Convolution formulas for multivariate arithmetic Tutte polynomials

TL;DR

This paper derives convolution formulas for the multivariate arithmetic Tutte polynomial of arithmetic matroids, providing shorter proofs and new formulas for characteristic polynomials, enhancing combinatorial understanding.

Contribution

It introduces new convolution formulas for the multivariate arithmetic Tutte polynomial of arithmetic matroids and simplifies existing proofs.

Findings

Derived convolution formulas for the multivariate arithmetic Tutte polynomial.

Provided shorter, combinatorial proofs for known convolution formulas.

Established a new convolution formula for the characteristic polynomial of an arithmetic matroid.

Abstract

The multivariate arithmetic Tutte polynomial of arithmetic matroids is a generalization of the multivariate Tutte polynomial of matroids. In this note, we give the convolution formulas for the multivariate arithmetic Tutte polynomial of the product of two arithmetic matroids. In particular, the convolution formulas for the multivariate arithmetic Tutte polynomial of an arithmetic matroid are obtained. Applying our results, several known convolution formulas including [5, Theorem 10.9 and Corollary 10.10] and [1, Theorems 1 and 4] are proved by a purely combinatorial proof. The proofs presented here are significantly shorter than the previous ones. In addition, we obtain a convolution formula for the characteristic polynomial of an arithmetic matroid.