Tropical differential equations

The Fundamental Theorem of Tropical Partial Differential Algebraic Geometry

Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of Tropical Differential Algebraic Geometry states that the support of solutions of systems of ordinary differential equations with formal power series coefficients over an uncountable algebraically closed field of characteristic zero can be obtained by solving a so-called tropicalized differential system. Tropicalized differential equations work on a completely d…

Exploring tropical differential equations

The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations, both in the archimedean (complex analytic) and in the non-archimedean (e.g., $p$-adic) settings. A third and subsidiary aim is to show how tropical differential algebraic geometry is a natural application of semiring theory…

Tropical Differential Groebner Basis

In this paper, the tropical differential Gröbner basis is studied, which is a natural generalization of the tropical Gröbner basis to the recently introduced tropical differential algebra. Like the differential Gröbner basis, the tropical differential Gröbner basis generally contains an infinite number of elements. We give a Buchberger style criterion for the tropical differential Gröbner basis. For ideals generated by homogeneous linear differential polynomials with constant coefficients, we give a complete algorithm to compute the tropical differential Gröbner basis.

Detecting tropical defects of polynomial equations

We introduce the notion of tropical defects, certificates that a system of polynomial equations is not a tropical basis, and provide two algorithms for finding them in affine spaces of complementary dimension to the zero set. We use these techniques to solve open problems regarding del Pezzo surfaces of degree 3 and realizability of valuated gaussoids on 4 elements.

热带微分代数中的无限拟阵

Infinite matroids in tropical differential algebra

本文研究了有限维向量空间的支撑集与拟阵的关系，证明了最小支撑集构成余有限拟阵，大基数域下所有支撑集构成拟阵，并将结果应用于热带微分代数几何，给出解支撑集为拟阵的条件及反例。

The Newton Polytope of the Implicit Equation

We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.