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# pushforwardMultiplicity -- pushforward of Minkowski weights

## Synopsis

• Usage:
pushforwardMultiplicity(X,X',mult,k)
• Inputs:
• Outputs:
• prod, a list, list of intersection products

## Description

The list mult is the list of multiplicities of the tropicalization of the intersection of a subvariety Y of X with the torus. The tropicalization has the fan structure of the fan of X', and k is the dimension of Y. This function calculates the intersection products of Y and V(sigma), where sigma is a cone of X of dimension k. This is done by calculating the pullback of V(sigma) in X', and substituting V(sigma') with m(sigma'), where sigma' is a cone of dimension k in X'. This is the same as taking the class in X' corresponding to the Minkowski weights given by Y and computing the Minkowski weights corresponding to the pushforward of this class on X.

 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 : mult = refineMultiplicity(T,X'); i8 : pushforwardMultiplicity(X,X',mult,dim T) o8 = {3, 3, 3, 3, 3, 3} o8 : List

## Ways to use pushforwardMultiplicity :

• pushforwardMultiplicity(NormalToricVariety,NormalToricVariety,List,ZZ)

## For the programmer

The object pushforwardMultiplicity is .