Algorithmic aspects of arithmetical structures

Arithmetical structures on dominated polynomials

“…In other words, every u ∈ min D ≥0 (f ) is greater or equal than a vector presented by step (5). Therefore let u ∈ min D ≥0 (f ) and let d ≤ f be such vector given at step (5). Then, assume that there is no vector d…”

“…Note that f d (X) has positive constant term for almost every vector d ∈ Π, except for (1,5,4). That is, only the vector (1,5,4) has the chance to be an arithmetical structure of f . Indeed, since…”

“…Afterwards, by erasing every vector greater than said vector from the set, then the minimality of the set is assured. The proof of the correctness of Algorithm 2.4 will be similar to the one given for [5,Algorithm 3.2]. Thus we begin by extending [5, Lemma 3.1] for the polynomial case.…”

“…Recently arithmetical structures aroused some interest, see for instance [2,CV18b,4,1,6]. In [5] was discussed some algorithmic aspects of arithmetical structures on matrices. The main goal of this article is to generalizes the concept of arithmetical structure in the context of dominated polynomials and we get an algorithm to computes arithmetical structures on dominated polynomials.…”

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In [5] was given an algorithm that computes arithmetical structures on matrices. We use some of the ideas contained there to get an algorithm that computes arithmetical structures over dominated polynomials.

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