A general framework for Noetherian well ordered polynomial reductions

Polynomial reduction is one of the main tools in computational algebra with innumerable applications in many areas, both pure and applied. Since many years both the theory and an efficient design of the related algorithm have been solidly established. This paper presents a general definition of polynomial reduction structure, studies its features and highlights the aspects needed in order to grant and to efficiently test the main properties (noetherianity, confluence, ideal membership). The most significant aspect of this analysis is a negative reappraisal of the role of the notion of term order which is usually considered a central and crucial tool in the theory. In fact, as it was already established in the computer science context in relation with termination of algorithms, most of the properties can be obtained simply considering a well-founded ordering, while the classical requirement that it be preserved by multiplication is irrelevant. The last part of the paper shows how the polynomial basis concepts present in literature are interpreted in our language and their properties are consequences of the general results established in the first part of the paper.Comment: 36 pages. New title and substantial improvements to the presentation according to the comments of the reviewer

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Artículo de la sección Tribuna: investigación y profesiónTribune Article section: Research and professio

Bar code for monomial ideals

The aim of this paper is to count 0-dimensional stable and strongly stable ideals in 2 and 3 variables, given their (constant) affine Hilbert polynomial p, by means of a bijection between these ideals and some integer partitions of p, which can be counted via determinantal formulas. This will be achieved by the Bar Code, a bidimensional diagram that allows to represent any finite set of terms M and desume many properties of the corresponding monomial ideal I, if M is an order ideal

A computational approach to the theory of adjoints

International audienceThe word "adjoint" refers to several definitions which are not all equivalent: we will deal with any of them. The aim of this work is to provide an algorithm which, given two plane curves D, H allows to decide whether H is adjoint to D. With a slight modification to the main procedure, we will be able to deal with special adjoints and true adjoints.Le mot "adjoint" fait référence à plusieurs définitions qui ne sont pas toutes équivalentes: nous en traiterons toutes. Le but de ce travail est de fournir un algorithme qui, étant donné les deux courbes planes D, H permet de décider si H est adjoint à D. Avec une légère modification de la procédure principale, nous pourrons traiter des ajoints spéciaux et les vrais adjoints