\begin{patch}{LexTriangularPackageXmpPagePatch1} \begin{paste}{LexTriangularPackageXmpPageFull1}{LexTriangularPackageXmpPageEmpty1} \pastebutton{LexTriangularPackageXmpPageFull1}{\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}{LexTriangularPackageXmpPageEmpty1} \begin{paste}{LexTriangularPackageXmpPageEmpty1}{LexTriangularPackageXmpPagePatch1} \pastebutton{LexTriangularPackageXmpPageEmpty1}{\showpaste} \tab{5}\spadcommand{R := Integer\bound{R }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch2} \begin{paste}{LexTriangularPackageXmpPageFull2}{LexTriangularPackageXmpPageEmpty2} \pastebutton{LexTriangularPackageXmpPageFull2}{\hidepaste} \tab{5}\spadcommand{ls : List Symbol := [a,b,c,d,e,f]\bound{ls }} \indentrel{3}\begin{verbatim} (2) [a,b,c,d,e,f] Type: List Symbol \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty2} \begin{paste}{LexTriangularPackageXmpPageEmpty2}{LexTriangularPackageXmpPagePatch2} \pastebutton{LexTriangularPackageXmpPageEmpty2}{\showpaste} \tab{5}\spadcommand{ls : List Symbol := [a,b,c,d,e,f]\bound{ls }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch3} \begin{paste}{LexTriangularPackageXmpPageFull3}{LexTriangularPackageXmpPageEmpty3} \pastebutton{LexTriangularPackageXmpPageFull3}{\hidepaste} \tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }} \indentrel{3}\begin{verbatim} (3) OrderedVariableList [a,b,c,d,e,f] Type: Domain \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty3} \begin{paste}{LexTriangularPackageXmpPageEmpty3}{LexTriangularPackageXmpPagePatch3} \pastebutton{LexTriangularPackageXmpPageEmpty3}{\showpaste} \tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch4} \begin{paste}{LexTriangularPackageXmpPageFull4}{LexTriangularPackageXmpPageEmpty4} \pastebutton{LexTriangularPackageXmpPageFull4}{\hidepaste} \tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }} \indentrel{3}\begin{verbatim} (4) NewSparseMultivariatePolynomial(Integer,OrderedVariable List [a,b,c,d,e,f]) Type: Domain \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty4} \begin{paste}{LexTriangularPackageXmpPageEmpty4}{LexTriangularPackageXmpPagePatch4} \pastebutton{LexTriangularPackageXmpPageEmpty4}{\showpaste} \tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch5} \begin{paste}{LexTriangularPackageXmpPageFull5}{LexTriangularPackageXmpPageEmpty5} \pastebutton{LexTriangularPackageXmpPageFull5}{\hidepaste} \tab{5}\spadcommand{p1: P := a*b*c*d*e*f - 1\free{P }\bound{p1 }} \indentrel{3}\begin{verbatim} (5) f e d c b a - 1 Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f]) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty5} \begin{paste}{LexTriangularPackageXmpPageEmpty5}{LexTriangularPackageXmpPagePatch5} \pastebutton{LexTriangularPackageXmpPageEmpty5}{\showpaste} \tab{5}\spadcommand{p1: P := a*b*c*d*e*f - 1\free{P }\bound{p1 }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch6} \begin{paste}{LexTriangularPackageXmpPageFull6}{LexTriangularPackageXmpPageEmpty6} \pastebutton{LexTriangularPackageXmpPageFull6}{\hidepaste} \tab{5}\spadcommand{p2: P := a*b*c*d*e +a*b*c*d*f +a*b*c*e*f +a*b*d*e*f +a*c*d*e*f +b*c*d*e*f\free{P }\bound{p2 }} \indentrel{3}\begin{verbatim} (6) ((((e + f)d + f e)c + f e d)b + f e d c)a + f e d c b Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f]) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty6} \begin{paste}{LexTriangularPackageXmpPageEmpty6}{LexTriangularPackageXmpPagePatch6} \pastebutton{LexTriangularPackageXmpPageEmpty6}{\showpaste} \tab{5}\spadcommand{p2: P := a*b*c*d*e +a*b*c*d*f +a*b*c*e*f +a*b*d*e*f +a*c*d*e*f +b*c*d*e*f\free{P }\bound{p2 }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch7} \begin{paste}{LexTriangularPackageXmpPageFull7}{LexTriangularPackageXmpPageEmpty7} \pastebutton{LexTriangularPackageXmpPageFull7}{\hidepaste} \tab{5}\spadcommand{p3: P := a*b*c*d + a*b*c*f + a*b*e*f + a*d*e*f + b*c*d*e + c*d*e*f\free{P }\bound{p3 }} \indentrel{3}\begin{verbatim} (7) (((d + f)c + f e)b + f e d)a + e d c b + f e d c Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f]) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty7} \begin{paste}{LexTriangularPackageXmpPageEmpty7}{LexTriangularPackageXmpPagePatch7} \pastebutton{LexTriangularPackageXmpPageEmpty7}{\showpaste} \tab{5}\spadcommand{p3: P := a*b*c*d + a*b*c*f + a*b*e*f + a*d*e*f + b*c*d*e + c*d*e*f\free{P }\bound{p3 }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch8} \begin{paste}{LexTriangularPackageXmpPageFull8}{LexTriangularPackageXmpPageEmpty8} \pastebutton{LexTriangularPackageXmpPageFull8}{\hidepaste} \tab{5}\spadcommand{p4: P := a*b*c + a*b*f + a*e*f + b*c*d + c*d*e + d*e*f\free{P }\bound{p4 }} \indentrel{3}\begin{verbatim} (8) ((c + f)b + f e)a + d c b + e d c + f e d Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f]) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty8} \begin{paste}{LexTriangularPackageXmpPageEmpty8}{LexTriangularPackageXmpPagePatch8} \pastebutton{LexTriangularPackageXmpPageEmpty8}{\showpaste} \tab{5}\spadcommand{p4: P := a*b*c + a*b*f + a*e*f + b*c*d + c*d*e + d*e*f\free{P }\bound{p4 }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch9} \begin{paste}{LexTriangularPackageXmpPageFull9}{LexTriangularPackageXmpPageEmpty9} \pastebutton{LexTriangularPackageXmpPageFull9}{\hidepaste} \tab{5}\spadcommand{p5: P := a*b + a*f + b*c + c*d + d*e + e*f\free{P }\bound{p5 }} \indentrel{3}\begin{verbatim} (9) (b + f)a + c b + d c + e d + f e Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f]) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty9} \begin{paste}{LexTriangularPackageXmpPageEmpty9}{LexTriangularPackageXmpPagePatch9} \pastebutton{LexTriangularPackageXmpPageEmpty9}{\showpaste} \tab{5}\spadcommand{p5: P := a*b + a*f + b*c + c*d + d*e + e*f\free{P }\bound{p5 }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch10} \begin{paste}{LexTriangularPackageXmpPageFull10}{LexTriangularPackageXmpPageEmpty10} \pastebutton{LexTriangularPackageXmpPageFull10}{\hidepaste} \tab{5}\spadcommand{p6: P := a + b + c + d + e + f\free{P }\bound{p6 }} \indentrel{3}\begin{verbatim} (10) a + b + c + d + e + f Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f]) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty10} \begin{paste}{LexTriangularPackageXmpPageEmpty10}{LexTriangularPackageXmpPagePatch10} \pastebutton{LexTriangularPackageXmpPageEmpty10}{\showpaste} \tab{5}\spadcommand{p6: P := a + b + c + d + e + f\free{P }\bound{p6 }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch11} \begin{paste}{LexTriangularPackageXmpPageFull11}{LexTriangularPackageXmpPageEmpty11} \pastebutton{LexTriangularPackageXmpPageFull11}{\hidepaste} \tab{5}\spadcommand{lp := [p1, p2, p3, p4, p5, p6]\free{p1 }\free{p2 }\free{p3 }\free{p4 }\free{p5 }\free{p6 }\bound{lp }} \indentrel{3}\begin{verbatim} (11) [f e d c b a - 1, ((((e + f)d + f e)c + f e d)b + f e d c)a + f e d c b, (((d + f)c + f e)b + f e d)a + e d c b + f e d c, ((c + f)b + f e)a + d c b + e d c + f e d, (b + f)a + c b + d c + e d + f e, a + b + c + d + e + f] Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f]) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty11} \begin{paste}{LexTriangularPackageXmpPageEmpty11}{LexTriangularPackageXmpPagePatch11} \pastebutton{LexTriangularPackageXmpPageEmpty11}{\showpaste} \tab{5}\spadcommand{lp := [p1, p2, p3, p4, p5, p6]\free{p1 }\free{p2 }\free{p3 }\free{p4 }\free{p5 }\free{p6 }\bound{lp }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch12} \begin{paste}{LexTriangularPackageXmpPageFull12}{LexTriangularPackageXmpPageEmpty12} \pastebutton{LexTriangularPackageXmpPageFull12}{\hidepaste} \tab{5}\spadcommand{lextripack := LEXTRIPK(R,ls)\free{R }\free{ls }\bound{lextripack }} \indentrel{3}\begin{verbatim} (12) LexTriangularPackage(Integer,[a,b,c,d,e,f]) Type: Domain \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty12} \begin{paste}{LexTriangularPackageXmpPageEmpty12}{LexTriangularPackageXmpPagePatch12} \pastebutton{LexTriangularPackageXmpPageEmpty12}{\showpaste} \tab{5}\spadcommand{lextripack := LEXTRIPK(R,ls)\free{R }\free{ls }\bound{lextripack }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch13} \begin{paste}{LexTriangularPackageXmpPageFull13}{LexTriangularPackageXmpPageEmpty13} \pastebutton{LexTriangularPackageXmpPageFull13}{\hidepaste} \tab{5}\spadcommand{lg := groebner(lp)$lextripack\free{lp }\free{lextripack }\bound{lg }} \indentrel{3}\begin{verbatim} (13) [a + b + c + d + e + f, 2 3968379498283200b + 15873517993132800f b + 2 3968379498283200d + 15873517993132800f d + 3 5 4 4 3968379498283200f e - 15873517993132800f e + 5 3 23810276989699200f e + 6 2 (206355733910726400f + 230166010900425600)e + 43 37 - 729705987316687f + 1863667496867205421f + 31 291674853771731104461f + 25 365285994691106921745f + 19 549961185828911895f + 13 - 365048404038768439269f + 7 - 292382820431504027669f - 2271898467631865497f * e + 44 38 - 3988812642545399f + 10187423878429609997f + 32 1594377523424314053637f + 26 20 1994739308439916238065f + 1596840088052642815f + 14 - 1993494118301162145413f + 8 2 - 1596049742289689815053f - 11488171330159667449f , (23810276989699200c - 23810276989699200f)b + 2 23810276989699200c + 71430830969097600f c + 2 - 23810276989699200d - 95241107958796800f d + 3 5 4 4 - 55557312975964800f e + 174608697924460800f e + 5 3 - 174608697924460800f e + 6 2 (- 2428648252949318400f - 2611193709870345600)e + 43 37 8305444561289527f - 21212087151945459641f + 31 - 3319815883093451385381f + 25 - 4157691646261657136445f + 19 - 6072721607510764095f + 13 4154986709036460221649f + 7 3327761311138587096749f + 25885340608290841637f * e + 44 38 45815897629010329f - 117013765582151891207f + 32 - 18313166848970865074187f + 26 - 22909971239649297438915f + 20 - 16133250761305157265f + 14 22897305857636178256623f + 8 2 18329944781867242497923f + 130258531002020420699f , (7936758996566400d - 7936758996566400f)b + 3 5 - 7936758996566400f d - 7936758996566400f e + 4 4 5 3 23810276989699200f e - 23810276989699200f e + 6 2 (- 337312257354072000f - 369059293340337600)e + 43 37 1176345388640471f - 3004383582891473073f + 31 - 470203502707246105653f + 25 - 588858183402644348085f + 19 - 856939308623513535f + 13 588472674242340526377f + 7 471313241958371103517f + 3659742549078552381f * e + 44 38 6423170513956901f - 16404772137036480803f + 32 - 2567419165227528774463f + 26 - 3211938090825682172335f + 20 - 2330490332697587485f + 14 3210100109444754864587f + 8 2 2569858315395162617847f + 18326089487427735751f , (11905138494849600e - 11905138494849600f)b + 3 5 4 4 - 3968379498283200f e + 15873517993132800f e + 5 3 - 27778656487982400f e + 6 2 (- 208339923659868000f - 240086959646133600)e + 43 37 786029984751110f - 2007519008182245250f + 31 - 314188062908073807090f + 25 - 393423667537929575250f + 19 - 550329120654394950f + 13 393196408728889612770f + 7 314892372799176495730f + 2409386515146668530f * e + 44 38 4177638546747827f - 10669685294602576381f + 32 - 1669852980419949524601f + 26 - 2089077057287904170745f + 20 - 1569899763580278795f + 14 2087864026859015573349f + 8 2 1671496085945199577969f + 11940257226216280177f , 6 (11905138494849600f - 11905138494849600)b + 2 5 3 4 - 15873517993132800f e + 39683794982832000f e + 4 3 - 39683794982832000f e + 11 5 2 (- 686529653202993600f - 607162063237329600f )e + 42 36 65144531306704f - 166381280901088652f + 30 - 26033434502470283472f + 24 - 31696259583860650140f + 18 12 971492093167581360f + 32220085033691389548f + 6 25526177666070529808f + 138603268355749244 * e + 43 37 167620036074811f - 428102417974791473f + 31 25 - 66997243801231679313f - 83426716722148750485f + 19 13 203673895369980765f + 83523056326010432457f + 7 66995789640238066937f + 478592855549587901f , 3 2 801692827936c + 2405078483808f c + 2 45 - 2405078483808f c - 13752945467f + 39 33 35125117815561f + 5496946957826433f + 27 21 6834659447749117f - 44484880462461f + 15 9 - 6873406230093057f - 5450844938762633f + 3 1216586044571f , (23810276989699200d - 23810276989699200f)c + 2 23810276989699200d + 71430830969097600f d + 3 5 4 4 7936758996566400f e - 31747035986265600f e + 5 3 31747035986265600f e + 6 2 (404774708824886400f + 396837949828320000)e + 43 37 - 1247372229446701f + 3185785654596621203f + 31 498594866849974751463f + 25 624542545845791047935f + 19 931085755769682885f + 13 - 624150663582417063387f + 7 - 499881859388360475647f - 3926885313819527351f * e + 44 38 - 7026011547118141f + 17944427051950691243f + 32 2808383522593986603543f + 26 20 3513624142354807530135f + 2860757006705537685f + 14 - 3511356735642190737267f + 8 2 - 2811332494697103819887f - 20315011631522847311f , (7936758996566400e - 7936758996566400f)c + 43 37 - 4418748183673f + 11285568707456559f + 31 25 1765998617294451019f + 2173749283622606155f + 19 13 - 55788292195402895f - 2215291421788292951f + 7 - 1718142665347430851f + 30256569458230237f * e + 44 38 4418748183673f - 11285568707456559f + 32 26 - 1765998617294451019f - 2173749283622606155f + 20 14 55788292195402895f + 2215291421788292951f + 8 2 1718142665347430851f - 30256569458230237f , 6 (72152354514240f - 72152354514240)c + 43 37 40950859449f - 104588980990367f + 31 25 - 16367227395575307f - 20268523416527355f + 19 13 442205002259535f + 20576059935789063f + 7 15997133796970563f - 275099892785581f , 3 2 1984189749141600d + 5952569247424800f d + 2 4 5 - 5952569247424800f d - 3968379498283200f e + 5 4 3 15873517993132800f e + 17857707742274400e + 7 2 (- 148814231185620000f - 162703559429611200f)e + 44 38 - 390000914678878f + 996062704593756434f + 32 155886323972034823914f + 26 20 194745956143985421330f + 6205077595574430f + 14 - 194596512653299068786f + 8 2 - 155796897940756922666f - 1036375759077320978f * e + 45 39 - 374998630035991f + 957747106595453993f + 33 27 149889155566764891693f + 187154171443494641685f + 21 15 - 127129015426348065f - 187241533243115040417f + 9 3 - 149719983567976534037f - 836654081239648061f , (5952569247424800e - 5952569247424800f)d + 3 5 4 4 - 3968379498283200f e + 9920948745708000f e + 5 3 - 3968379498283200f e + 6 2 (- 148814231185620000f - 150798420934761600)e + 43 37 492558110242553f - 1257992359608074599f + 31 - 196883094539368513959f + 25 - 246562115745735428055f + 19 - 325698701993885505f + 13 246417769883651808111f + 7 197327352068200652911f + 1523373796389332143f * e + 44 38 2679481081803026f - 6843392695421906608f + 32 - 1071020459642646913578f + 26 20 - 1339789169692041240060f - 852746750910750210f + 14 1339105101971878401312f + 8 2 1071900289758712984762f + 7555239072072727756f , 6 (11905138494849600f - 11905138494849600)d + 2 5 3 4 - 7936758996566400f e + 31747035986265600f e + 4 3 - 31747035986265600f e + 11 5 2 (- 420648226818019200f - 404774708824886400f )e + 42 36 15336187600889f - 39169739565161107f + 30 - 6127176127489690827f + 24 - 7217708742310509615f + 18 12 538628483890722735f + 7506804353843507643f + 6 5886160769782607203f + 63576108396535879 * e + 43 37 71737781777066f - 183218856207557938f + 31 25 - 28672874271132276078f - 35625223686939812010f + 19 13 164831339634084390f + 35724160423073052642f + 7 28627022578664910622f + 187459987029680506f , 6 5 1322793166094400e - 3968379498283200f e + 2 4 3 3 3968379498283200f e - 5291172664377600f e + 10 4 2 (- 230166010900425600f - 226197631402142400f )e + 47 - 152375364610443885f + 41 389166626064854890415f + 35 60906097841360558987335f + 29 76167367934608798697275f + 23 27855066785995181125f + 17 - 76144952817052723145495f + 11 - 60933629892463517546975f + 5 - 411415071682002547795f * e + 42 36 - 209493533143822f + 535045979490560586f + 30 24 83737947964973553146f + 104889507084213371570f + 18 12 167117997269207870f - 104793725781390615514f + 6 - 83842685189903180394f - 569978796672974242 , 6 3 (25438330117200f + 25438330117200)e + 7 2 (76314990351600f + 76314990351600f)e + 44 38 - 1594966552735f + 4073543370415745f + 32 26 637527159231148925f + 797521176113606525f + 20 14 530440941097175f - 797160527306433145f + 8 2 - 638132320196044965f - 4510507167940725f * e + 45 39 - 6036376800443f + 15416903421476909f + 33 27 2412807646192304449f + 3017679923028013705f + 21 15 1422320037411955f - 3016560402417843941f + 9 3 - 2414249368183033161f - 16561862361763873f , 12 2 (1387545279120f - 1387545279120)e + 43 37 4321823003f - 11037922310209f + 31 25 - 1727510711947989f - 2165150991154425f + 19 13 - 5114342560755f + 2162682824948601f + 7 1732620732685741f + 13506088516033f * e + 44 38 24177661775f - 61749727185325f + 32 26 - 9664106795754225f - 12090487758628245f + 20 14 - 8787672733575f + 12083693383005045f + 8 2 9672870290826025f + 68544102808525f , 48 42 36 30 18 f - 2554f - 399710f - 499722f + 499722f + 12 6 399710f + 2554f - 1 ] Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f]) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty13} \begin{paste}{LexTriangularPackageXmpPageEmpty13}{LexTriangularPackageXmpPagePatch13} \pastebutton{LexTriangularPackageXmpPageEmpty13}{\showpaste} \tab{5}\spadcommand{lg := groebner(lp)$lextripack\free{lp }\free{lextripack }\bound{lg }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch14} \begin{paste}{LexTriangularPackageXmpPageFull14}{LexTriangularPackageXmpPageEmpty14} \pastebutton{LexTriangularPackageXmpPageFull14}{\hidepaste} \tab{5}\spadcommand{lexTriangular(lg,false)$lextripack\free{lg }\free{lextripack }} \indentrel{3}\begin{verbatim} (14) [ 6 {f + 1, 6 5 2 4 3 3 4 2 5 e - 3f e + 3f e - 4f e + 3f e - 3f e - 1, 2 5 3 4 4 3 5 2 3d + f e - 4f e + 4f e - 2f e - 2e + 2f, c + f, 2 5 3 4 4 3 5 