Term-ordering free involutive bases

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1 Dipartimento di Matematica Torino 2 Dipartimento di Matematica Genova

Abstract : In this paper, we consider a monomial ideal $J \triangleleft P:=Ax 1,\dots ,x n$, over a commutative ring $A$, and we face the problem of the characterization for the family $\MfJ$ of all homogeneous ideals $I \triangleleft P$ such that the $A$-module $P-I$ is free with basis given by the set of terms in the \Gr\ escalier $\cNJ$ of $J$. This family is in general wider than that of the ideals having $J$ as initial ideal w.r.t. any term-ordering, hence more suited to a computational approach to the study of Hilbert schemes.\\ For this purpose, we exploit and enhance the concepts of multiplicative variables, complete sets and involutive bases introduced by \cite{Riq1,Riq2, Riq3} and in \cite{J1,J2,J3} and we generalize the construction of $J$-marked bases and term-ordering free reduction process introduced and deeply studied in \cite{BCLR,CR} for the special case of a strongly stable monomial ideal $J$.\\ Here, we introduce and characterize for every monomial ideal $J$ a particular complete set of generators $\mathcal FJ$, called stably complete, that allows an explicit description of the family $\MfJ$. We obtain stronger results if $J$ is quasi stable, proving that $\mathcal FJ$ is a Pommaret basis and $\MfJ$ has a natural structure of affine scheme. The final section presents a detailed analysis of the origin and the historical evolution of the main notions we refer to.

Autor: Michela Ceria - Teo Mora - Margherita Roggero -

Fuente: https://hal.archives-ouvertes.fr/

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