The fundamental theorem of tropical differential algebraic geometry

Fuensanta Aroca, Cristhian Garay and Zeinab Toghani

Vol. 283 (2016), No. 2, 257–270

DOI: 10.2140/pjm.2016.283.257

Abstract

Let $I$ be an ideal of the ring of Laurent polynomials $K\left[x_1^{\pm 1},\dots ,x_n^{\pm 1}\right]$ with coefficients in a real-valued field $\left(K,v\right)$. The fundamental theorem of tropical algebraic geometry states the equality $trop\left(V\left(I\right)\right)=V\left(trop\left(I\right)\right)$ between the tropicalization $trop\left(V\left(I\right)\right)$ of the closed subscheme $V\left(I\right)\subset {\left(K^{\ast }\right)}^n$ and the tropical variety $V\left(trop\left(I\right)\right)$ associated to the tropicalization of the ideal $trop\left(I\right)$.

In this work we prove an analogous result for a differential ideal $G$ of the ring of differential polynomials $K\left[\left[t\right]\right]\left\{x_1,\dots ,x_n\right\}$, where $K$ is an uncountable algebraically closed field of characteristic zero. We define the tropicalization $trop\left(Sol\left(G\right)\right)$ of the set of solutions $Sol\left(G\right)\subset K{\left[\left[t\right]\right]}^n$ of $G$, and the set of solutions $Sol\left(trop\left(G\right)\right)\subset \mathcal{P}{\left({\mathbb{Z}}_{\ge 0}\right)}^n$ associated to the tropicalization of the ideal $trop\left(G\right)$. These two sets are linked by a tropicalization morphism $trop:Sol\left(G\right)\to Sol\left(trop\left(G\right)\right)$.

We show the equality $trop\left(Sol\left(G\right)\right)=Sol\left(trop\left(G\right)\right)$, answering a question recently raised by D. Grigoriev.

Keywords

differential algebra, tropical geometry, arc spaces, power series solutions of ODEs

Mathematical Subject Classification 2010

Primary: 13N99, 13P15, 14T99

Milestones

Received: 12 January 2016

Revised: 9 March 2016

Accepted: 23 March 2016

Published: 22 June 2016