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|
\begin{patch}{RegularTriangularSetXmpPagePatch1}
\begin{paste}{RegularTriangularSetXmpPageFull1}{RegularTriangularSetXmpPageEmpty1}
\pastebutton{RegularTriangularSetXmpPageFull1}{\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}{RegularTriangularSetXmpPageEmpty1}
\begin{paste}{RegularTriangularSetXmpPageEmpty1}{RegularTriangularSetXmpPagePatch1}
\pastebutton{RegularTriangularSetXmpPageEmpty1}{\showpaste}
\tab{5}\spadcommand{R := Integer\bound{R }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch2}
\begin{paste}{RegularTriangularSetXmpPageFull2}{RegularTriangularSetXmpPageEmpty2}
\pastebutton{RegularTriangularSetXmpPageFull2}{\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}{RegularTriangularSetXmpPageEmpty2}
\begin{paste}{RegularTriangularSetXmpPageEmpty2}{RegularTriangularSetXmpPagePatch2}
\pastebutton{RegularTriangularSetXmpPageEmpty2}{\showpaste}
\tab{5}\spadcommand{ls : List Symbol := [x,y,z,t]\bound{ls }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch3}
\begin{paste}{RegularTriangularSetXmpPageFull3}{RegularTriangularSetXmpPageEmpty3}
\pastebutton{RegularTriangularSetXmpPageFull3}{\hidepaste}
\tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }}
\indentrel{3}\begin{verbatim}
(3) OrderedVariableList [x,y,z,t]
Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty3}
\begin{paste}{RegularTriangularSetXmpPageEmpty3}{RegularTriangularSetXmpPagePatch3}
\pastebutton{RegularTriangularSetXmpPageEmpty3}{\showpaste}
\tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch4}
\begin{paste}{RegularTriangularSetXmpPageFull4}{RegularTriangularSetXmpPageEmpty4}
\pastebutton{RegularTriangularSetXmpPageFull4}{\hidepaste}
\tab{5}\spadcommand{E := IndexedExponents V\free{V }\bound{E }}
\indentrel{3}\begin{verbatim}
(4) IndexedExponents OrderedVariableList [x,y,z,t]
Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty4}
\begin{paste}{RegularTriangularSetXmpPageEmpty4}{RegularTriangularSetXmpPagePatch4}
\pastebutton{RegularTriangularSetXmpPageEmpty4}{\showpaste}
\tab{5}\spadcommand{E := IndexedExponents V\free{V }\bound{E }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch5}
\begin{paste}{RegularTriangularSetXmpPageFull5}{RegularTriangularSetXmpPageEmpty5}
\pastebutton{RegularTriangularSetXmpPageFull5}{\hidepaste}
\tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }}
\indentrel{3}\begin{verbatim}
(5)
NewSparseMultivariatePolynomial(Integer,OrderedVariable
List [x,y,z,t])
Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty5}
\begin{paste}{RegularTriangularSetXmpPageEmpty5}{RegularTriangularSetXmpPagePatch5}
\pastebutton{RegularTriangularSetXmpPageEmpty5}{\showpaste}
\tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch6}
\begin{paste}{RegularTriangularSetXmpPageFull6}{RegularTriangularSetXmpPageEmpty6}
\pastebutton{RegularTriangularSetXmpPageFull6}{\hidepaste}
\tab{5}\spadcommand{x: P := 'x\free{P }\bound{x }}
\indentrel{3}\begin{verbatim}
(6) x
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty6}
\begin{paste}{RegularTriangularSetXmpPageEmpty6}{RegularTriangularSetXmpPagePatch6}
\pastebutton{RegularTriangularSetXmpPageEmpty6}{\showpaste}
\tab{5}\spadcommand{x: P := 'x\free{P }\bound{x }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch7}
\begin{paste}{RegularTriangularSetXmpPageFull7}{RegularTriangularSetXmpPageEmpty7}
\pastebutton{RegularTriangularSetXmpPageFull7}{\hidepaste}
