To obtain coprime polynomial pairs, it is often used an approach that consists of taking two random polynomials of the required degrees and testing whether they are relatively prime. If they are not, they are discarded and another pair is tried. The test can obviously be carried out very efficiently. Although in many cases the probability of success for the test is close to 1, this test approac...

Inspired by Faugère and Mou’s sparse FGLM algorithm, we show how using linear recurrent multi-dimensional sequences can allow one to perform operations such as the primary decomposition of an ideal, by computing the annihilator of one or several such sequences.

Functional generalized linear models (FGLM) have been widely used to study the relations between non-Gaussian response and functional covariates. However, most existing works for FGLM assume independence among observations therefore they are of limited applicability correlated data. A particularly important example is spatial data, where we observe functions over domains, such as age population...

In this paper, a modied version of LLL algorithm, which is a an algorithm with output-sensitivecomplexity, is presented to convert a given Grobner basis with respect to a specic order of a polynomialideal I in arbitrary dimensions to a Grobner basis of I with respect to another term order.Also a comparison with the FGLM conversion and Buchberger method is considered.

Let K be a field equipped with valuation. Tropical varieties over can defined theory of Gröbner bases taking into account the valuation K. Because use valuation, tropical has proved to provide settings for computations polynomial rings p-adic that are more stable than classical bases. Beforehand, these strategies were only available homogeneous polynomials. In this article, we extend F5 strateg...

We present two algorithms for the arithmetic of cubic curves with a totally ramified prime at infinity. The first algorithm, inspired by Cantor’s reduction for hyperelliptic curves, is easily implemented with a few lines of code, making use of a polynomial arithmetic package. We prove explicit reducedness criteria for superelliptic curves of genus 3 and 4, which show the correctness of the algo...

Association schemes are combinatorial objects that allow us solve problems in several branches of mathematics. They have been used in the study of permutation groups and graphs and also in the design of experiments, coding theory, partition designs etc. In this paper we show some techniques for computing properties of association schemes. The main framework arises from the fact that we can char...

Consider a zero-dimensional ideal I in K[X1, . . . ,Xn]. Inspired by Faugère and Mou’s Sparse FGLM algorithm, we use Krylov sequences based on multiplication matrices of I in order to compute a description of its zero set by means of univariate polynomials. Steel recently showed how to use Coppersmith’s block-Wiedemann algorithm in this context; he describes an algorithm that can be easily para...