - %A - 1]
       ]
     ,

                     4     3     2
     [complexRoots= ?  + 2?  - 8?  + 48,

       coordinates =
                 3
         [8x - %A  + 4%A - 8, 2y + %A,
                 3                   3
          8z + %A  - 8%A + 8, 8t - %A  + 4%A - 8]
       ]
     ,

                     5    4     3     2
     [complexRoots= ?  + ?  - 2?  - 4?  + 5? + 8,

       coordinates =
                 3            3            3
         [3x + %A  - 1,3y + %A  - 1,3z + %A  - 1,t - %A]
       ]
     ,

                     3
     [complexRoots= ?  + 3? - 1,
      coordinates= [x - %A,y - %A,z - %A,t - %A]]
     ]
Type: List Record(complexRoots: SparseUnivariatePolynomial Integer,coordinates: List Polynomial Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty21}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty21}{ZeroDimensionalSolvePackageXmpPagePatch21}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty21}{\showpaste}
\tab{5}\spadcommand{univariateSolve(lf)$pack\free{lf }\free{pack }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch22}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull22}{ZeroDimensionalSolvePackageXmpPageEmpty22}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull22}{\hidepaste}
\tab{5}\spadcommand{ts := lts.1\free{lts }\bound{ts }}
\indentrel{3}\begin{verbatim}
   (22)
     2           3        3
   {t  + t + 1, z  - z - t  + t,

               3      2
       (3z + 3t  - 3)y
     + 
          2      3           6     3            3      2
       (3z  + (6t  - 6)z + 3t  - 6t  + 3)y + (3t  - 3)z
     + 
          6     3          9     6     3
       (3t  - 6t  + 3)z + t  - 3t  + 5t  - 3t
     ,
    x + y + z}
                  Type: RegularChain(Integer,[x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty22}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty22}{ZeroDimensionalSolvePackageXmpPagePatch22}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty22}{\showpaste}
\tab{5}\spadcommand{ts := lts.1\free{lts }\bound{ts }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch23}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull23}{ZeroDimensionalSolvePackageXmpPageEmpty23}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull23}{\hidepaste}
\tab{5}\spadcommand{univariateSolve(ts)$pack\free{ts }\free{pack }}
\indentrel{3}\begin{verbatim}
   (23)
   [
                     4     3      2
     [complexRoots= ?  + 5?  + 16?  + 30? + 57,

       coordinates =
                     3       2
         [151x + 15%A  + 54%A  + 104%A + 93,
                     3       2
          151y - 10%A  - 36%A  - 19%A - 62,
                    3       2
          151z - 5%A  - 18%A  - 85%A - 31,
                    3       2
          151t - 5%A  - 18%A  - 85%A - 31]
       ]
     ,

                     4    3     2
     [complexRoots= ?  - ?  - 2?  + 3,

       coordinates =
                3                  3
         [x - %A  + 2%A + 1, y + %A  - %A - 1, z - %A,
                3
          t + %A  - %A - 1]
       ]
     ,

                     4     3     2
     [complexRoots= ?  + 2?  - 8?  + 48,

       coordinates =
                 3
         [8x - %A  + 4%A - 8, 2y + %A,
                 3                   3
          8z + %A  - 8%A + 8, 8t - %A  + 4%A - 8]
       ]
     ]
Type: List Record(complexRoots: SparseUnivariatePolynomial Integer,coordinates: List Polynomial Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty23}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty23}{ZeroDimensionalSolvePackageXmpPagePatch23}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty23}{\showpaste}
\tab{5}\spadcommand{univariateSolve(ts)$pack\free{ts }\free{pack }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch24}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull24}{ZeroDimensionalSolvePackageXmpPageEmpty24}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull24}{\hidepaste}
\tab{5}\spadcommand{realSolve(ts)$pack\free{ts }\free{pack }}
\indentrel{3}\begin{verbatim}
   (24)  []
           Type: List List RealClosure Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty24}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty24}{ZeroDimensionalSolvePackageXmpPagePatch24}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty24}{\showpaste}
\tab{5}\spadcommand{realSolve(ts)$pack\free{ts }\free{pack }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch25}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull25}{ZeroDimensionalSolvePackageXmpPageEmpty25}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull25}{\hidepaste}
\tab{5}\spadcommand{lr2 := realSolve(lf)$pack\free{lf }\free{pack }\bound{lr2 }}
\indentrel{3}\begin{verbatim}
   (25)
   [[0,- 1,1,1], [0,0,1,0], [1,0,0,0], [0,0,0,1],
    [0,1,0,0], [1,0,%R37,- %R37], [1,0,%R38,- %R38],
    [0,1,%R35,- %R35], [0,1,%R36,- %R36], [- 1,0,1,1],