\tab{5}\spadcommand{y: P := 'y\free{P }\bound{y }}
\indentrel{3}\begin{verbatim}
(7) y
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty7}
\begin{paste}{RegularTriangularSetXmpPageEmpty7}{RegularTriangularSetXmpPagePatch7}
\pastebutton{RegularTriangularSetXmpPageEmpty7}{\showpaste}
\tab{5}\spadcommand{y: P := 'y\free{P }\bound{y }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch8}
\begin{paste}{RegularTriangularSetXmpPageFull8}{RegularTriangularSetXmpPageEmpty8}
\pastebutton{RegularTriangularSetXmpPageFull8}{\hidepaste}
\tab{5}\spadcommand{z: P := 'z\free{P }\bound{z }}
\indentrel{3}\begin{verbatim}
(8) z
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty8}
\begin{paste}{RegularTriangularSetXmpPageEmpty8}{RegularTriangularSetXmpPagePatch8}
\pastebutton{RegularTriangularSetXmpPageEmpty8}{\showpaste}
\tab{5}\spadcommand{z: P := 'z\free{P }\bound{z }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch9}
\begin{paste}{RegularTriangularSetXmpPageFull9}{RegularTriangularSetXmpPageEmpty9}
\pastebutton{RegularTriangularSetXmpPageFull9}{\hidepaste}
\tab{5}\spadcommand{t: P := 't\free{P }\bound{t }}
\indentrel{3}\begin{verbatim}
(9) t
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty9}
\begin{paste}{RegularTriangularSetXmpPageEmpty9}{RegularTriangularSetXmpPagePatch9}
\pastebutton{RegularTriangularSetXmpPageEmpty9}{\showpaste}
\tab{5}\spadcommand{t: P := 't\free{P }\bound{t }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch10}
\begin{paste}{RegularTriangularSetXmpPageFull10}{RegularTriangularSetXmpPageEmpty10}
\pastebutton{RegularTriangularSetXmpPageFull10}{\hidepaste}
\tab{5}\spadcommand{T := REGSET(R,E,V,P)\free{R }\free{E }\free{V }\free{P }\bound{T }}
\indentrel{3}\begin{verbatim}
(10)
RegularTriangularSet(Integer,IndexedExponents OrderedVa
riableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewS
parseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t]))
Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty10}
\begin{paste}{RegularTriangularSetXmpPageEmpty10}{RegularTriangularSetXmpPagePatch10}
\pastebutton{RegularTriangularSetXmpPageEmpty10}{\showpaste}
\tab{5}\spadcommand{T := REGSET(R,E,V,P)\free{R }\free{E }\free{V }\free{P }\bound{T }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch11}
\begin{paste}{RegularTriangularSetXmpPageFull11}{RegularTriangularSetXmpPageEmpty11}
\pastebutton{RegularTriangularSetXmpPageFull11}{\hidepaste}
\tab{5}\spadcommand{p1 := x ** 31 - x ** 6 - x - y\free{x }\free{y }\bound{p1 }}
\indentrel{3}\begin{verbatim}
31 6
(11) x - x - x - y
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty11}
\begin{paste}{RegularTriangularSetXmpPageEmpty11}{RegularTriangularSetXmpPagePatch11}
\pastebutton{RegularTriangularSetXmpPageEmpty11}{\showpaste}
\tab{5}\spadcommand{p1 := x ** 31 - x ** 6 - x - y\free{x }\free{y }\bound{p1 }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch12}
\begin{paste}{RegularTriangularSetXmpPageFull12}{RegularTriangularSetXmpPageEmpty12}
\pastebutton{RegularTriangularSetXmpPageFull12}{\hidepaste}
\tab{5}\spadcommand{p2 := x ** 8 - z\free{x }\free{z }\bound{p2 }}
\indentrel{3}\begin{verbatim}
8
(12) x - z
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty12}
\begin{paste}{RegularTriangularSetXmpPageEmpty12}{RegularTriangularSetXmpPagePatch12}
\pastebutton{RegularTriangularSetXmpPageEmpty12}{\showpaste}
\tab{5}\spadcommand{p2 := x ** 8 - z\free{x }\free{z }\bound{p2 }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch13}
