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# refineMultiplicity -- Compute the multiplicities of a refinement of a tropical variety

## Synopsis

• Usage:
refineMultiplicity(T,X)
• Inputs:
• T, ,
• X, , which fan refines T
• Outputs:
• mult, a list, list of the multiplicities, indexed by cones of X of the same dimension of T

## Description

This function calculate the multiplicities of a refinement of a tropical variety T. The output is a list of integers indexed by orbits(X,n-k), where k = dim T, n = dim X and X is the normal toric variety of the refinement.

 i1 : X = toricProjectiveSpace 3; i2 : R = QQ[x_1 .. x_3]; i3 : f = x_1*x_2*x_3 + x_1*x_2 + x_1*x_3 + x_2*x_3; i4 : T = tropicalVariety(ideal f); i5 : F = gfanFanCommonRefinement(fan X, fan T); i6 : X' = makeSimplicial (normalToricVariety F); i7 : multiplicities T o7 = {1, 1, 1, 1, 1, 1} o7 : List i8 : refineMultiplicity(T,X') o8 = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1} o8 : List

## Ways to use refineMultiplicity :

• refineMultiplicity(List,Fan,NormalToricVariety)
• refineMultiplicity(TropicalCycle,NormalToricVariety)

## For the programmer

The object refineMultiplicity is .