The Difference Kapranov Theorem

TL;DR

This paper extends Kapranov's Theorem to Laurent difference polynomials within tropical geometry and difference algebra, establishing a difference analogue that broadens the theoretical framework of these fields.

Contribution

It introduces a difference version of Kapranov's Theorem for Laurent difference polynomials over valued difference fields with ACFA residue fields.

Findings

Established a difference analogue of Kapranov's Theorem.

Developed a difference version of Newton's Lemma.

Linked tropical geometry with difference algebra.

Abstract

This paper creates a link between \textit{Tropical Geometry} and \textit{Difference Algebra}. The main result is a difference version of \textit{Kapranov's Theorem}. In this theorem, we extend Kapranov's Theorem to the case of a Laurent difference polynomial with coefficients from a multiplicative valued difference field, where the residue field is an algebraically closed field with a generic automorphism (ACFA). A result of this paper that plays a critical role in the proof of the Difference Kapranov Theorem is a difference version of \textit{Newton's Lemma}.