\begin{paste}{RegularTriangularSetXmpPageFull13}{RegularTriangularSetXmpPageEmpty13}
\pastebutton{RegularTriangularSetXmpPageFull13}{\hidepaste}
\tab{5}\spadcommand{p3 := x ** 10 - t\free{x }\free{t }\bound{p3 }}
\indentrel{3}\begin{verbatim}
10
(13) x - t
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty13}
\begin{paste}{RegularTriangularSetXmpPageEmpty13}{RegularTriangularSetXmpPagePatch13}
\pastebutton{RegularTriangularSetXmpPageEmpty13}{\showpaste}
\tab{5}\spadcommand{p3 := x ** 10 - t\free{x }\free{t }\bound{p3 }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch14}
\begin{paste}{RegularTriangularSetXmpPageFull14}{RegularTriangularSetXmpPageEmpty14}
\pastebutton{RegularTriangularSetXmpPageFull14}{\hidepaste}
\tab{5}\spadcommand{lp := [p1, p2, p3]\free{p1 }\free{p2 }\free{p3 }\bound{lp }}
\indentrel{3}\begin{verbatim}
31 6 8 10
(14) [x - x - x - y,x - z,x - t]
Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty14}
\begin{paste}{RegularTriangularSetXmpPageEmpty14}{RegularTriangularSetXmpPagePatch14}
\pastebutton{RegularTriangularSetXmpPageEmpty14}{\showpaste}
\tab{5}\spadcommand{lp := [p1, p2, p3]\free{p1 }\free{p2 }\free{p3 }\bound{lp }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch15}
\begin{paste}{RegularTriangularSetXmpPageFull15}{RegularTriangularSetXmpPageEmpty15}
\pastebutton{RegularTriangularSetXmpPageFull15}{\hidepaste}
\tab{5}\spadcommand{zeroSetSplit(lp)$T\free{lp }\free{T }}
\indentrel{3}\begin{verbatim}
(15)
[
5 4 2 3 8 5 3 2
{z - t , t z y + 2z y - t + 2t + t - t ,
4 2
(t - t)x - t y - z }
]
Type: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty15}
\begin{paste}{RegularTriangularSetXmpPageEmpty15}{RegularTriangularSetXmpPagePatch15}
\pastebutton{RegularTriangularSetXmpPageEmpty15}{\showpaste}
\tab{5}\spadcommand{zeroSetSplit(lp)$T\free{lp }\free{T }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch16}
\begin{paste}{RegularTriangularSetXmpPageFull16}{RegularTriangularSetXmpPageEmpty16}
\pastebutton{RegularTriangularSetXmpPageFull16}{\hidepaste}
\tab{5}\spadcommand{lts := zeroSetSplit(lp,false)$T\free{lp }\free{T }\bound{lts }}
\indentrel{3}\begin{verbatim}
(16)
[
5 4 2 3 8 5 3 2
{z - t , t z y + 2z y - t + 2t + t - t ,
4 2
(t - t)x - t y - z }
,
3 5 2 3 2
{t - 1,z - t,t z y + 2z y + 1,z x - t},
{t,z,y,x}]
Type: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty16}
\begin{paste}{RegularTriangularSetXmpPageEmpty16}{RegularTriangularSetXmpPagePatch16}
\pastebutton{RegularTriangularSetXmpPageEmpty16}{\showpaste}
\tab{5}\spadcommand{lts := zeroSetSplit(lp,false)$T\free{lp }\free{T }\bound{lts }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch17}
\begin{paste}{RegularTriangularSetXmpPageFull17}{RegularTriangularSetXmpPageEmpty17}
\pastebutton{RegularTriangularSetXmpPageFull17}{\hidepaste}
\tab{5}\spadcommand{[coHeight(ts) for ts in lts]\free{lts }}
\indentrel{3}\begin{verbatim}
(17) [1,0,0]
Type: List NonNegativeInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty17}
\begin{paste}{RegularTriangularSetXmpPageEmpty17}{RegularTriangularSetXmpPagePatch17}
\pastebutton{RegularTriangularSetXmpPageEmpty17}{\showpaste}
\tab{5}\spadcommand{[coHeight(ts) for ts in lts]\free{lts }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch18}
\begin{paste}{RegularTriangularSetXmpPageFull18}{RegularTriangularSetXmpPageEmpty18}
\pastebutton{RegularTriangularSetXmpPageFull18}{\hidepaste}