     [%R32,

          1     15    2     14    1     13    4     12
         �� %R32   + �� %R32   + �� %R32   - �� %R32
         27          27          27          27
       + 
           11     11    4     10    1     9   14     8
         - �� %R32   - �� %R32   + �� %R32  + �� %R32
           27          27          27         27
       + 
          1     7   2     6   1     5   2     4       3
         �� %R32  + � %R32  + � %R32  + � %R32  + %R32
         27         9         3         9
       + 
         4     2
         � %R32  - %R32 - 2
         3
       ,

            1     15    1     14    1     13    2     12
         - �� %R32   - �� %R32   - �� %R32   + �� %R32
           54          27          54          27
       + 
         11     11    2     10    1     9    7     8
         �� %R32   + �� %R32   - �� %R32  - �� %R32
         54          27          54         27
       + 
            1     7   1     6   1     5   1     4       3
         - �� %R32  - � %R32  - � %R32  - � %R32  - %R32
           54         9         6         9
       + 
           2     2   1        3
         - � %R32  + � %R32 + �
           3         2        2
       ,

            1     15    1     14    1     13    2     12
         - �� %R32   - �� %R32   - �� %R32   + �� %R32
           54          27          54          27
       + 
         11     11    2     10    1     9    7     8
         �� %R32   + �� %R32   - �� %R32  - �� %R32
         54          27          54         27
       + 
            1     7   1     6   1     5   1     4       3
         - �� %R32  - � %R32  - � %R32  - � %R32  - %R32
           54         9         6         9
       + 
           2     2   1        3
         - � %R32  + � %R32 + �
           3         2        2
       ]
     ,

     [%R33,

          1     15    2     14    1     13    4     12
         �� %R33   + �� %R33   + �� %R33   - �� %R33
         27          27          27          27
       + 
           11     11    4     10    1     9   14     8
         - �� %R33   - �� %R33   + �� %R33  + �� %R33
           27          27          27         27
       + 
          1     7   2     6   1     5   2     4       3
         �� %R33  + � %R33  + � %R33  + � %R33  + %R33
         27         9         3         9
       + 
         4     2
         � %R33  - %R33 - 2
         3
       ,

            1     15    1     14    1     13    2     12
         - �� %R33   - �� %R33   - �� %R33   + �� %R33
           54          27          54          27
       + 
         11     11    2     10    1     9    7     8
         �� %R33   + �� %R33   - �� %R33  - �� %R33
         54          27          54         27
       + 
            1     7   1     6   1     5   1     4       3
         - �� %R33  - � %R33  - � %R33  - � %R33  - %R33
           54         9         6         9
       + 
           2     2   1        3
         - � %R33  + � %R33 + �
           3         2        2
       ,

            1     15    1     14    1     13    2     12
         - �� %R33   - �� %R33   - �� %R33   + �� %R33
           54          27          54          27
       + 
         11     11    2     10    1     9    7     8
         �� %R33   + �� %R33   - �� %R33  - �� %R33
         54          27          54         27
       + 
            1     7   1     6   1     5   1     4       3
         - �� %R33  - � %R33  - � %R33  - � %R33  - %R33
           54         9         6         9
       + 
           2     2   1        3
         - � %R33  + � %R33 + �
           3         2        2
       ]
     ,

     [%R34,

          1     15    2     14    1     13    4     12
         �� %R34   + �� %R34   + �� %R34   - �� %R34
         27          27          27          27
       + 
           11     11    4     10    1     9   14     8
         - �� %R34   - �� %R34   + �� %R34  + �� %R34
           27          27          27         27
       + 
          1     7   2     6   1     5   2     4       3
         �� %R34  + � %R34  + � %R34  + � %R34  + %R34
         27         9         3         9
       + 
         4     2
         � %R34  - %R34 - 2
         3
       ,

            1     15    1     14    1     13    2     12
         - �� %R34   - �� %R34   - �� %R34   + �� %R34
           54          27          54          27
       + 
         11     11    2     10    1     9    7     8
         �� %R34   + �� %R34   - �� %R34  - �� %R34
         54          27          54         27
       + 
            1     7   1     6   1     5   1     4       3
         - �� %R34  - � %R34  - � %R34  - � %R34  - %R34
           54         9         6         9
       + 
           2     2   1        3
         - � %R34  + � %R34 + �
           3         2        2
       ,

            1     15    1     14    1     13    2     12
         - �� %R34   - �� %R34   - �� %R34   + �� %R34
           54          27          54          27
       + 
         11     11    2     10    1     9    7     8
         �� %R34   + �� %R34   - �� %R34  - �� %R34
         54          27          54         27
       + 
            1     7   1     6   1     5   1     4       3
         - �� %R34  - � %R34  - � %R34  - � %R34  - %R34
           54         9         6         9
       + 
           2     2   1        3
         - � %R34  + � %R34 + �
           3         2        2
       ]
     ,
    [- 1,1,0,1], [- 1,1,1,0],

