The dequantization transform and generalized Newton polytopes

TL;DR

This paper introduces a dequantization transform for functions on complex and positive real spaces, linking polynomial subdifferentials to Newton polytopes and extending these concepts to broader functions and convex sets.

Contribution

It develops a dequantization transform that connects polynomial subdifferentials with Newton polytopes and generalizes this framework to various functions and convex sets.

Findings

Dequantization transform relates polynomial subdifferentials to Newton polytopes.

The transform acts as a homomorphism to the semiring of convex polytopes.

Generalization of these concepts to a wide class of functions and convex sets.

Abstract

For functions defined on C^n or (R_+)^n we construct a dequantization transform, which is closely related to the Maslov dequantization. The subdifferential at the origin of a dequantized polynomial coincides with its Newton polytope. For the semiring of polynomials with nonnegative coefficients, the dequantization transform is a homomorphism of this semiring to the idempotent semiring of convex polytopes with the well-known Minkowski operations. Using the dequantization transform we generalize these results to a wide class of functions and convex sets.