A tropical version of Hilbert polynomial (in dimension one)

Nikita Elizarov; Dima Grigoriev

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arXiv · 2111.14742

A tropical version of Hilbert polynomial (in dimension one)

Nikita Elizarov·Dima Grigoriev

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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.

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Nikita Elizarov, Dima Grigoriev. 2021-11-29. A tropical version of Hilbert polynomial (in dimension one). https://arxiv.org/abs/2111.14742

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