     [%R23,

            1     15    1     14    1     13    2     12
         - �� %R23   - �� %R23   - �� %R23   + �� %R23
           54          27          54          27
       + 
         11     11    2     10    1     9    7     8
         �� %R23   + �� %R23   - �� %R23  - �� %R23
         54          27          54         27
       + 
            1     7   1     6   1     5   1     4       3
         - �� %R23  - � %R23  - � %R23  - � %R23  - %R23
           54         9         6         9
       + 
           2     2   1        3
         - � %R23  + � %R23 + �
           3         2        2
       ,
      %R30,

                   1     15    1     14    1     13
         - %R30 + �� %R23   + �� %R23   + �� %R23
                  54          27          54
       + 
            2     12   11     11    2     10    1     9
         - �� %R23   - �� %R23   - �� %R23   + �� %R23
           27          54          27          54
       + 
          7     8    1     7   1     6   1     5   1     4
         �� %R23  + �� %R23  + � %R23  + � %R23  + � %R23
         27         54         9         6         9
       + 
         2     2   1        1
         � %R23  - � %R23 - �
         3         2        2
       ]
     ,

     [%R23,

            1     15    1     14    1     13    2     12
         - �� %R23   - �� %R23   - �� %R23   + �� %R23
           54          27          54          27
       + 
         11     11    2     10    1     9    7     8
         �� %R23   + �� %R23   - �� %R23  - �� %R23
         54          27          54         27
       + 
            1     7   1     6   1     5   1     4       3
         - �� %R23  - � %R23  - � %R23  - � %R23  - %R23
           54         9         6         9
       + 
           2     2   1        3
         - � %R23  + � %R23 + �
           3         2        2
       ,
      %R31,

                   1     15    1     14    1     13
         - %R31 + �� %R23   + �� %R23   + �� %R23
                  54          27          54
       + 
            2     12   11     11    2     10    1     9
         - �� %R23   - �� %R23   - �� %R23   + �� %R23
           27          54          27          54
       + 
          7     8    1     7   1     6   1     5   1     4
         �� %R23  + �� %R23  + � %R23  + � %R23  + � %R23
         27         54         9         6         9
       + 
         2     2   1        1
         � %R23  - � %R23 - �
         3         2        2
       ]
     ,

     [%R24,

            1     15    1     14    1     13    2     12
         - �� %R24   - �� %R24   - �� %R24   + �� %R24
           54          27          54          27
       + 
         11     11    2     10    1     9    7     8
         �� %R24   + �� %R24   - �� %R24  - �� %R24
         54          27          54         27
       + 
            1     7   1     6   1     5   1     4       3
         - �� %R24  - � %R24  - � %R24  - � %R24  - %R24
           54         9         6         9
       + 
           2     2   1        3
         - � %R24  + � %R24 + �
           3         2        2
       ,
      %R28,

                   1     15    1     14    1     13
         - %R28 + �� %R24   + �� %R24   + �� %R24
                  54          27          54
       + 
            2     12   11     11    2     10    1     9
         - �� %R24   - �� %R24   - �� %R24   + �� %R24
           27          54          27          54
       + 
          7     8    1     7   1     6   1     5   1     4
         �� %R24  + �� %R24  + � %R24  + � %R24  + � %R24
         27         54         9         6         9
       + 
         2     2   1        1
         � %R24  - � %R24 - �
         3         2        2
       ]
     ,

     [%R24,

            1     15    1     14    1     13    2     12
         - �� %R24   - �� %R24   - �� %R24   + �� %R24
           54          27          54          27
       + 
         11     11    2     10    1     9    7     8
         �� %R24   + �� %R24   - �� %R24  - �� %R24
         54          27          54         27
       + 
            1     7   1     6   1     5   1     4       3
         - �� %R24  - � %R24  - � %R24  - � %R24  - %R24
           54         9         6         9
       + 
           2     2   1        3
         - � %R24  + � %R24 + �
           3         2        2
       ,
      %R29,

                   1     15    1     14    1     13
         - %R29 + �� %R24   + �� %R24   + �� %R24
                  54          27          54
       + 
            2     12   11     11    2     10    1     9
         - �� %R24   - �� %R24   - �� %R24   + �� %R24
           27          54          27          54
       + 
          7     8    1     7   1     6   1     5   1     4
         �� %R24  + �� %R24  + � %R24  + � %R24  + � %R24
         27         54         9         6         9
       + 
         2     2   1        1
         � %R24  - � %R24 - �
         3         2        2
       ]
     ,