2 3b + 2f e - 5f e + 5f e - 10f e - 4e + 7f, 2 5 3 4 4 3 5 2 a - f e + 3f e - 3f e + 4f e + 3e - 3f} , 6 2 2 {f - 1, e - f, d - f, c + 4f c + f , 2 (c - f)b - f c - 5f , a + b + c + 3f} , 6 2 2 {f - 1,e - f,d - f,c - f,b + 4f b + f ,a + b + 4f}, 6 2 2 {f - 1, e - f, d + 4f d + f , 2 (d - f)c - f d - 5f , b - f, a + c + d + 3f} , { 36 30 24 18 12 f - 2554f - 399709f - 502276f - 399709f + 6 - 2554f + 1 , 12 2 (161718564f - 161718564)e + 31 25 19 - 504205f + 1287737951f + 201539391380f + 13 7 253982817368f + 201940704665f + 1574134601f * e + 32 26 20 - 2818405f + 7198203911f + 1126548149060f + 14 8 2 1416530563364f + 1127377589345f + 7988820725f , 6 (693772639560f - 693772639560)d + 2 5 3 4 - 462515093040f e + 1850060372160f e + 4 3 - 1850060372160f e + 11 5 2 (- 24513299931120f - 23588269745040f )e + 30 24 - 890810428f + 2275181044754f + 18 12 355937263869776f + 413736880104344f + 6 342849304487996f + 3704966481878 * e + 31 25 - 4163798003f + 10634395752169f + 19 13 1664161760192806f + 2079424391370694f + 7 1668153650635921f + 10924274392693f , 6 31 (12614047992f - 12614047992)c - 7246825f + 25 19 18508536599f + 2896249516034f + 13 7 3581539649666f + 2796477571739f - 48094301893f , 6 (693772639560f - 693772639560)b + 2 5 3 4 - 925030186080f e + 2312575465200f e + 4 3 - 2312575465200f e + 11 5 2 (- 40007555547960f - 35382404617560f )e + 30 24 - 3781280823f + 9657492291789f + 18 12 1511158913397906f + 1837290892286154f + 6 1487216006594361f + 8077238712093 * e + 31 25 - 9736390478f + 24866827916734f + 19 13 3891495681905296f + 4872556418871424f + 7 3904047887269606f + 27890075838538f , a + b + c + d + e + f} , 6 2 2 2 {f - 1, e + 4f e + f , (e - f)d - f e - 5f , c - f, b - f, a + d + e + 3f} ] Type: List RegularChain(Integer,[a,b,c,d,e,f]) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty14} \begin{paste}{LexTriangularPackageXmpPageEmpty14}{LexTriangularPackageXmpPagePatch14} \pastebutton{LexTriangularPackageXmpPageEmpty14}{\showpaste} \tab{5}\spadcommand{lexTriangular(lg,false)$lextripack\free{lg }\free{lextripack }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch15} \begin{paste}{LexTriangularPackageXmpPageFull15}{LexTriangularPackageXmpPageEmpty15} \pastebutton{LexTriangularPackageXmpPageFull15}{\hidepaste} \tab{5}\spadcommand{lts := lexTriangular(lg,true)$lextripack\free{lg }\free{lextripack }\bound{lts }} \indentrel{3}\begin{verbatim} (15) [ 6 {f + 1, 6 5 2 4 3 3 4 2 5 e - 3f e + 3f e - 4f e + 3f e - 3f e - 1, 2 5 3 4 4 3 5 2 3d + f e - 4f e + 4f e - 2f e - 2e + 2f, c + f, 2 5 3 4 4 3 5 2 3b + 2f e - 5f e + 5f e - 10f e - 4e + 7f, 2 5 3 4 4 3 5 2 a - f e + 3f e - 3f e + 4f e + 3e - 3f} , 6 2 2 {f - 1,e - f,d - f,c + 4f c + f ,b + c + 4f,a - f}, 6 2 2 {f - 1,e - f,d - f,c - f,b + 4f b + f ,a + b + 4f}, 6 2 2 {f - 1,e - f,d + 4f d + f ,c + d + 4f,b - f,a - f}, { 36 30 24 18 12 f - 2554f - 399709f - 502276f - 399709f + 6 - 2554f + 1 , 2 1387545279120e + 31 25 4321823003f - 11037922310209f + 19 13 - 1727506390124986f - 2176188913464634f + 7 - 1732620732685741f - 13506088516033f * e + 32 26 24177661775f - 61749727185325f + 20 14 - 9664082618092450f - 12152237485813570f + 8 2 - 9672870290826025f - 68544102808525f , 1387545279120d + 30 24 - 1128983050f + 2883434331830f + 18 12 451234998755840f + 562426491685760f + 6 447129055314890f - 165557857270 * e + 31 25 - 1816935351f + 4640452214013f + 19 13 726247129626942f + 912871801716798f + 7 726583262666877f + 4909358645961f , 31 25 1387545279120c + 778171189f - 1987468196267f + 19 13 - 310993556954378f - 383262822316802f + 7 - 300335488637543f + 5289595037041f , 1387545279120b + 30 24 1128983050f - 2883434331830f + 18 12 - 451234998755840f - 562426491685760f + 6 - 447129055314890f + 165557857270 * e + 31 25 - 3283058841f + 8384938292463f + 19 13 1312252817452422f + 1646579934064638f + 7 1306372958656407f + 4694680112151f , 31 1387545279120a + 1387545279120e + 4321823003f + 25 19 - 11037922310209f - 1727506390124986f + 13 7 - 2176188913464634f - 1732620732685741f + - 13506088516033f } , 6 2 2 {f - 1,e + 4f e + f ,d + e + 4f,c - f,b - f,a - f}] Type: List RegularChain(Integer,[a,b,c,d,e,f]) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty15} \begin{paste}{LexTriangularPackageXmpPageEmpty15}{LexTriangularPackageXmpPagePatch15} \pastebutton{LexTriangularPackageXmpPageEmpty15}{\showpaste} \tab{5}\spadcommand{lts := lexTriangular(lg,true)$lextripack\free{lg }\free{lextripack }\bound{lts }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch16} \begin{paste}{LexTriangularPackageXmpPageFull16}{LexTriangularPackageXmpPageEmpty16} \pastebutton{LexTriangularPackageXmpPageFull16}{\hidepaste} \tab{5}\spadcommand{[[init(p) for p in (ts :: List(P))] for ts in lts]\free{lts }} \indentrel{3}\begin{verbatim} (16) [[1,3,1,3,1,1], [1,1,1,1,1,1], [1,1,1,1,1,1], [1,1,1,1,1,1], [1387545279120, 1387545279120, 1387545279120, 1387545279120, 1387545279120, 1] , [1,1,1,1,1,1]] Type: List List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f]) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty16} \begin{paste}{LexTriangularPackageXmpPageEmpty16}{LexTriangularPackageXmpPagePatch16} \pastebutton{LexTriangularPackageXmpPageEmpty16}{\showpaste} \tab{5}\spadcommand{[[init(p) for p in (ts :: List(P))] for ts in lts]\free{lts }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch17} \begin{paste}{LexTriangularPackageXmpPageFull17}{LexTriangularPackageXmpPageEmpty17} \pastebutton{LexTriangularPackageXmpPageFull17}{\hidepaste} \tab{5}\spadcommand{squareFreeLexTriangular(lg,true)$lextripack\free{lg }\free{lextripack }} \indentrel{3}\begin{verbatim} (17) [ 6 {f + 1, 6 5 2 4 3 3 4 2 5 e - 3f e + 3f e - 4f e + 3f e - 3f e - 1, 2 5 3 4 4 3 5 2 3d + f e - 4f e + 4f e - 2f e - 2e + 2f, c + f, 2 5 3 4 4 3 5 2 3b + 2f e - 5f e + 5f e - 10f e - 4e + 7f, 2 5 3 4 4 3 5 2 a - f e + 3f