\tab{5}\spadcommand{f1 := y**2*z+2*x*y*t-2*x-z\free{z }\free{x }\free{y }\free{t }\bound{f1 }}
\indentrel{3}\begin{verbatim}
2
(18) (2t y - 2)x + z y - z
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty18}
\begin{paste}{RegularTriangularSetXmpPageEmpty18}{RegularTriangularSetXmpPagePatch18}
\pastebutton{RegularTriangularSetXmpPageEmpty18}{\showpaste}
\tab{5}\spadcommand{f1 := y**2*z+2*x*y*t-2*x-z\free{z }\free{x }\free{y }\free{t }\bound{f1 }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch19}
\begin{paste}{RegularTriangularSetXmpPageFull19}{RegularTriangularSetXmpPageEmpty19}
\pastebutton{RegularTriangularSetXmpPageFull19}{\hidepaste}
\tab{5}\spadcommand{f2 := -x**3*z+ 4*x*y**2*z+ 4*x**2*y*t+ 2*y**3*t+ 4*x**2- 10*y**2+ 4*x*z- 10*y*t+ 2\free{z }\free{x }\free{y }\free{t }\bound{f2 }}
\indentrel{3}\begin{verbatim}
(19)
3 2 2 3 2
- z x + (4t y + 4)x + (4z y + 4z)x + 2t y - 10y
+
- 10t y + 2
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty19}
\begin{paste}{RegularTriangularSetXmpPageEmpty19}{RegularTriangularSetXmpPagePatch19}
\pastebutton{RegularTriangularSetXmpPageEmpty19}{\showpaste}
\tab{5}\spadcommand{f2 := -x**3*z+ 4*x*y**2*z+ 4*x**2*y*t+ 2*y**3*t+ 4*x**2- 10*y**2+ 4*x*z- 10*y*t+ 2\free{z }\free{x }\free{y }\free{t }\bound{f2 }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch20}
\begin{paste}{RegularTriangularSetXmpPageFull20}{RegularTriangularSetXmpPageEmpty20}
\pastebutton{RegularTriangularSetXmpPageFull20}{\hidepaste}
\tab{5}\spadcommand{f3 := 2*y*z*t+x*t**2-x-2*z\free{z }\free{x }\free{y }\free{t }\bound{f3 }}
\indentrel{3}\begin{verbatim}
2
(20) (t - 1)x + 2t z y - 2z
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty20}
\begin{paste}{RegularTriangularSetXmpPageEmpty20}{RegularTriangularSetXmpPagePatch20}
\pastebutton{RegularTriangularSetXmpPageEmpty20}{\showpaste}
\tab{5}\spadcommand{f3 := 2*y*z*t+x*t**2-x-2*z\free{z }\free{x }\free{y }\free{t }\bound{f3 }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch21}
\begin{paste}{RegularTriangularSetXmpPageFull21}{RegularTriangularSetXmpPageEmpty21}
\pastebutton{RegularTriangularSetXmpPageFull21}{\hidepaste}
\tab{5}\spadcommand{f4 := -x*z**3+ 4*y*z**2*t+ 4*x*z*t**2+ 2*y*t**3+ 4*x*z+ 4*z**2-10*y*t- 10*t**2+2\free{z }\free{x }\free{y }\free{t }\bound{f4 }}
\indentrel{3}\begin{verbatim}
(21)
3 2 2 3 2
(- z + (4t + 4)z)x + (4t z + 2t - 10t)y + 4z
+
2
- 10t + 2
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty21}
\begin{paste}{RegularTriangularSetXmpPageEmpty21}{RegularTriangularSetXmpPagePatch21}
\pastebutton{RegularTriangularSetXmpPageEmpty21}{\showpaste}
\tab{5}\spadcommand{f4 := -x*z**3+ 4*y*z**2*t+ 4*x*z*t**2+ 2*y*t**3+ 4*x*z+ 4*z**2-10*y*t- 10*t**2+2\free{z }\free{x }\free{y }\free{t }\bound{f4 }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch22}
\begin{paste}{RegularTriangularSetXmpPageFull22}{RegularTriangularSetXmpPageEmpty22}
\pastebutton{RegularTriangularSetXmpPageFull22}{\hidepaste}
\tab{5}\spadcommand{lf := [f1, f2, f3, f4]\free{f1 }\free{f2 }\free{f3 }\free{f4 }\bound{lf }}
\indentrel{3}\begin{verbatim}
(22)
2
[(2t y - 2)x + z y - z,
3 2 2 3
- z x + (4t y + 4)x + (4z y + 4z)x + 2t y
+
2
- 10y - 10t y + 2
,
2
(t - 1)x + 2t z y - 2z,
3 2 2 3 2
(- z + (4t + 4)z)x + (4t z + 2t - 10t)y + 4z
+
2
- 10t + 2
]
Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty22}
\begin{paste}{RegularTriangularSetXmpPageEmpty22}{RegularTriangularSetXmpPagePatch22}
