The Fundamental Theorem of Tropical Differential Algebraic Geometry

TL;DR

This paper extends the fundamental theorem of tropical algebraic geometry to differential algebra, proving an analogous equality between the tropicalization of solution sets and the solution sets of tropicalized differential ideals.

Contribution

It establishes a tropical fundamental theorem for differential algebraic geometry, linking solution sets of differential ideals with their tropical counterparts.

Findings

Proves the equality $ ext{trop}( ext{Sol}(G))= ext{Sol}( ext{trop}(G))$ for differential ideals.

Answers a recent question by D. Grigoriev on tropical differential algebraic geometry.

Extends classical tropical geometry results to the differential algebra setting.

Abstract

Let $I$ be an ideal of the ring of Laurent polynomials $K[x_1^{\pm1},\ldots,x_n^{\pm1}]$ with coefficients in a real-valued field $(K,v)$. The fundamental theorem of tropical algebraic geometry states the equality $\text{trop}(V(I))=V(\text{trop}(I))$ between the tropicalization $\text{trop}(V(I))$ of the closed subscheme $V(I)\subset (K^*)^n$ and the tropical variety $V(\text{trop}(I))$ associated to the tropicalization of the ideal $\text{trop}(I)$. In this work we prove an analogous result for a differential ideal $G$ of the ring of differential polynomials $K[[t]]\{x_1,\ldots,x_n\}$, where $K$ is an uncountable algebraically closed field of characteristic zero. We define the tropicalization $\text{trop}(\text{Sol}(G))$ of the set of solutions $\text{Sol}(G)\subset K[[t]]^n$ of $G$, and the set of solutions associated to the tropicalization of the ideal $\text{trop}(G)$. These two sets are linked by a tropicalization morphism $\text{trop}:\text{Sol}(G)\longrightarrow \text{Sol}(\text{trop}(G))$. We show the equality $\text{trop}(\text{Sol}(G))=\text{Sol}(\text{trop}(G))$, answering a question raised by D. Grigoriev earlier this year.