The fundamental theorem of tropical differential algebraic geometry

Abstract

Let I be an ideal of the ring of Laurent polynomials K[x ±1 1 , . . . , x ±1 n ] with coefficients in a real-valued field (K, v). The fundamental theorem of tropical algebraic geometry states the equality trop(V (I)) = V (trop(I)) between the tropicalization trop(V (I)) of the closed subscheme V (I) ⊂ (K * ) n and the tropical variety V (trop(I)) associated to the tropicalization of the ideal trop(I).In this work we prove an analogous result for a differential ideal G of the ring of differential polynomials K[[t]]{x1, . . . , xn}, where K is an uncountable algebraically closed field of characteristic zero. We define the tropicalization trop(Sol(G)) of the set of solutions Sol(G) ⊂ K[[t]] n of G, and the set of solutions Sol(trop(G)) ⊂ P(Z ≥0 ) n associated to the tropicalization of the ideal trop(G). These two sets are linked by a tropicalization morphism trop : Sol(G) −→ Sol(trop(G)).We show the equality trop(Sol(G)) = Sol(trop(G)), answering a question raised by D. Grigoriev earlier this year.

On the Relationship Between Differential Algebra and Tropical Differential Algebraic Geometry

“…Last, the proof of the converse inclusion relies on an approximation theorem. The two versions of this approximation theorem given in [1,Proposition 7.3] and [7, Proposition 6.3] assume F to be uncountable. Our new version (Theorem 1) relies on weaker hypotheses.…”

“…Let us denote all vertex sets as T m = {vert | X ⊂ N m }. Then, the composition of taking the support and then its vertex set of the formal power series defines a non-degenerate valuation such that some ideas of [1] can be recovered.…”

“…Tropical differential algebraic geometry is a much more recent theory, founded by Grigoriev [8] aiming at applying the concepts of tropical algebra (aka min-plus algebra) to the study of formal power series solutions of systems of ODE. Tropical differential algebra obtained an important impulse by the proof of the fundamental theorem of tropical differential algebraic geometry [1] which was recently extended to the partial case in [7]. The common topic of both theories is the existence problem of formal power series solutions of polynomial differential equations on which an important paper [6] by Denef and Lipshitz was published in 1984.…”

“…In both [1] and [7], the fundamental theorem applies to a polynomial differential system Σ with coefficients in formal power series rings F [[x]] (ordinary case) or F [[x 1 , . .…”

Newton’s lemma for differential equations

“…The first successful extension has been proposed by Grigoriev (see [10], [3]). The aim of his approach is to describe the subsets that arise as sets of exponents of solutions in K[[t]] M .…”