e - 3f e + 4f e + 3e - 3f} , 6 2 2 {f - 1,e - f,d - f,c + 4f c + f ,b + c + 4f,a - f}, 6 2 2 {f - 1,e - f,d - f,c - f,b + 4f b + f ,a + b + 4f}, 6 2 2 {f - 1,e - f,d + 4f d + f ,c + d + 4f,b - f,a - f}, { 36 30 24 18 12 f - 2554f - 399709f - 502276f - 399709f + 6 - 2554f + 1 , 2 1387545279120e + 31 25 4321823003f - 11037922310209f + 19 13 - 1727506390124986f - 2176188913464634f + 7 - 1732620732685741f - 13506088516033f * e + 32 26 24177661775f - 61749727185325f + 20 14 - 9664082618092450f - 12152237485813570f + 8 2 - 9672870290826025f - 68544102808525f , 1387545279120d + 30 24 - 1128983050f + 2883434331830f + 18 12 451234998755840f + 562426491685760f + 6 447129055314890f - 165557857270 * e + 31 25 - 1816935351f + 4640452214013f + 19 13 726247129626942f + 912871801716798f + 7 726583262666877f + 4909358645961f , 31 25 1387545279120c + 778171189f - 1987468196267f + 19 13 - 310993556954378f - 383262822316802f + 7 - 300335488637543f + 5289595037041f , 1387545279120b + 30 24 1128983050f - 2883434331830f + 18 12 - 451234998755840f - 562426491685760f + 6 - 447129055314890f + 165557857270 * e + 31 25 - 3283058841f + 8384938292463f + 19 13 1312252817452422f + 1646579934064638f + 7 1306372958656407f + 4694680112151f , 31 1387545279120a + 1387545279120e + 4321823003f + 25 19 - 11037922310209f - 1727506390124986f + 13 7 - 2176188913464634f - 1732620732685741f + - 13506088516033f } , 6 2 2 {f - 1,e + 4f e + f ,d + e + 4f,c - f,b - f,a - f}] Type: List SquareFreeRegularTriangularSet(Integer,IndexedExponents OrderedVariableList [a,b,c,d,e,f],OrderedVariableList [a,b,c,d,e,f],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty17} \begin{paste}{LexTriangularPackageXmpPageEmpty17}{LexTriangularPackageXmpPagePatch17} \pastebutton{LexTriangularPackageXmpPageEmpty17}{\showpaste} \tab{5}\spadcommand{squareFreeLexTriangular(lg,true)$lextripack\free{lg }\free{lextripack }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch18} \begin{paste}{LexTriangularPackageXmpPageFull18}{LexTriangularPackageXmpPageEmpty18} \pastebutton{LexTriangularPackageXmpPageFull18}{\hidepaste} \tab{5}\spadcommand{reduce(+,[degree(ts) for ts in lts])\free{lts }} \indentrel{3}\begin{verbatim} (18) 156 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty18} \begin{paste}{LexTriangularPackageXmpPageEmpty18}{LexTriangularPackageXmpPagePatch18} \pastebutton{LexTriangularPackageXmpPageEmpty18}{\showpaste} \tab{5}\spadcommand{reduce(+,[degree(ts) for ts in lts])\free{lts }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch19} \begin{paste}{LexTriangularPackageXmpPageFull19}{LexTriangularPackageXmpPageEmpty19} \pastebutton{LexTriangularPackageXmpPageFull19}{\hidepaste} \tab{5}\spadcommand{ls2 : List Symbol := concat(ls,new()$Symbol)\free{ls }\bound{ls2 }} \indentrel{3}\begin{verbatim} (19) [a,b,c,d,e,f,%A] Type: List Symbol \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty19} \begin{paste}{LexTriangularPackageXmpPageEmpty19}{LexTriangularPackageXmpPagePatch19} \pastebutton{LexTriangularPackageXmpPageEmpty19}{\showpaste} \tab{5}\spadcommand{ls2 : List Symbol := concat(ls,new()$Symbol)\free{ls }\bound{ls2 }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch20} \begin{paste}{LexTriangularPackageXmpPageFull20}{LexTriangularPackageXmpPageEmpty20} \pastebutton{LexTriangularPackageXmpPageFull20}{\hidepaste} \tab{5}\spadcommand{zdpack := ZDSOLVE(R,ls,ls2)\free{R }\free{ls }\free{ls2 }\bound{zdpack }} \indentrel{3}\begin{verbatim} (20) ZeroDimensionalSolvePackage(Integer,[a,b,c,d,e,f],[a,b, c,d,e,f,%A]) Type: Domain \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty20} \begin{paste}{LexTriangularPackageXmpPageEmpty20}{LexTriangularPackageXmpPagePatch20} \pastebutton{LexTriangularPackageXmpPageEmpty20}{\showpaste} \tab{5}\spadcommand{zdpack := ZDSOLVE(R,ls,ls2)\free{R }\free{ls }\free{ls2 }\bound{zdpack }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch21} \begin{paste}{LexTriangularPackageXmpPageFull21}{LexTriangularPackageXmpPageEmpty21} \pastebutton{LexTriangularPackageXmpPageFull21}{\hidepaste} \tab{5}\spadcommand{concat [univariateSolve(ts)$zdpack for ts in lts]\free{lts }\free{zdpack }} \indentrel{3}\begin{verbatim} (21) [ 4 2 [complexRoots= ? - 13? + 49, coordinates = 3 3 [7a + %A - 6%A, 21b + %A + %A, 3 3 21c - 2%A + 19%A, 7d - %A + 6%A, 3 3 21e - %A - %A, 21f + 2%A - 19%A] ] , 4 2 [complexRoots= ? + 11? + 49, coordinates = 3 3 [35a + 3%A + 19%A, 35b + %A + 18%A, 3 3 35c - 2%A - %A, 35d - 3%A - 19%A, 3 3 35e - %A - 18%A, 35f + 2%A + %A] ] , [ complexRoots = 8 7 6 5 4 3 ? - 12? + 58? - 120? + 207? - 360? + 2 802? - 1332? + 1369 , coordinates = [ 7 6 5 43054532a + 33782%A - 546673%A + 3127348%A + 4 3 2 - 6927123%A + 4365212%A - 25086957%A + 39582814%A - 107313172 , 7 6 5 43054532b - 33782%A + 546673%A - 3127348%A + 4 3 2 6927123%A - 4365212%A + 25086957%A + - 39582814%A + 107313172 , 7 6 5 21527266c - 22306%A + 263139%A - 1166076%A + 4 3 2 1821805%A - 2892788%A + 10322663%A + - 9026596%A + 12950740 , 7 6 5 43054532d + 22306%A - 263139%A + 1166076%A + 4 3 2 - 1821805%A + 2892788%A - 10322663%A + 30553862%A - 12950740 , 7 6 5 43054532e - 22306%A + 263139%A - 1166076%A + 4 3 2 1821805%A - 2892788%A + 10322663%A + - 30553862%A + 12950740 , 7 6 5 21527266f + 22306%A - 263139%A + 1166076%A + 4 3 2 - 1821805%A + 2892788%A - 10322663%A + 9026596%A - 12950740 ] ] , [ complexRoots = 8 7 6 5 4 3 ? + 12? + 58? + 120? + 207? + 360? + 2 802? + 1332? + 1369 , coordinates = [ 7 6 5 43054532a + 33782%A + 546673%A + 3127348%A + 4 3 2 6927123%A + 4365212%A + 25086957%A + 39582814%A + 107313172 , 7 6 5 43054532b - 33782%A - 546673%A - 3127348%A + 4 3 2 - 6927123%A - 4365212%A - 25086957%A + - 39582814%A - 107313172 , 7 6 5 21527266c - 22306%A - 263139%A - 1166076%A + 4 3 2 - 1821805%A - 2892788%A - 10322663%A + - 9026596%A - 12950740 , 7 6 5 43054532d + 22306%A + 263139%A + 1166076%A + 4 3 2 1821805%A + 2892788%A + 10322663%A + 30553862%A + 12950740 , 7 6 5 43054532e - 22306%A - 263139%A - 1166076%A + 4 3 2 - 1821805%A - 2892788%A - 10322663%A + - 30553862%A - 12950740 , 7 6 5 21527266f + 22306%A + 263139%A + 1166076%A + 4 3 2 1821805%A + 2892788%A + 10322663%A + 9026596%A + 12950740 ] ] , 4 2 [complexRoots= ? - ? + 1, coordinates = 3 3 [a - %A, b + %A - %A, c + %A , d + %A, 3 3 e - %A + %A, f - %A ] ] , 8 6 4 2 [complexRoots= ? + 4? + 12? + 16? + 4, coordinates = 7 5 3 [4a - 2%A - 7%A - 20%A - 22%A, 7 5 3 4b + 2%A + 7%A + 20%A + 22%A, 7 5 3 4c + %A + 3%A + 10%A + 10%A, 7 5 3 4d + %A + 3%A + 10%A + 6%A, 7 5 3 4e - %A - 3%A - 10%A - 6%A, 7 5 3 4f - %A - 3%A - 10%A - 10%A] ] , 4 3 2 [complexRoots= ? + 6? + 30? + 36? + 36, coordinates = 3 2 [30a - %A - 5%A - 30%A - 6, 3 2 6b + %A + 5%A + 24%A + 6, 3 2 30c - %A - 5%A - 6, 3 2 30d - %A - 5%A - 30%A - 6, 3 2 30e - %A - 5%A - 30%A - 6, 3 2 30f - %A - 5%A - 30%A - 6] ] , 4 3 2 [complexRoots= ? - 6? + 30? - 36? + 36, coordinates = 3 2 [30a - %A + 5%A - 30%A + 6, 3 2 6b + %A - 5%A + 24%A - 6, 3 2 30c - %A + 5%A + 6, 3 2 30d - %A + 5%A - 30%A + 6, 3 2 30e - %A + 5%A - 30%A + 6, 3 2 30f - %A + 5%A - 30%A + 6] ] , 2 [complexRoots= ? + 6? + 6, coordinates = [a + 1,b - %A - 5,c + %A + 1,d + 1,e + 1,f + 1] ] , 2 [complexRoots= ? - 6? + 6, coordinates = [a - 1,b - %A + 5,c + %A - 1,d - 1,e - 1,f - 1] ] , 4 3 2 [complexRoots= ? + 6? + 30? + 36? + 36, coordinates = 3 2 [6a + %A + 5%A + 24%A + 6, 3 2 30b - %A - 5%A - 6, 3 2 30c - %A - 5%A - 30%A - 6, 3 2 30d - %A - 5%A - 30%A - 6, 3 2 30e - %A - 5%A - 30%A - 6, 3 2 30f - %A - 5%A - 30%A - 6] ] , 4 3 2 [complexRoots= ? - 6? + 30? - 36? + 36, coordinates = 3 2 [6a + %A - 5%A + 24%A - 6, 3 2 30b - %A + 5%A + 6, 3 2 30c - %A + 5%A - 30%A + 6, 3 2 30d - %A + 5%A - 30%A + 6, 3 2 30e - %A + 5%A - 30%A + 6, 3 2 30f - %A + 5%A - 30%A + 6] ] , 2 [complexRoots= ? + 6? + 6, coordinates = [a - %A - 5,b + %A + 1,c + 1,d + 1,e + 1,f + 1] ] , 2 [complexRoots= ? - 6? + 6, coordinates = [a - %A + 5,b + %A - 1,c - 1,d - 1,e - 1,f - 1] ] , 4 3 2 [complexRoots= ? + 6? + 30? + 36? + 36, coordinates = 3 2 [30a - %A - 5%A - 30%A - 6, 3 2 30b - %A - 5%A - 30%A - 6, 3 2 6c + %A + 5%A + 24%A + 6, 3 2 30d - %A - 5%A - 6, 3 2 30e - %A - 5%A - 30%A - 6, 3 2 30f - %A - 5%A - 30%A - 6] ] , 4 3 2 [complexRoots= ? - 6? + 30? - 36? + 36, coordinates = 3 2 [30a - %A + 5%A - 30%A + 6, 3 2 30b - %A + 5%A - 30%A + 6, 3 2 6c + %A - 5%A + 24%A - 6, 3 2 30d - %A + 5%A + 6, 3 2 30e - %A + 5%A - 30%A + 6, 3 2 30f - %A + 5%A - 30%A + 6] ] , 2 [complexRoots= ? + 6? + 6, coordinates = [a + 1,b + 1,c - %A - 5,d + %A + 1,e + 1,f + 1] ] , 2 [complexRoots= ? - 6? + 6, coordinates = [a - 1,b - 1,c - %A + 5,d + %A - 1,e - 1,f - 1] ] , [ complexRoots = 8 7 6 5 4 2 ? + 6? + 16? + 24? + 18? - 8? + 4 , coordinates = [ 7 6 5 4 3 2a + 2%A + 9%A + 18%A + 19%A + 4%A + 2 - 10%A - 2%A + 4 , 7 6 5 4 3 2b + 2%A + 9%A + 18%A + 19%A + 4%A + 2 - 10%A - 4%A + 4 , 7 6 5 4 3 2c - %A - 4%A - 8%A - 9%A - 4%A - 2%A - 4, 7 6 5 4 3 2d + %A + 4%A + 8%A + 9%A + 4%A + 2%A + 4, 7 6 5 4 3 2e - 2%A - 9%A - 18%A - 19%A - 4%A + 2 10%A + 4%A - 4 , 7 6 5 4 3 2f - 2%A - 9%A - 18%A - 19%A - 4%A + 2 10%A + 2%A - 4 ] ] , [ complexRoots = 8 7 6 5 4 3 ? + 12? + 64? + 192? + 432? + 768? + 2 1024? + 768? + 256 , coordinates = [ 7 6 5 4 1408a - 19%A - 200%A - 912%A - 2216%A + 3 2 - 4544%A - 6784%A - 6976%A - 1792 , 7 6 5 4 1408b - 37%A - 408%A - 1952%A - 5024%A + 3 2 - 10368%A - 16768%A - 17920%A - 5120 , 7 6 5 4 1408c + 37%A + 408%A + 1952%A + 5024%A + 3 2 10368%A + 16768%A + 17920%A + 5120 , 7 6 5 4 1408d + 19%A + 200%A + 912%A + 2216%A + 3 2 4544%A + 6784%A + 6976%A + 1792 , 2e + %A, 2f - %A] ] , 8 6 4 2 [complexRoots= ? + 4? + 12? + 16? + 4, coordinates = 7 5 3 [4a - %A - 3%A - 10%A - 6%A, 7 5 3 4b - %A - 3%A - 10%A - 10%A, 7 5 3 4c - 2%A - 7%A - 20%A - 22%A, 7 5 3 4d + 2%A + 7%A + 20%A + 22%A, 7 5 3 4e + %A + 3%A + 10%A + 10%A, 7 5 3 4f + %A + 3%A + 10%A + 6%A] ] , 8 6 4 2 [complexRoots= ? + 16? - 96? + 256? + 256, coordinates = 7 5 3 [512a - %A - 12%A + 176%A - 448%A, 7 5 3 128b - %A - 16%A + 96%A - 256%A, 7 5 3 128c + %A + 16%A - 96%A + 256%A, 7 5 3 512d + %A + 12%A - 176%A + 448%A, 2e + %A, 2f - %A] ] , [ complexRoots = 8 7 6 5 4 3 ? - 12? + 64? - 192? + 432? - 768? + 2 1024? - 768? + 256 , coordinates = [ 7 6 5 4 1408a - 19%A + 200%A - 912%A + 2216%A + 3 2 - 4544%A + 6784%A - 6976%A + 1792 , 7 6 5 4 1408b - 37%A + 408%A - 1952%A + 5024%A + 3 2 - 10368%A + 16768%A - 17920%A + 5120 , 7 6 5 4 1408c + 37%A - 408%A + 1952%A - 5024%A + 3 2 10368%A - 16768%A + 17920%A - 5120 , 7 6 5 4 1408d + 19%A - 200%A + 912%A - 2216%A + 3 2 4544%A - 6784%A + 6976%A - 1792 , 2e + %A, 2f - %A] ] , [ complexRoots = 8 7 6 5 4 2 ? - 6? + 16? - 24? + 18? - 8? + 4 , coordinates = [ 7 6 5 4 3 2a + 2%A - 9%A + 18%A - 19%A + 4%A + 2 10%A - 2%A - 4 , 7 6 5 4 3 2b + 2%A - 9%A + 18%A - 19%A + 4%A + 2 10%A - 4%A - 4 , 7 6 5 4 3 2c - %A + 4%A - 8%A + 9%A - 4%A - 2%A + 4, 7 6 5 4 3 2d + %A - 4%A + 8%A - 9%A + 4%A + 2%A - 4, 7 6 5 4 3 2e - 2%A + 9%A - 18%A + 19%A - 4%A + 2 - 10%A + 4%A + 4 , 7 6 5 4 3 2f - 2%A + 9%A - 18%A + 19%A - 4%A + 2 - 10%A + 2%A + 4 ] ] , 4 2 [complexRoots= ? + 12? + 144, coordinates = 2 2 2 [12a - %A - 12, 12b - %A - 12, 12c - %A - 12, 2 2 12d - %A - 12, 6e + %A + 3%A + 12, 2 6f + %A - 3%A + 12] ] , 4 3 2 [complexRoots= ? + 6? + 30? + 36? + 36, coordinates = 3 2 [6a - %A - 5%A - 24%A - 6, 3 2 30b + %A + 5%A + 30%A + 6, 3 2 30c + %A + 5%A + 30%A + 6, 3 2 30d + %A + 5%A + 30%A + 6, 3 2 30e + %A + 5%A + 30%A + 6, 3 2 30f + %A + 5%A + 6] ] , 4 3 2 [complexRoots= ? - 6? + 30? - 36? + 36, coordinates = 3 2 [6a - %A + 5%A - 24%A + 6, 3 2 30b + %A - 5%A + 30%A - 6, 3 2 30c + %A - 5%A + 30%A - 6, 3 2 30d + %A - 5%A + 30%A - 6, 3 2 30e + %A - 5%A + 30%A - 6, 3 2 30f + %A - 