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# gfanMinkowskiSum -- the Minkowski sum of Newton polytopes

## Synopsis

• Usage:
P = gfanMinkowskiSum L
P = gfanMinkowskiSum I
• Inputs:
• Optional inputs:
• disableSymmetryTest
• nocones
• symmetry
• Outputs:
• P, an instance of the type Fan, the normal fan of the Minkowski sum of the newton polytopes of the polynomials in L or generators of I.

## Description

The Newton polytope of a polynomial is the convex hull of the exponent vectors of the terms. This method produces the normal fan of the Minkowski sum of these polytopes, which is the same as the common refinement of the normal fans.

 i1 : QQ[x,y] o1 = QQ[x..y] o1 : PolynomialRing i2 : gfanMinkowskiSum { x + y + x*y } o2 = Fan{...1...} o2 : Fan i3 : gfanMinkowskiSum { x + y + x*y + 1} o3 = Fan{...1...} o3 : Fan i4 : gfanMinkowskiSum { x + y + x*y, x + y + x*y + 1} o4 = Fan{...1...} o4 : Fan

gfan Documentation

This is a program for computing the normal fan of the Minkowski sum of the Newton polytopes of a list of polynomials.
Options:
--symmetry:
Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup.

--disableSymmetryTest:
When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.

--nocones:
Tell the program to not list cones in the output.


## Ways to use gfanMinkowskiSum :

• gfanMinkowskiSum(Ideal)
• gfanMinkowskiSum(List)

## For the programmer

The object gfanMinkowskiSum is .