A tropical version of Hilbert polynomial (in dimension one)

TL;DR

This paper introduces a tropical Hilbert function for univariate polynomials, showing it eventually behaves as a linear plus periodic function, with the leading term linked to tropical entropy, and provides bounds on this entropy.

Contribution

It defines a tropical Hilbert function for univariate polynomials and characterizes its asymptotic behavior, linking it to tropical entropy and establishing bounds.

Findings

Tropical Hilbert function equals a linear plus periodic function for large degrees.

The leading coefficient of the linear part matches the tropical entropy.

Sharp bounds are established for the tropical entropy.

Abstract

For a tropical univariate polynomial $f$ we define its tropical Hilbert function as the dimension of a tropical linear prevariety of solutions of the tropical Macauley matrix of the polynomial up to a (growing) degree. We show that the tropical Hilbert function equals (for sufficiently large degrees) a sum of a linear function and a periodic function with an integer period. The leading coefficient of the linear function coincides with the tropical entropy of $f$. Also we establish sharp bounds on the tropical entropy.