Curve valuations and mixed volumes in the implicitization of rational varieties

TL;DR

This paper explores the tropicalization of rational varieties using curve valuations, extending existing results to non-generic cases and providing new formulas for degrees and orders based on combinatorial support conditions.

Contribution

It introduces a generalized approach to tropicalization of rational varieties with non-trivial denominators, offering new formulas and bounds for degrees and orders.

Findings

Derived new formulas for the degree of the image based on support positions.

Provided bounds on the degree when polynomial supports differ.

Established bounds for the order at the origin for hypersurface images.

Abstract

We address the description of the tropicalization of families of rational varieties under parametrizations with prescribed support, via curve valuations. We recover and extend results by Sturmfels, Tevelev and Yu for generic coefficients, considering rational parametrizations with non-trivial denominator. The advantage of our point of view is that it can be generalized to deal with non-generic parametrizations. We provide a detailed analysis of the degree of the closed image, based on combinatorial conditions on the relative positions of the supports of the polynomials defining the parametrization. We obtain a new formula and finer bounds on the degree, when the supports of the polynomials are different. We also present a new formula and bounds for the order at the origin in case the closed image is a hypersurface.