\pastebutton{RegularTriangularSetXmpPageEmpty22}{\showpaste}
\tab{5}\spadcommand{lf := [f1, f2, f3, f4]\free{f1 }\free{f2 }\free{f3 }\free{f4 }\bound{lf }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch23}
\begin{paste}{RegularTriangularSetXmpPageFull23}{RegularTriangularSetXmpPageEmpty23}
\pastebutton{RegularTriangularSetXmpPageFull23}{\hidepaste}
\tab{5}\spadcommand{zeroSetSplit(lf)$T\free{lf }\free{T }}
\indentrel{3}\begin{verbatim}
(23)
[
2 8 6 2
{t - 1, z - 16z + 256z - 256, t y - 1,
3 2
(z - 8z)x - 8z + 16}
,
2 2 2
{3t + 1,z - 7t - 1,y + t,x + z},
8 6 2 3 2
{t - 10t + 10t - 1,z,(t - 5t)y - 5t + 1,x},
2 2
{t + 3,z - 4,y + t,x - z}]
Type: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty23}
\begin{paste}{RegularTriangularSetXmpPageEmpty23}{RegularTriangularSetXmpPagePatch23}
\pastebutton{RegularTriangularSetXmpPageEmpty23}{\showpaste}
\tab{5}\spadcommand{zeroSetSplit(lf)$T\free{lf }\free{T }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch24}
\begin{paste}{RegularTriangularSetXmpPageFull24}{RegularTriangularSetXmpPageEmpty24}
\pastebutton{RegularTriangularSetXmpPageFull24}{\hidepaste}
\tab{5}\spadcommand{lts2 := zeroSetSplit(lf,false)$T\free{lf }\free{T }\bound{lts2 }}
\indentrel{3}\begin{verbatim}
(24)
8 6 2 3 2
[{t - 10t + 10t - 1,z,(t - 5t)y - 5t + 1,x},
2 8 6 2
{t - 1, z - 16z + 256z - 256, t y - 1,
3 2
(z - 8z)x - 8z + 16}
,
2 2 2
{3t + 1,z - 7t - 1,y + t,x + z},
2 2
{t + 3,z - 4,y + t,x - z}]
Type: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty24}
\begin{paste}{RegularTriangularSetXmpPageEmpty24}{RegularTriangularSetXmpPagePatch24}
\pastebutton{RegularTriangularSetXmpPageEmpty24}{\showpaste}
\tab{5}\spadcommand{lts2 := zeroSetSplit(lf,false)$T\free{lf }\free{T }\bound{lts2 }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch25}
\begin{paste}{RegularTriangularSetXmpPageFull25}{RegularTriangularSetXmpPageEmpty25}
\pastebutton{RegularTriangularSetXmpPageFull25}{\hidepaste}
\tab{5}\spadcommand{[coHeight(ts) for ts in lts2]\free{lts2 }}
\indentrel{3}\begin{verbatim}
(25) [0,0,0,0]
Type: List NonNegativeInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty25}
\begin{paste}{RegularTriangularSetXmpPageEmpty25}{RegularTriangularSetXmpPagePatch25}
\pastebutton{RegularTriangularSetXmpPageEmpty25}{\showpaste}
\tab{5}\spadcommand{[coHeight(ts) for ts in lts2]\free{lts2 }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch26}
\begin{paste}{RegularTriangularSetXmpPageFull26}{RegularTriangularSetXmpPageEmpty26}
\pastebutton{RegularTriangularSetXmpPageFull26}{\hidepaste}
\tab{5}\spadcommand{degrees := [degree(ts) for ts in lts2]\free{lts2 }\bound{degrees }}
\indentrel{3}\begin{verbatim}
(26) [8,16,4,4]
Type: List NonNegativeInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty26}
\begin{paste}{RegularTriangularSetXmpPageEmpty26}{RegularTriangularSetXmpPagePatch26}
\pastebutton{RegularTriangularSetXmpPageEmpty26}{\showpaste}
\tab{5}\spadcommand{degrees := [degree(ts) for ts in lts2]\free{lts2 }\bound{degrees }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch27}
\begin{paste}{RegularTriangularSetXmpPageFull27}{RegularTriangularSetXmpPageEmpty27}
\pastebutton{RegularTriangularSetXmpPageFull27}{\hidepaste}
\tab{5}\spadcommand{reduce(+,degrees)\free{degrees }}
\indentrel{3}\begin{verbatim}
(27) 32
Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty27}
\begin{paste}{RegularTriangularSetXmpPageEmpty27}{RegularTriangularSetXmpPagePatch27}
