Entropy of radical ideal of a tropical prevariety

TL;DR

This paper introduces the concept of entropy for tropical ideals, demonstrating that the entropy of radicals of certain tropical prevarieties is zero, and providing examples of non-radical ideals with positive entropy.

Contribution

It defines entropy for tropical ideals and establishes key properties, including entropy vanishing for radicals of specific tropical prevarieties and presenting a counterexample.

Findings

Entropy of the radical of a tropical bivariate polynomial with vanishing coefficients is zero.

Entropy of a zero-dimensional tropical prevariety is zero.

A non-radical tropical ideal can have positive entropy.

Abstract

The entropy of a tropical ideal is introduced. The radical of a tropical ideal consists of all tropical polynomials vanishing on the tropical prevariety determined by the ideal. We prove that the entropy of the radical of a tropical bivariate polynomial with vanishing coefficients equals zero. Also we prove that the entropy of a zero-dimensional tropical prevariety vanishes. An example of a non-radical tropical ideal having a positive entropy is exhibited.