     [%R25,

            1     15    1     14    1     13    2     12
         - �� %R25   - �� %R25   - �� %R25   + �� %R25
           54          27          54          27
       + 
         11     11    2     10    1     9    7     8
         �� %R25   + �� %R25   - �� %R25  - �� %R25
         54          27          54         27
       + 
            1     7   1     6   1     5   1     4       3
         - �� %R25  - � %R25  - � %R25  - � %R25  - %R25
           54         9         6         9
       + 
           2     2   1        3
         - � %R25  + � %R25 + �
           3         2        2
       ,
      %R26,

                   1     15    1     14    1     13
         - %R26 + �� %R25   + �� %R25   + �� %R25
                  54          27          54
       + 
            2     12   11     11    2     10    1     9
         - �� %R25   - �� %R25   - �� %R25   + �� %R25
           27          54          27          54
       + 
          7     8    1     7   1     6   1     5   1     4
         �� %R25  + �� %R25  + � %R25  + � %R25  + � %R25
         27         54         9         6         9
       + 
         2     2   1        1
         � %R25  - � %R25 - �
         3         2        2
       ]
     ,

     [%R25,

            1     15    1     14    1     13    2     12
         - �� %R25   - �� %R25   - �� %R25   + �� %R25
           54          27          54          27
       + 
         11     11    2     10    1     9    7     8
         �� %R25   + �� %R25   - �� %R25  - �� %R25
         54          27          54         27
       + 
            1     7   1     6   1     5   1     4       3
         - �� %R25  - � %R25  - � %R25  - � %R25  - %R25
           54         9         6         9
       + 
           2     2   1        3
         - � %R25  + � %R25 + �
           3         2        2
       ,
      %R27,

                   1     15    1     14    1     13
         - %R27 + �� %R25   + �� %R25   + �� %R25
                  54          27          54
       + 
            2     12   11     11    2     10    1     9
         - �� %R25   - �� %R25   - �� %R25   + �� %R25
           27          54          27          54
       + 
          7     8    1     7   1     6   1     5   1     4
         �� %R25  + �� %R25  + � %R25  + � %R25  + � %R25
         27         54         9         6         9
       + 
         2     2   1        1
         � %R25  - � %R25 - �
         3         2        2
       ]
     ,
    [1,%R21,- %R21,0], [1,%R22,- %R22,0],
    [1,%R19,0,- %R19], [1,%R20,0,- %R20],
            1     3   1   1     3   1   1     3   1
    [%R17,- � %R17  + �,- � %R17  + �,- � %R17  + �],
            3         3   3         3   3         3
            1     3   1   1     3   1   1     3   1
    [%R18,- � %R18  + �,- � %R18  + �,- � %R18  + �]]
            3         3   3         3   3         3
           Type: List List RealClosure Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty25}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty25}{ZeroDimensionalSolvePackageXmpPagePatch25}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty25}{\showpaste}
\tab{5}\spadcommand{lr2 := realSolve(lf)$pack\free{lf }\free{pack }\bound{lr2 }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch26}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull26}{ZeroDimensionalSolvePackageXmpPageEmpty26}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull26}{\hidepaste}
\tab{5}\spadcommand{\#lr2\free{lr2 }}
\indentrel{3}\begin{verbatim}
   (26)  27
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty26}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty26}{ZeroDimensionalSolvePackageXmpPagePatch26}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty26}{\showpaste}
\tab{5}\spadcommand{\#lr2\free{lr2 }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch27}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull27}{ZeroDimensionalSolvePackageXmpPageEmpty27}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull27}{\hidepaste}
\tab{5}\spadcommand{lpr2 := positiveSolve(lf)$pack\free{lf }\free{pack }\bound{lpr2 }}
\indentrel{3}\begin{verbatim}
   (27)
            1     3   1   1     3   1   1     3   1
   [[%R40,- � %R40  + �,- � %R40  + �,- � %R40  + �]]
            3         3   3         3   3         3
           Type: List List RealClosure Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty27}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty27}{ZeroDimensionalSolvePackageXmpPagePatch27}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty27}{\showpaste}
\tab{5}\spadcommand{lpr2 := positiveSolve(lf)$pack\free{lf }\free{pack }\bound{lpr2 }}
\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch28}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull28}{ZeroDimensionalSolvePackageXmpPageEmpty28}
\pastebutton{ZeroDimensionalSolvePackageXmpPageFull28}{\hidepaste}
\tab{5}\spadcommand{[approximate(r,1/10**21)::Float for r in lpr2.1]\free{lpr2 }}
\indentrel{3}\begin{verbatim}
   (28)
   [0.3221853546 2608559291, 0.3221853546 2608559291,
    0.3221853546 2608559291, 0.3221853546 2608559291]
                                       Type: List Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty28}
\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty28}{ZeroDimensionalSolvePackageXmpPagePatch28}
\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty28}{\showpaste}
\tab{5}\spadcommand{[approximate(r,1/10**21)::Float for r in lpr2.1]\free{lpr2 }}
\end{paste}\end{patch}