\pastebutton{RegularTriangularSetXmpPageEmpty27}{\showpaste}
\tab{5}\spadcommand{reduce(+,degrees)\free{degrees }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch28}
\begin{paste}{RegularTriangularSetXmpPageFull28}{RegularTriangularSetXmpPageEmpty28}
\pastebutton{RegularTriangularSetXmpPageFull28}{\hidepaste}
\tab{5}\spadcommand{u : R := 2\free{R }\bound{u }}
\indentrel{3}\begin{verbatim}
(28) 2
Type: Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty28}
\begin{paste}{RegularTriangularSetXmpPageEmpty28}{RegularTriangularSetXmpPagePatch28}
\pastebutton{RegularTriangularSetXmpPageEmpty28}{\showpaste}
\tab{5}\spadcommand{u : R := 2\free{R }\bound{u }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch29}
\begin{paste}{RegularTriangularSetXmpPageFull29}{RegularTriangularSetXmpPageEmpty29}
\pastebutton{RegularTriangularSetXmpPageFull29}{\hidepaste}
\tab{5}\spadcommand{q1 := 2*(u-1)**2+ 2*(x-z*x+z**2)+ y**2*(x-1)**2- 2*u*x+ 2*y*t*(1-x)*(x-z)+ 2*u*z*t*(t-y)+ u**2*t**2*(1-2*z)+ 2*u*t**2*(z-x)+ 2*u*t*y*(z-1)+ 2*u*z*x*(y+1)+ (u**2-2*u)*z**2*t**2+ 2*u**2*z**2+ 4*u*(1-u)*z+ t**2*(z-x)**2\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q1 }}
\indentrel{3}\begin{verbatim}
(29)
2 2 2
(y - 2t y + t )x
+
2 2 2
(- 2y + ((2t + 4)z + 2t)y + (- 2t + 2)z - 4t - 2)x
+
2 2 2 2
y + (- 2t z - 4t)y + (t + 10)z - 8z + 4t + 2
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty29}
\begin{paste}{RegularTriangularSetXmpPageEmpty29}{RegularTriangularSetXmpPagePatch29}
\pastebutton{RegularTriangularSetXmpPageEmpty29}{\showpaste}
\tab{5}\spadcommand{q1 := 2*(u-1)**2+ 2*(x-z*x+z**2)+ y**2*(x-1)**2- 2*u*x+ 2*y*t*(1-x)*(x-z)+ 2*u*z*t*(t-y)+ u**2*t**2*(1-2*z)+ 2*u*t**2*(z-x)+ 2*u*t*y*(z-1)+ 2*u*z*x*(y+1)+ (u**2-2*u)*z**2*t**2+ 2*u**2*z**2+ 4*u*(1-u)*z+ t**2*(z-x)**2\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q1 }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch30}
\begin{paste}{RegularTriangularSetXmpPageFull30}{RegularTriangularSetXmpPageEmpty30}
\pastebutton{RegularTriangularSetXmpPageFull30}{\hidepaste}
\tab{5}\spadcommand{q2 := t*(2*z+1)*(x-z)+ y*(z+2)*(1-x)+ u*(u-2)*t+ u*(1-2*u)*z*t+ u*y*(x+u-z*x-1)+ u*(u+1)*z**2*t\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q2 }}
\indentrel{3}\begin{verbatim}
2
(30) (- 3z y + 2t z + t)x + (z + 4)y + 4t z - 7t z
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty30}
\begin{paste}{RegularTriangularSetXmpPageEmpty30}{RegularTriangularSetXmpPagePatch30}
\pastebutton{RegularTriangularSetXmpPageEmpty30}{\showpaste}
\tab{5}\spadcommand{q2 := t*(2*z+1)*(x-z)+ y*(z+2)*(1-x)+ u*(u-2)*t+ u*(1-2*u)*z*t+ u*y*(x+u-z*x-1)+ u*(u+1)*z**2*t\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q2 }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch31}
\begin{paste}{RegularTriangularSetXmpPageFull31}{RegularTriangularSetXmpPageEmpty31}
\pastebutton{RegularTriangularSetXmpPageFull31}{\hidepaste}
\tab{5}\spadcommand{q3 := -u**2*(z-1)**2+ 2*z*(z-x)-2*(x-1)\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q3 }}
\indentrel{3}\begin{verbatim}
2
(31) (- 2z - 2)x - 2z + 8z - 2
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty31}
\begin{paste}{RegularTriangularSetXmpPageEmpty31}{RegularTriangularSetXmpPagePatch31}
\pastebutton{RegularTriangularSetXmpPageEmpty31}{\showpaste}
\tab{5}\spadcommand{q3 := -u**2*(z-1)**2+ 2*z*(z-x)-2*(x-1)\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q3 }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch32}
