Computing Tropical Resultants

TL;DR

This paper introduces the tropical resultant, a tropical geometric object representing common solutions of tropical polynomials, and provides algorithms for its computation, structure analysis, and applications to Newton polytope recovery.

Contribution

It defines the tropical resultant, proves its relation to algebraic solvability, and develops algorithms for its traversal and for recovering Newton polytopes from tropical data.

Findings

TR(A) is the tropicalization of the algebraic variety of solvable systems.

Polynomial-time computation of the dimension of TR(A).

Algorithms for Newton polytope recovery from tropical hypersurfaces.

Abstract

We fix the supports A=(A_1,...,A_k) of a list of tropical polynomials and define the tropical resultant TR(A) to be the set of choices of coefficients such that the tropical polynomials have a common solution. We prove that TR(A) is the tropicalization of the algebraic variety of solvable systems and that its dimension can be computed in polynomial time. The tropical resultant inherits a fan structure from the secondary fan of the Cayley configuration of A and we present algorithms for the traversal of TR(A) in this structure. We also present a new algorithm for recovering a Newton polytope from the support of its tropical hypersurface. We use this to compute the Newton polytope of the sparse resultant polynomial in the case when TR(A) is of codimension 1. Finally we consider the more general setting of specialized tropical resultants and report on experiments with our implementations.