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

- Usage:
    `tropDef(P,C)`
- Inputs:
  - C, an embedded complex, a simplicial complex
  - P, an embedded complex, the deformation polytope of C as returned by PT1 C.
- Outputs:
  - an embedded co-complex,

## 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 Polyhedra to compute convex hulls and its faces instead of ConvexInterface you are limited to rather simple examples.

## See also

- PT1 -- Compute the deformation polytope associated to a Stanley-Reisner complex.
- saveDeformations -- Store the deformation data of a complex in a file.
- convHull(String) -- The convex hull complex.
- deform -- Compute the deformations associated to a Stanley-Reisner complex.

## Ways to use tropDef:

- `tropDef(Complex,Complex)`

## For the programmer

The object tropDef is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/SRdeformations.m2:7689:0.