\begin{paste}{RegularTriangularSetXmpPageFull32}{RegularTriangularSetXmpPageEmpty32}
\pastebutton{RegularTriangularSetXmpPageFull32}{\hidepaste}
\tab{5}\spadcommand{q4 := u**2+4*(z-x**2)+3*y**2*(x-1)**2- 3*t**2*(z-x)**2 +3*u**2*t**2*(z-1)**2+u**2*z*(z-2)+6*u*t*y*(z+x+z*x-1)\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q4 }}
\indentrel{3}\begin{verbatim}
(32)
2 2 2 2 2
(3y - 3t - 4)x + (- 6y + (12t z + 12t)y + 6t z)x
+
2 2 2 2
3y + (12t z - 12t)y + (9t + 4)z + (- 24t - 4)z
+
2
12t + 4
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty32}
\begin{paste}{RegularTriangularSetXmpPageEmpty32}{RegularTriangularSetXmpPagePatch32}
\pastebutton{RegularTriangularSetXmpPageEmpty32}{\showpaste}
\tab{5}\spadcommand{q4 := u**2+4*(z-x**2)+3*y**2*(x-1)**2- 3*t**2*(z-x)**2 +3*u**2*t**2*(z-1)**2+u**2*z*(z-2)+6*u*t*y*(z+x+z*x-1)\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q4 }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch33}
\begin{paste}{RegularTriangularSetXmpPageFull33}{RegularTriangularSetXmpPageEmpty33}
\pastebutton{RegularTriangularSetXmpPageFull33}{\hidepaste}
\tab{5}\spadcommand{lq := [q1, q2, q3, q4]\free{q1 }\free{q2 }\free{q3 }\free{q4 }\bound{lq }}
\indentrel{3}\begin{verbatim}
(33)
[
2 2 2
(y - 2t y + t )x
+
2 2 2
- 2y + ((2t + 4)z + 2t)y + (- 2t + 2)z - 4t
+
- 2
*
x
+
2 2 2 2
y + (- 2t z - 4t)y + (t + 10)z - 8z + 4t + 2
,
2
(- 3z y + 2t z + t)x + (z + 4)y + 4t z - 7t z,
2
(- 2z - 2)x - 2z + 8z - 2,
2 2 2
(3y - 3t - 4)x
+
2 2 2
(- 6y + (12t z + 12t)y + 6t z)x + 3y
+
2 2 2 2
(12t z - 12t)y + (9t + 4)z + (- 24t - 4)z + 12t
+
4
]
Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty33}
\begin{paste}{RegularTriangularSetXmpPageEmpty33}{RegularTriangularSetXmpPagePatch33}
\pastebutton{RegularTriangularSetXmpPageEmpty33}{\showpaste}
\tab{5}\spadcommand{lq := [q1, q2, q3, q4]\free{q1 }\free{q2 }\free{q3 }\free{q4 }\bound{lq }}
\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPagePatch34}
\begin{paste}{RegularTriangularSetXmpPageFull34}{RegularTriangularSetXmpPageEmpty34}
\pastebutton{RegularTriangularSetXmpPageFull34}{\hidepaste}
\tab{5}\spadcommand{zeroSetSplit(lq,true,true)$T\free{lq }\free{T }}
\indentrel{3}\begin{verbatim}
*** QCMPACK Statistics ***
Table size: 36
Entries reused: 255
*** REGSETGCD: Gcd Statistics ***
Table size: 125
Entries reused: 0
*** REGSETGCD: Inv Set Statistics ***
Table size: 30
Entries reused: 0
(34)
[
{
24 23
960725655771966t + 386820897948702t
+
22 21
8906817198608181t + 2704966893949428t
+
20 19
37304033340228264t + 7924782817170207t
+
18 17
93126799040354990t + 13101273653130910t
+
16 15
156146250424711858t + 16626490957259119t
+
14 13
190699288479805763t + 24339173367625275t
+
12 11
180532313014960135t + 35288089030975378t
+
10 9
135054975747656285t + 34733736952488540t
+
8 7
75947600354493972t + 19772555692457088t
+
6 5
28871558573755428t + 5576152439081664t
+
4 3
6321711820352976t + 438314209312320t
+
2
581105748367008t - 60254467992576t + 1449115951104
,
266042108694913023855152657370520823616684_
74181372891857784
*
23
t
+
443104378424686086067294899528296664238693_
556855017735265295
*
22
t
+
279078393286701234679141342358988327155321_
305829547090310242
*
21
t
+
339027636141323246510761717661554305462062_
6391823613392185226
*
20
t
+
941478179503540575554198645220352803719793_
196473813837434129
*
19
t
+
115478551946794752422116967496739493525857_
47674184320988144390
*
18
t
+
134360956676559778988170165669941321646721_
5660333356417241432
