Arithmetical structures on dominated polynomials

Arithmetical structures on matrices were introduced in Corrales H, Valencia CE (Arithmetical structures on graphs. Linear Algebra Appl, 536:120–151, 2018), which are finite whenever the matrix is irreducible. We generalize the algorithm that computes arithmetical structures on matrices given in Valencia CE, Villagrán RR (Algorithmic aspects of arithmetical structures. Linear Algeb Appl, 640:191–208, 2022), to an algorithm that computes arithmetical structures on dominated polynomials. A dominated polynomial is an integer multivariate polynomial, such that it contains a monomial, which is divided by all of its monomials. We give an example of a dominated polynomial which is not the determinant of an integer matrix and show how the algorithm works on it.