The integer 'nu' needs to be a 'good degree' for the first parameter '{f0,f1,f2,f3}' that can be verified by doing isGoodDegree(polinomialList,nu)
implicitEq({f0,f1,f2,f3},nu) computes the determinant of the maximal minors of representationMatrix({f0,f1,f2,f3},nu). Equivalently, it computes the determinant of the Z-complex in degree 'nu'. This is:
implicitEq({f0,f1,f2,f3}, nu)=det((Z(f0,f1,f2,f3))_nu)
We also have that: degree(implicitEq({f0,f1,f2,f3}, nu))=degreeImplicitEq({f0,f1,f2,f3}, nu)
i1 : S = QQ[s,t];
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i2 : f0 = 2*s+s^2*t;
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i3 : f1 = s^4*t^2+2*s;
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i4 : f2 = s^2*t-3*t;
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i5 : f3 = s*t+5*s^4*t^2
4 2
o5 = 5s t + s*t
o5 : S
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i6 : implicitEq ({f0,f1,f2,f3},2)
14 13 12 2
o6 = - 7192693440X + 100218401280X X - 648311999040X X +
0 0 1 0 1
------------------------------------------------------------------------
11 3 10 4 9 5
2580849100800X X - 7063324804800X X + 14058802644480X X -
0 1 0 1 0 1
------------------------------------------------------------------------
8 6 7 7 6 8
20986693738560X X + 23869078548480X X - 20784453940800X X +
0 1 0 1 0 1
------------------------------------------------------------------------
5 9 4 10 3 11
13789149580800X X - 6861085007040X X + 2482793441280X X -
0 1 0 1 0 1
------------------------------------------------------------------------
2 12 13 14 13
617669605440X X + 94561344000X X - 6721272000X + 9270720X X +
0 1 0 1 1 0 2
------------------------------------------------------------------------
12 11 2 10 3
368058240X X X - 5124044160X X X + 29421216000X X X -
0 1 2 0 1 2 0 1 2
------------------------------------------------------------------------
9 4 8 5 7 6
99991108800X X X + 227312300160X X X - 365840513280X X X +
0 1 2 0 1 2 0 1 2
------------------------------------------------------------------------
6 7 5 8 4 9
428251000320X X X - 367735896000X X X + 230012956800X X X -
0 1 2 0 1 2 0 1 2
------------------------------------------------------------------------
3 10 2 11 12
102233450880X X X + 30655607040X X X - 5568932160X X X +
0 1 2 0 1 2 0 1 2
------------------------------------------------------------------------
13 12 2 11 2 10 2 2
463536000X X - 9270720X X + 98887680X X X - 483782400X X X +
1 2 0 2 0 1 2 0 1 2
------------------------------------------------------------------------
9 3 2 8 4 2 7 5 2
1438560000X X X - 2904292800X X X + 4209546240X X X -
0 1 2 0 1 2 0 1 2
------------------------------------------------------------------------
6 6 2 5 7 2 4 8 2 3 9 2
4511324160X X X + 3618777600X X X - 2165832000X X X + 946252800X X X
0 1 2 0 1 2 0 1 2 0 1 2
------------------------------------------------------------------------
2 10 2 11 2 12 2 11 3
- 286859520X X X + 54132480X X X - 4795200X X - 63936X X +
0 1 2 0 1 2 1 2 0 2
------------------------------------------------------------------------
10 3 9 2 3 8 3 3 7 4 3
639360X X X - 2877120X X X + 7672320X X X - 13426560X X X +
0 1 2 0 1 2 0 1 2 0 1 2
------------------------------------------------------------------------
6 5 3 5 6 3 4 7 3 3 8 3
16111872X X X - 13426560X X X + 7672320X X X - 2877120X X X +
0 1 2 0 1 2 0 1 2 0 1 2
------------------------------------------------------------------------
2 9 3 10 3 13 12
639360X X X - 63936X X X - 7000970688X X + 90824050368X X X -
0 1 2 0 1 2 0 3 0 1 3
------------------------------------------------------------------------
11 2 10 3 9 4
543814467840X X X + 1989849438720X X X - 4964295695040X X X +
0 1 3 0 1 3 0 1 3
------------------------------------------------------------------------
8 5 7 6 6 7
8917168049856X X X - 11864839827456X X X + 11840156951040X X X -
0 1 3 0 1 3 0 1 3
------------------------------------------------------------------------
5 8 4 9 3 10
8861631577920X X X + 4912873035840X X X - 1961052749568X X X +
0 1 3 0 1 3 0 1 3
------------------------------------------------------------------------
2 11 12 13
533716927488X X X - 88767144000X X X + 6813979200X X -
0 1 3 0 1 3 1 3
------------------------------------------------------------------------
12 11 10 2
188014464X X X + 2301504192X X X X - 12906014400X X X X +
0 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
9 3 8 4 7 5
43840595520X X X X - 100475743680X X X X + 163677310848X X X X -
0 1 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
6 6 5 7 4 8
194336924544X X X X + 169452777600X X X X - 107695077120X X X X +
0 1 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
3 9 2 10 11
48653484480X X X X - 14831169984X X X X + 2739039552X X X X -
0 1 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
12 11 2 10 2 9 2 2
231768000X X X - 618048X X X + 8098560X X X X - 46992960X X X X +
1 2 3 0 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
8 3 2 7 4 2 6 5 2
