# Combinatorics and Arithmetic for Physics: special days

30 November 2021 to 2 December 2021
Le Bois-Marie
Europe/Paris timezone

## A Tropical Version of Hilbert Polynomial (remote)

2 Dec 2021, 14:00
50m
Centre de conférences Marilyn et James Simons (Le Bois-Marie)

### Centre de conférences Marilyn et James Simons

#### Le Bois-Marie

35, route de Chartres 91440 Bures-sur-Yvette

### Speaker

Dimitri Grigoryev (CNRS Painlevé Lab, Univ. Lille)

### Description

We define Hilbert function of a semiring ideal of tropical polynomials in n variables. For $n = 1$ we prove that it is the sum of a linear function and a periodic function (for sufficiently large values). The leading coefficient of the linear function equals the tropical entropy of the ideal. For an arbitrary n we discuss a conjecture that the tropical Hilbert function of a radical ideal is a polynomial of degree at most $n − 1$ (for sufficiently large values). For $n = 1$ the conjecture is true, also we have proved it for zero- dimensional ideals and for planar tropical curves.

### Presentation Materials

 cap2021-Grigoryev.pdf Vidéo