*
17
t
+
232338138681478735039335516171756408598991_
02987800663566699334
*
16
t
+
869574020537672336950845440508790740850931_
336484983573386433
*
15
t
+
315615543058769348754194614869699265542417_
50065103460820476969
*
14
t
+
127140099028771748744206595254773187955482_
3889855386072264931
*
13
t
+
319450899138637360448025269640795401983370_
49550503295825160523
*
12
t
+
373873570428814450987137156023284588443910_
2270778010470931960
*
11
t
+
252939975123914120261446014357711315875619_
05532992045692885927
*
10
t
+
521023900984606712346926279987005277341047_
1135950175008046524
*
9
t
+
150838879869302971662598705686082704274031_
87606238713491129188
*
8
t
+
352208723469293012638368627077577955348176_
9125670839075109000
*
7
t
+
607994520039568101308653379256888649110124_
4247440034969288588
*
6
t
+
109063485243390088819991375624798602319698_
7723469934933603680
*
5
t
+
140581943087190710229443253753833540210283_
8994019667487458352
*
4
t
+
880715279503204500725366712655077488783478_
28884933605202432
*
3
t
+
135882489433640933229781177155977768016065_
765482378657129440
*
2
t
+
-
139572834428822622305598946074003140825_
16690749975646520320
*
t
+
33463769297318929927725832570930847259211711_
2855749713920
*
z
+
8567175484043952879756725964506833932149637101_
090521164936
*
23
t
+
1497923928642017918457083740327289424987975192_
51667250945721
*
22
t
+
7725837178364582215741086158215976413812300307_
4190374021550
*
21
t
+
1108862254126854214498918940708612211184560556_
764334742191654
*
20
t
+
2132504944606788652197744801068260537838157896_
21501732672327
*
19
t
+
3668929075160666195729177894178343514501987898_
410131431699882
*
18
t
+
1713889064710018728794901243687482363147654590_
39567820048872
*
17
t
+
7192430746914602166660233477331022483144921771_
645523139658986
*
16
t
+
-
1287986746896900728128799656330902919596631_
43108437362453385
*
15
t
+
9553010858341425909306423132921134040856028790_
803526430270671
*
14
t
+
-
1329609624567549287453868764630043782465845_
8709144441096603
*
13
t
+
9475806805814145326383085518325333106881690568_
644274964864413
*
12
t
+
8032346879251334588616598556640849276062987947_
99856265539336
*
11
t
+
7338202759292865165994622349207516400662174302_
614595173333825
*
10
t
+
1308004628480367351164369613111971668880538855_
640917200187108
*
9
t
+
4268059455741255498880229598973705747098216067_
697754352634748
*
8
t
+
8928935268585140957913187759040933001030456015_
14470613580600
*
7
t
+
1679152575460683956631925852181341501981598137_
465328797013652
*
6
t
+
2697574157679229803789671541433578355441131582_
80591408043936
*
5
t
+
3809515278646575290335808298012827240813453726_
80202920198224
*
4
t
+
1978554529422849503299882693760134113272503533_
9452913286656
*
3
t
+
3647741205738478294236663530339663776330392817_
4935079178528
*
2
t
+
-
3722212879279038648713080422224976273210890_
229485838670848
*
t
+
890797248531143483612306344840138620247285999068_
74105856
,
3 2 3 2
(3z - 11z + 8z + 4)y + 2t z + 4t z - 5t z - t,
2
(z + 1)x + z - 4z + 1}
]
Type: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}
\begin{patch}{RegularTriangularSetXmpPageEmpty34}
\begin{paste}{RegularTriangularSetXmpPageEmpty34}{RegularTriangularSetXmpPagePatch34}
\pastebutton{RegularTriangularSetXmpPageEmpty34}{\showpaste}
\tab{5}\spadcommand{zeroSetSplit(lq,true,true)$T\free{lq }\free{T }}
\end{paste}\end{patch}
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