5%A - 6] ] , 4 2 [complexRoots= ? + 12? + 144, coordinates = 2 2 2 [12a + %A + 12, 12b + %A + 12, 12c + %A + 12, 2 2 12d + %A + 12, 6e - %A + 3%A - 12, 2 6f - %A - 3%A - 12] ] , 2 [complexRoots= ? - 12, coordinates = [a - 1,b - 1,c - 1,d - 1,2e + %A + 4,2f - %A + 4] ] , 2 [complexRoots= ? + 6? + 6, coordinates = [a + %A + 5,b - 1,c - 1,d - 1,e - 1,f - %A - 1] ] , 2 [complexRoots= ? - 6? + 6, coordinates = [a + %A - 5,b + 1,c + 1,d + 1,e + 1,f - %A + 1] ] , 2 [complexRoots= ? - 12, coordinates = [a + 1,b + 1,c + 1,d + 1,2e + %A - 4,2f - %A - 4] ] , 4 3 2 [complexRoots= ? + 6? + 30? + 36? + 36, coordinates = 3 2 [30a - %A - 5%A - 30%A - 6, 3 2 30b - %A - 5%A - 30%A - 6, 3 2 30c - %A - 5%A - 30%A - 6, 3 2 6d + %A + 5%A + 24%A + 6, 3 2 30e - %A - 5%A - 6, 3 2 30f - %A - 5%A - 30%A - 6] ] , 4 3 2 [complexRoots= ? - 6? + 30? - 36? + 36, coordinates = 3 2 [30a - %A + 5%A - 30%A + 6, 3 2 30b - %A + 5%A - 30%A + 6, 3 2 30c - %A + 5%A - 30%A + 6, 3 2 6d + %A - 5%A + 24%A - 6, 3 2 30e - %A + 5%A + 6, 3 2 30f - %A + 5%A - 30%A + 6] ] , 2 [complexRoots= ? + 6? + 6, coordinates = [a + 1,b + 1,c + 1,d - %A - 5,e + %A + 1,f + 1] ] , 2 [complexRoots= ? - 6? + 6, coordinates = [a - 1,b - 1,c - 1,d - %A + 5,e + %A - 1,f - 1] ] ] Type: List Record(complexRoots: SparseUnivariatePolynomial Integer,coordinates: List Polynomial Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty21} \begin{paste}{LexTriangularPackageXmpPageEmpty21}{LexTriangularPackageXmpPagePatch21} \pastebutton{LexTriangularPackageXmpPageEmpty21}{\showpaste} \tab{5}\spadcommand{concat [univariateSolve(ts)$zdpack for ts in lts]\free{lts }\free{zdpack }} \end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPagePatch22} \begin{paste}{LexTriangularPackageXmpPageFull22}{LexTriangularPackageXmpPageEmpty22} \pastebutton{LexTriangularPackageXmpPageFull22}{\hidepaste} \tab{5}\spadcommand{concat [realSolve(ts)$zdpack for ts in lts]\free{lts }\free{zdpack }} \indentrel{3}\begin{verbatim} (22) [[%R1,%R1,%R1,%R5,- %R5 - 4%R1,%R1], [%R1,%R1,%R1,%R6,- %R6 - 4%R1,%R1], [%R2,%R2,%R2,%R3,- %R3 - 4%R2,%R2], [%R2,%R2,%R2,%R4,- %R4 - 4%R2,%R2], [%R7,%R7,%R7,%R7,%R11,- %R11 - 4%R7], [%R7,%R7,%R7,%R7,%R12,- %R12 - 4%R7], [%R8,%R8,%R8,%R8,%R9,- %R9 - 4%R8], [%R8,%R8,%R8,%R8,%R10,- %R10 - 4%R8], [%R13,%R13,%R17,- %R17 - 4%R13,%R13,%R13], [%R13,%R13,%R18,- %R18 - 4%R13,%R13,%R13], [%R14,%R14,%R15,- %R15 - 4%R14,%R14,%R14], [%R14,%R14,%R16,- %R16 - 4%R14,%R14,%R14], [%R19, %R29, 7865521 31 6696179241 25 컴컴컴컴컴 %R19 - 컴컴컴컴컴 %R19 6006689520 2002229840 + 25769893181 19 1975912990729 13 - 컴컴컴컴컴� %R19 - 컴컴컴컴컴컴� %R19 49235160 3003344760 + 1048460696489 7 21252634831 - 컴컴컴컴컴컴� %R19 - 컴컴컴컴컴� %R19 2002229840 6006689520 , 778171189 31 1987468196267 25 - 컴컴컴컴컴컴� %R19 + 컴컴컴컴컴컴� %R19 1387545279120 1387545279120 + 155496778477189 19 191631411158401 13 컴컴컴컴컴컴컴� %R19 + 컴컴컴컴컴컴컴� %R19 693772639560 693772639560 + 300335488637543 7 755656433863 컴컴컴컴컴컴컴� %R19 - 컴컴컴컴컴컴 %R19 1387545279120 198220754160 , 1094352947 31 2794979430821 25 컴컴컴컴컴컴 %R19 - 컴컴컴컴컴컴� %R19 462515093040 462515093040 + 218708802908737 19 91476663003591 13 - 컴컴컴컴컴컴컴� %R19 - 컴컴컴컴컴컴컴 %R19 231257546520 77085848840 + 145152550961823 7 1564893370717 - 컴컴컴컴컴컴컴� %R19 - 컴컴컴컴컴컴� %R19 154171697680 462515093040 , 4321823003 31 - %R29 - 컴컴컴컴컴컴� %R19 1387545279120 + 180949546069 25 863753195062493 19 컴컴컴컴컴컴 %R19 + 컴컴컴컴컴컴컴� %R19 22746643920 693772639560 + 1088094456732317 13 1732620732685741 7 컴컴컴컴컴컴컴컴 %R19 + 컴컴컴컴컴컴컴컴 %R19 693772639560 1387545279120 + 13506088516033 컴컴컴컴컴컴컴 %R19 1387545279120 ] , [%R19, %R30, 7865521 31 6696179241 25 컴컴컴컴컴 %R19 - 컴컴컴컴컴 %R19 6006689520 2002229840 + 25769893181 19 1975912990729 13 - 컴컴컴컴컴� %R19 - 컴컴컴컴컴컴� %R19 49235160 3003344760 + 1048460696489 7 21252634831 - 컴컴컴컴컴컴� %R19 - 컴컴컴컴컴� %R19 2002229840 6006689520 , 778171189 31 1987468196267 25 - 컴컴컴컴컴컴� %R19 + 컴컴컴컴컴컴� %R19 1387545279120 1387545279120 + 155496778477189 19 191631411158401 13 컴컴컴컴컴컴컴� %R19 + 컴컴컴컴컴컴컴� %R19 693772639560 693772639560 + 300335488637543 7 755656433863 컴컴컴컴컴컴컴� %R19 - 컴컴컴컴컴컴 %R19 1387545279120 198220754160 , 1094352947 31 2794979430821 25 컴컴컴컴컴컴 %R19 - 컴컴컴컴컴컴� %R19 462515093040 462515093040 + 218708802908737 19 91476663003591 13 - 컴컴컴컴컴컴컴� %R19 - 컴컴컴컴컴컴컴 %R19 231257546520 77085848840 + 145152550961823 7 1564893370717 - 컴컴컴컴컴컴컴� %R19 - 컴컴컴컴컴컴� %R19 154171697680 462515093040 , 4321823003 31 - %R30 - 컴컴컴컴컴컴� %R19 1387545279120 + 180949546069 25 863753195062493 19 컴컴컴컴컴컴 %R19 + 컴컴컴컴컴컴컴� %R19 22746643920 693772639560 + 1088094456732317 13 1732620732685741 7 컴컴컴컴컴컴컴컴 %R19 + 컴컴컴컴컴컴컴컴 %R19 693772639560 1387545279120 + 13506088516033 컴컴컴컴컴컴컴 %R19 1387545279120 ] , [%R20, %R27, 7865521 31 6696179241 25 컴컴컴컴컴 %R20 - 컴컴컴컴컴 %R20 6006689520 2002229840 + 25769893181 19 1975912990729 13 - 컴컴컴컴컴� %R20 - 컴컴컴컴컴컴� %R20 49235160 3003344760 + 1048460696489 7 21252634831 - 컴컴컴컴컴컴� %R20 - 컴컴컴컴컴� %R20 2002229840 6006689520 , 778171189 31 1987468196267 25 - 컴컴컴컴컴컴� %R20 + 컴컴컴컴컴컴� %R20 1387545279120 1387545279120 + 155496778477189 19 191631411158401 13 컴컴컴컴컴컴컴� %R20 + 컴컴컴컴컴컴컴� %R20 693772639560 693772639560 + 300335488637543 7 755656433863 컴컴컴컴컴컴컴� %R20 - 컴컴컴컴컴컴 %R20 1387545279120 198220754160 , 1094352947 31 2794979430821 25 컴컴컴컴컴컴 %R20 - 컴컴컴컴컴컴� %R20 462515093040 462515093040 + 218708802908737 19 91476663003591 13 - 컴컴컴컴컴컴컴� %R20 - 컴컴컴컴컴컴컴 %R20 231257546520 77085848840 + 145152550961823 7 1564893370717 - 컴컴컴컴컴컴컴� %R20 - 컴컴컴컴컴컴� %R20 154171697680 462515093040 , 4321823003 31 - %R27 - 컴컴컴컴컴컴� %R20 1387545279120 + 180949546069 25 863753195062493 19 컴컴컴컴컴컴 %R20 + 컴컴컴컴컴컴컴� %R20 22746643920 693772639560 + 1088094456732317 13 1732620732685741 7 컴컴컴컴컴컴컴컴 %R20 + 컴컴컴컴컴컴컴컴 %R20 693772639560 1387545279120 + 13506088516033 컴컴컴컴컴컴컴 %R20 1387545279120 ] , [%R20, %R28, 7865521 31 6696179241 25 컴컴컴컴컴 %R20 - 컴컴컴컴컴 %R20 6006689520 2002229840 + 25769893181 19 1975912990729 13 - 컴컴컴컴컴� %R20 - 