160479360X X X X - 359959680X X X X + 558544896X X X X -
0 1 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
5 6 2 4 7 2 3 8 2
613146240X X X X + 476962560X X X X - 257981760X X X X +
0 1 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
2 9 2 10 2 11 2 12 2
92494080X X X X - 19798848X X X X + 1918080X X X - 2762674560X X
0 1 2 3 0 1 2 3 1 2 3 0 3
------------------------------------------------------------------------
11 2 10 2 2 9 3 2
+ 33152094720X X X - 182312864640X X X + 607551840000X X X -
0 1 3 0 1 3 0 1 3
------------------------------------------------------------------------
8 4 2 7 5 2 6 6 2
1366459372800X X X + 2185199493120X X X - 2547743466240X X X +
0 1 3 0 1 3 0 1 3
------------------------------------------------------------------------
5 7 2 4 8 2 3 9 2
2182076858880X X X - 1362556080000X X X + 604949644800X X X -
0 1 3 0 1 3 0 1 3
------------------------------------------------------------------------
2 10 2 11 2 12 2
181271986560X X X + 32915531520X X X - 2739018240X X -
0 1 3 0 1 3 1 3
------------------------------------------------------------------------
11 2 10 2 9 2 2
65470464X X X + 682516800X X X X - 3247948800X X X X +
0 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
8 3 2 7 4 2 6 5 2
9320909760X X X X - 17937884160X X X X + 24326241408X X X X -
0 1 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
5 6 2 4 7 2 3 8 2
23738158080X X X X + 16677705600X X X X - 8270760960X X X X +
0 1 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
2 9 2 10 2 11 2
2757879360X X X X - 556498944X X X X + 51468480X X X -
0 1 2 3 0 1 2 3 1 2 3
------------------------------------------------------------------------
10 2 2 9 2 2 8 2 2 2 7 3 2 2
383616X X X + 3644352X X X X - 15536448X X X X + 39128832X X X X -
0 2 3 0 1 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
6 4 2 2 5 5 2 2 4 6 2 2
64447488X X X X + 72503424X X X X - 56391552X X X X +
0 1 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
3 7 2 2 2 8 2 2 9 2 2 10 2 2
29922048X X X X - 10357632X X X X + 2109888X X X X - 191808X X X -
0 1 2 3 0 1 2 3 0 1 2 3 1 2 3
------------------------------------------------------------------------
11 3 10 3 9 2 3
552534912X X + 6075838080X X X - 30364229376X X X +
0 3 0 1 3 0 1 3
------------------------------------------------------------------------
8 3 3 7 4 3 6 5 3
91033611264X X X - 181920681216X X X + 254444053248X X X -
0 1 3 0 1 3 0 1 3
------------------------------------------------------------------------
5 6 3 4 7 3 3 8 3
254159410176X X X + 181310731776X X X - 90525320064X X X +
0 1 3 0 1 3 0 1 3
------------------------------------------------------------------------
2 9 3 10 3 11 3 10 3
30127026816X X X - 6014843136X X X + 545757696X X - 3708288X X X
0 1 3 0 1 3 1 3 0 2 3
------------------------------------------------------------------------
9 3 8 2 3 7 3 3
+ 40982976X X X X - 201973824X X X X + 585398016X X X X -
0 1 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
6 4 3 5 5 3 4 6 3
1106348544X X X X + 1425900672X X X X - 1270152576X X X X +
0 1 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
3 7 3 2 8 3 9 3
772602624X X X X - 307276416X X X X + 72183744X X X X -
0 1 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
10 3 10 4 9 4 8 2 4
7608384X X X - 55048896X X + 549338112X X X - 2466842688X X X +
1 2 3 0 3 0 1 3 0 1 3
------------------------------------------------------------------------
7 3 4 6 4 4 5 5 4
6564436992X X X - 11463596928X X X + 13727314944X X X -
0 1 3 0 1 3 0 1 3
------------------------------------------------------------------------
4 6 4 3 7 4 2 8 4
11415261312X X X + 6509196288X X X - 2435769792X X X +
0 1 3 0 1 3 0 1 3
------------------------------------------------------------------------
9 4 10 4 9 4 8 4
540131328X X X - 53898048X X - 575424X X X + 5178816X X X X -
0 1 3 1 3 0 2 3 0 1 2 3
------------------------------------------------------------------------
7 2 4 6 3 4 5 4 4
20715264X X X X + 48335616X X X X - 72503424X X X X +
0 1 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
4 5 4 3 6 4 2 7 4 8 4
72503424X X X X - 48335616X X X X + 20715264X X X X - 5178816X X X X
0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3
------------------------------------------------------------------------
9 4 9 5 8 5 7 2 5
+ 575424X X X - 2109888X X + 18988992X X X - 75955968X X X +
1 2 3 0 3 0 1 3 0 1 3
------------------------------------------------------------------------
6 3 5 5 4 5 4 5 5 3 6 5
177230592X X X - 265845888X X X + 265845888X X X - 177230592X X X +
0 1 3 0 1 3 0 1 3 0 1 3
------------------------------------------------------------------------
2 7 5 8 5 9 5
75955968X X X - 18988992X X X + 2109888X X
0 1 3 0 1 3 1 3
o6 : QQ[X , X , X , X ]
0 1 2 3
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For the programmer: it computes the determinant of a maximal minor of 'representationMatrix (teToricRationalMap{f0,f1,f2,f3},2)' using the function 'maxMinor'