Infinite matroids in tropical differential algebra
Cristhian Garay López (CIMAT (Guanajuato))
Abstract: Given a set $\Sigma\subset K[x_1,\ldots,x_n]$ of homogeneous linear polynomials, a classical result in tropical algebraic geometry states that the tropicalization (with respect to the trivial valuation) of the corresponding variety $V(\Sigma)\subset K^n$ is a fan $B(V(\Sigma))\subset(\mathbb{R}\cup\{-\infty\})^n$ that depends only on the matroid over the set of labels $E=[n]$ associated to the ideal $(\Sigma)$. Moreover, this set is tropically convex in the sense that it is closed under tropical linear combinations.
We discuss an analogue of this result in the context of tropical differential algebraic geometry, namely, if $\Sigma\subset K[\![t_1,\ldots,t_m]\!][x_{1,J},\ldots,x_{n,J}\::\:J\in\mathbb{N}^m]$ is certain type of set of homogeneous linear differential polynomials with coefficients in $K[\![t_1,\ldots,t_m]\!]$, then the tropicalization (with respect to the trivial valuation) of the set of formal solutions $Sol(\Sigma)\subset K[\![t_1,\ldots,t_m]\!]^n$ is a matroid $B(Sol(\Sigma))$ over the set of labels $E=\mathbb{N}^{mn}$, where $m,n$ are positive integers. Moreover, this set is tropically convex in the sense that it is closed under boolean linear combinations, i.e., it is a commutative and idempotent monoid.
algebraic geometrycombinatorics
Audience: researchers in the topic
(LAGARTOS) Latin American Real and Tropical Geometry Seminar
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Organizers: | Alicia Dickenstein*, Ethan Cotterill*, Cristhian Garay López* |
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