Stable intersection of tropical hypersurfaces

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ren
Posts: 34
Joined: 03 May 2011, 15:21

Stable intersection of tropical hypersurfaces

Postby ren » 07 May 2021, 17:21

In the following example, the stable intersection T12345 is empty even though I think it should not be:

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application "tropical"; $f1 = toTropicalPolynomial("min(33+x1,72+x2)",qw( w x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 )); $f2 = toTropicalPolynomial("min(31+x2+x4,33+x1+w,57+x2+w,54+x3+w,2+x14+w)",qw( w x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 )); $f3 = toTropicalPolynomial("min(37+x3+x6,57+x2+w,54+x3+w,29+x15+w)",qw( w x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 )); $f4 = toTropicalPolynomial("min(31+x2+x4,42+x4+x10,27+x14+w,87+x16+w,40+x18+w)",qw( w x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 )); $f5 = toTropicalPolynomial("min(72+x5+x8,12+x5+w,31+x7+w,2+x14+w,28+x16+w)",qw( w x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 )); $T1 = new Hypersurface<Min>(POLYNOMIAL=>$f1); $T2 = new Hypersurface<Min>(POLYNOMIAL=>$f2); $T3 = new Hypersurface<Min>(POLYNOMIAL=>$f3); $T4 = new Hypersurface<Min>(POLYNOMIAL=>$f4); $T5 = new Hypersurface<Min>(POLYNOMIAL=>$f5); $T12 = intersect($T1,$T2); $T123 = intersect($T12,$T3); $T1234 = intersect($T123,$T4); $T12345 = intersect($T1234,$T5);
The ambient dimension is 20, the codimension of $T1234 is 4, and the codimension of $T5 is 1, the dimension of the intersection of the lineality spaces is 2 (see below). Hence their stable intersection should have codimension 4+1 (see [MaclaganSturmfels, Theorem 3.6.10]).

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application "polytope"; $c = new Cone(INEQUALITIES=>[], EQUATIONS=>$T1234->LINEALITY_SPACE/$T5->LINEALITY_SPACE); print $c->DIM;

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