# tropDef -- The co-complex of tropical faces of the deformation polytope.

## Synopsis

• Usage:
tropDef(P,C)
• Inputs:
• C, , a simplicial complex
• P, , the deformation polytope of C as returned by PT1 C.
• Outputs:

## Description

Computes the co-complex of tropical faces of the deformation polytope.

This is work in progress.

 i1 : R=QQ[x_0..x_3] o1 = R o1 : PolynomialRing i2 : I=ideal(x_0*x_1,x_2*x_3) o2 = ideal (x x , x x ) 0 1 2 3 o2 : Ideal of R i3 : C=idealToComplex I o3 = 1: x x x x x x x x 0 2 1 2 0 3 1 3 o3 : complex of dim 1 embedded in dim 3 (printing facets) equidimensional, simplicial, F-vector {1, 4, 4, 0, 0}, Euler = -1 i4 : PT1C=PT1 C o4 = 3: y y y y y y y y 0 1 2 3 4 5 6 7 o4 : complex of dim 3 embedded in dim 3 (printing facets) equidimensional, non-simplicial, F-vector {1, 8, 14, 8, 1}, Euler = 0 i5 : tropDefC=tropDef(C,PT1C) o5 = 1: y y y y y y y y 0 3 6 7 2 5 1 4 o5 : co-complex of dim 1 embedded in dim 3 (printing facets) equidimensional, non-simplicial, F-vector {0, 0, 4, 4, 1}, Euler = -1 i6 : tropDefC.grading o6 = | -1 0 0 | | 1 0 0 | | -1 2 0 | | 0 -1 -1 | | 2 -1 -1 | | 0 1 -1 | | 0 -1 1 | | -1 0 2 | 8 3 o6 : Matrix ZZ <--- ZZ

## Caveat

The implementation of testing whether a face is tropical so far uses a trick to emulate higher order. For complicated (non-complete intersections and non-Pfaffians) examples this may lead to an incorrect result. Use with care. This will be fixed at some point.

If using OldPolyhedra to compute convex hulls and its faces instead of ConvexInterface you are limited to rather simple examples.