The Difference Kapranov Theorem
Saba Aliyari
TL;DR
The paper develops a Difference Tropical Geometry framework to connect tropical and difference-algebraic objects, culminating in a Difference Kapranov Theorem for Laurent difference polynomials over multiplicative valued difference fields with ACFA residue fields and transcendental scaling $\rho$. Central tools include a Difference Newton Lemma for lifting roots from initial forms and a polyhedral description of difference tropical hypersurfaces via $(\Gamma,\mathbb{Q}(\rho))$-complexes. The authors establish that the tropicalization $\mathrm{trop}(V(f))$ coincides with the set of $w$ where the initial form is non-monomial and with the valuation-closure of root vectors, thereby extending Kapranov’s correspondence to the difference setting. This work lays groundwork toward a potential difference analogue of the Fundamental Theorem of Tropical Algebraic Geometry and bridges tropical methods with difference algebraic geometry for Laurent polynomials.
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}.