Odd-Cycle-Free Facet Complexes and the K\"onig property

We use the definition of a simplicial cycle to define an odd-cycle-free facet complex (hypergraph). These are facet complexes that do not contain any cycles of odd length. We show that besides one class of such facet complexes, all of them satisfy the K\"onig property. This new family of complexes includes the family of balanced hypergraphs, which are known to satisfy the K\"onig property. These facet complexes are, however, not Mengerian; we give an example to demonstrate this fact.Comment: 12 pages, 11 figure

ON THE HILBERT QUASI - POLYNOMIALS FOR NON - STANDARD GRADED RINGS

Abstract The Hilbert function, its generating function and the Hilbert polynomial of a graded ring K[x1, . . . , xk ] have been extensively studied since the famous paper of Hilbert: Ueber die Theorie der algebraischen Formen ([Hilbert, 1890]). In particular the coefficients and the degree of the Hilbert polynomial play an important role in Algebraic Geometry. If the ring graduation is non-standard, then its Hilbert function is not definitely equal to a polynomial but to a quasi-polynomial. It turns out that a Hilbert quasi-polynomial P of degree n splits into a polynomial S of degree n and a lower degree quasi-polynomial T . We have completely determined the degree of T and the first few coefficients of P . Moreover, the quasi-polynomial T has a periodic structure that we have described. We have also developed a software to compute effectively the Hilbert quasi-polynomial for any ring K[x1, . . . , xk ]/I

On the Hilbert quasi-polynomials for non-standard graded rings

The Hilbert function, its generating function and the Hilbert polynomial of a graded ring K[x1, . . . , xk ] have been extensively studied since the famous paper of Hilbert: Ueber die Theorie der algebraischen Formen ([Hilbert, 1890]). In particular, the coefficients and the degree of the Hilbert polynomial play an important role in Algebraic Geometry. If the ring grading is non-standard, then its Hilbert function is not eventually equal to a polynomial but to a quasi-polynomial. It turns out that a Hilbert quasi- polynomial P of degree n splits into a polynomial S of degree n and a lower degree quasi-polynomial T . We have completely determined the degree of T and the first few coefficients of P . Moreover, the quasi-polynomial T has a periodic structure that we have described. We have also developed a software to compute effectively the Hilbert quasi-polynomial for any ring K[x1 , . . . , xk]/I

Reducing the size and number of linear programs in a dynamic Gr\"obner basis algorithm

The dynamic algorithm to compute a Gr\"obner basis is nearly twenty years old, yet it seems to have arrived stillborn; aside from two initial publications, there have been no published followups. One reason for this may be that, at first glance, the added overhead seems to outweigh the benefit; the algorithm must solve many linear programs with many linear constraints. This paper describes two methods of reducing the cost substantially, answering the problem effectively.Comment: 11 figures, of which half are algorithms; submitted to journal for refereeing, December 201

Optimization of basic algorithms in commutative algebra

Dottorato di ricerca in algebra. 8 cicloConsiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7 Rome; Biblioteca Nazionale Centrale - P.za Cavalleggeri, 1, Florence / CNR - Consiglio Nazionale delle RichercheSIGLEITItal

A Classification of compatible module ordering

The aim of this paper is to give a complete classification for the module orderings which are compatible with their ring ordering and that gives the same ordering on every module component (Riquier orderings). These orderings have simple classifications, based on the classification of ring orderings. The orderings encountered in practice in commutative computer algebra usually are Riquier orderings. (C) 1999 Elsevier Science B.V. All rights reserved

On the Hilbert quasi-polynomials for non-standard graded rings

The Hilbert function, its generating function and the Hilbert polynomial of a graded ring K[x1, ..., xk] have been extensively studied since the famous paper of Hilbert: Ueber die Theorie der algebraischen Formen ([Hilbert, 1890]). In particular, the coefficients and the degree of the Hilbert polynomial play an important role in Algebraic Geometry. If the ring grading is non-standard, then its Hilbert function is not eventually equal to a polynomial but to a quasi-polynomial. It turns out that a Hilbert quasi-polynomial P of degree n splits into a polynomial S of degree n and a lower degree quasi-polynomial T. We have completely determined the degree of T and the first few coefficients of P. Moreover, the quasi-polynomial T has a periodic structure that we have described. We have also developed a software to compute effectively the Hilbert quasi-polynomial for any ring K[x1, ..., xk]/I