컴컴컴컴컴컴� %R20 49235160 3003344760 + 1048460696489 7 21252634831 - 컴컴컴컴컴컴� %R20 - 컴컴컴컴컴� %R20 2002229840 6006689520 , 778171189 31 1987468196267 25 - 컴컴컴컴컴컴� %R20 + 컴컴컴컴컴컴� %R20 1387545279120 1387545279120 + 155496778477189 19 191631411158401 13 컴컴컴컴컴컴컴� %R20 + 컴컴컴컴컴컴컴� %R20 693772639560 693772639560 + 300335488637543 7 755656433863 컴컴컴컴컴컴컴� %R20 - 컴컴컴컴컴컴 %R20 1387545279120 198220754160 , 1094352947 31 2794979430821 25 컴컴컴컴컴컴 %R20 - 컴컴컴컴컴컴� %R20 462515093040 462515093040 + 218708802908737 19 91476663003591 13 - 컴컴컴컴컴컴컴� %R20 - 컴컴컴컴컴컴컴 %R20 231257546520 77085848840 + 145152550961823 7 1564893370717 - 컴컴컴컴컴컴컴� %R20 - 컴컴컴컴컴컴� %R20 154171697680 462515093040 , 4321823003 31 - %R28 - 컴컴컴컴컴컴� %R20 1387545279120 + 180949546069 25 863753195062493 19 컴컴컴컴컴컴 %R20 + 컴컴컴컴컴컴컴� %R20 22746643920 693772639560 + 1088094456732317 13 1732620732685741 7 컴컴컴컴컴컴컴컴 %R20 + 컴컴컴컴컴컴컴컴 %R20 693772639560 1387545279120 + 13506088516033 컴컴컴컴컴컴컴 %R20 1387545279120 ] , [%R21, %R25, 7865521 31 6696179241 25 컴컴컴컴컴 %R21 - 컴컴컴컴컴 %R21 6006689520 2002229840 + 25769893181 19 1975912990729 13 - 컴컴컴컴컴� %R21 - 컴컴컴컴컴컴� %R21 49235160 3003344760 + 1048460696489 7 21252634831 - 컴컴컴컴컴컴� %R21 - 컴컴컴컴컴� %R21 2002229840 6006689520 , 778171189 31 1987468196267 25 - 컴컴컴컴컴컴� %R21 + 컴컴컴컴컴컴� %R21 1387545279120 1387545279120 + 155496778477189 19 191631411158401 13 컴컴컴컴컴컴컴� %R21 + 컴컴컴컴컴컴컴� %R21 693772639560 693772639560 + 300335488637543 7 755656433863 컴컴컴컴컴컴컴� %R21 - 컴컴컴컴컴컴 %R21 1387545279120 198220754160 , 1094352947 31 2794979430821 25 컴컴컴컴컴컴 %R21 - 컴컴컴컴컴컴� %R21 462515093040 462515093040 + 218708802908737 19 91476663003591 13 - 컴컴컴컴컴컴컴� %R21 - 컴컴컴컴컴컴컴 %R21 231257546520 77085848840 + 145152550961823 7 1564893370717 - 컴컴컴컴컴컴컴� %R21 - 컴컴컴컴컴컴� %R21 154171697680 462515093040 , 4321823003 31 - %R25 - 컴컴컴컴컴컴� %R21 1387545279120 + 180949546069 25 863753195062493 19 컴컴컴컴컴컴 %R21 + 컴컴컴컴컴컴컴� %R21 22746643920 693772639560 + 1088094456732317 13 1732620732685741 7 컴컴컴컴컴컴컴컴 %R21 + 컴컴컴컴컴컴컴컴 %R21 693772639560 1387545279120 + 13506088516033 컴컴컴컴컴컴컴 %R21 1387545279120 ] , [%R21, %R26, 7865521 31 6696179241 25 컴컴컴컴컴 %R21 - 컴컴컴컴컴 %R21 6006689520 2002229840 + 25769893181 19 1975912990729 13 - 컴컴컴컴컴� %R21 - 컴컴컴컴컴컴� %R21 49235160 3003344760 + 1048460696489 7 21252634831 - 컴컴컴컴컴컴� %R21 - 컴컴컴컴컴� %R21 2002229840 6006689520 , 778171189 31 1987468196267 25 - 컴컴컴컴컴컴� %R21 + 컴컴컴컴컴컴� %R21 1387545279120 1387545279120 + 155496778477189 19 191631411158401 13 컴컴컴컴컴컴컴� %R21 + 컴컴컴컴컴컴컴� %R21 693772639560 693772639560 + 300335488637543 7 755656433863 컴컴컴컴컴컴컴� %R21 - 컴컴컴컴컴컴 %R21 1387545279120 198220754160 , 1094352947 31 2794979430821 25 컴컴컴컴컴컴 %R21 - 컴컴컴컴컴컴� %R21 462515093040 462515093040 + 218708802908737 19 91476663003591 13 - 컴컴컴컴컴컴컴� %R21 - 컴컴컴컴컴컴컴 %R21 231257546520 77085848840 + 145152550961823 7 1564893370717 - 컴컴컴컴컴컴컴� %R21 - 컴컴컴컴컴컴� %R21 154171697680 462515093040 , 4321823003 31 - %R26 - 컴컴컴컴컴컴� %R21 1387545279120 + 180949546069 25 863753195062493 19 컴컴컴컴컴컴 %R21 + 컴컴컴컴컴컴컴� %R21 22746643920 693772639560 + 1088094456732317 13 1732620732685741 7 컴컴컴컴컴컴컴컴 %R21 + 컴컴컴컴컴컴컴컴 %R21 693772639560 1387545279120 + 13506088516033 컴컴컴컴컴컴컴 %R21 1387545279120 ] , [%R22, %R23, 7865521 31 6696179241 25 컴컴컴컴컴 %R22 - 컴컴컴컴컴 %R22 6006689520 2002229840 + 25769893181 19 1975912990729 13 - 컴컴컴컴컴� %R22 - 컴컴컴컴컴컴� %R22 49235160 3003344760 + 1048460696489 7 21252634831 - 컴컴컴컴컴컴� %R22 - 컴컴컴컴컴� %R22 2002229840 6006689520 , 778171189 31 1987468196267 25 - 컴컴컴컴컴컴� %R22 + 컴컴컴컴컴컴� %R22 1387545279120 1387545279120 + 155496778477189 19 191631411158401 13 컴컴컴컴컴컴컴� %R22 + 컴컴컴컴컴컴컴� %R22 693772639560 693772639560 + 300335488637543 7 755656433863 컴컴컴컴컴컴컴� %R22 - 컴컴컴컴컴컴 %R22 1387545279120 198220754160 , 1094352947 31 2794979430821 25 컴컴컴컴컴컴 %R22 - 컴컴컴컴컴컴� %R22 462515093040 462515093040 + 218708802908737 19 91476663003591 13 - 컴컴컴컴컴컴컴� %R22 - 컴컴컴컴컴컴컴 %R22 231257546520 77085848840 + 145152550961823 7 1564893370717 - 컴컴컴컴컴컴컴� %R22 - 컴컴컴컴컴컴� %R22 154171697680 462515093040 , 4321823003 31 - %R23 - 컴컴컴컴컴컴� %R22 1387545279120 + 180949546069 25 863753195062493 19 컴컴컴컴컴컴 %R22 + 컴컴컴컴컴컴컴� %R22 22746643920 693772639560 + 1088094456732317 13 1732620732685741 7 컴컴컴컴컴컴컴컴 %R22 + 컴컴컴컴컴컴컴컴 %R22 693772639560 1387545279120 + 13506088516033 컴컴컴컴컴컴컴 %R22 1387545279120 ] , [%R22, %R24, 7865521 31 6696179241 25 컴컴컴컴컴 %R22 - 컴컴컴컴컴 %R22 6006689520 2002229840 + 25769893181 19 1975912990729 13 - 컴컴컴컴컴� %R22 - 컴컴컴컴컴컴� %R22 49235160 3003344760 + 1048460696489 7 21252634831 - 컴컴컴컴컴컴� %R22 - 컴컴컴컴컴� %R22 2002229840 6006689520 , 778171189 31 1987468196267 25 - 컴컴컴컴컴컴� %R22 + 컴컴컴컴컴컴� %R22 1387545279120 1387545279120 + 155496778477189 19 191631411158401 13 컴컴컴컴컴컴컴� %R22 + 컴컴컴컴컴컴컴� %R22 693772639560 693772639560 + 300335488637543 7 755656433863 컴컴컴컴컴컴컴� %R22 - 컴컴컴컴컴컴 %R22 1387545279120 198220754160 , 1094352947 31 2794979430821 25 컴컴컴컴컴컴 %R22 - 컴컴컴컴컴컴� %R22 462515093040 462515093040 + 218708802908737 19 91476663003591 13 - 컴컴컴컴컴컴컴� %R22 - 컴컴컴컴컴컴컴 %R22 231257546520 77085848840 + 145152550961823 7 1564893370717 - 컴컴컴컴컴컴컴� %R22 - 컴컴컴컴컴컴� %R22 154171697680 462515093040 , 4321823003 31 - %R24 - 컴컴컴컴컴컴� %R22 1387545279120 + 180949546069 25 863753195062493 19 컴컴컴컴컴컴 %R22 + 컴컴컴컴컴컴컴� %R22 22746643920 693772639560 + 1088094456732317 13 1732620732685741 7 컴컴컴컴컴컴컴컴 %R22 + 컴컴컴컴컴컴컴컴 %R22 693772639560 1387545279120 + 13506088516033 컴컴컴컴컴컴컴 %R22 1387545279120 ] , [%R31,%R35,- %R35 - 4%R31,%R31,%R31,%R31], [%R31,%R36,- %R36 - 4%R31,%R31,%R31,%R31], [%R32,%R33,- %R33 - 4%R32,%R32,%R32,%R32], [%R32,%R34,- %R34 - 4%R32,%R32,%R32,%R32]] Type: List List RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{LexTriangularPackageXmpPageEmpty22} \begin{paste}{LexTriangularPackageXmpPageEmpty22}{LexTriangularPackageXmpPagePatch22} \pastebutton{LexTriangularPackageXmpPageEmpty22}{\showpaste} \tab{5}\spadcommand{concat [realSolve(ts)$zdpack for ts in lts]\free{lts }\free{zdpack }} \end{paste}\end{patch}