Border bases for lattice ideals

The main ingredient to construct an O-border basis of an ideal I $\subseteq$ K[x1,. .., xn] is the order ideal O, which is a basis of the K-vector space K[x1,. .., xn]/I. In this paper we give a procedure to find all the possible order ideals associated with a lattice ideal IM (where M is a lattice of Z n). The construction can be applied to ideals of any dimension (not only zero-dimensional) and shows that the possible order ideals are always in a finite number. For lattice ideals of positive dimension we also show that, although a border basis is infinite, it can be defined in finite terms. Furthermore we give an example which proves that not all border bases of a lattice ideal come from Gr\"obner bases. Finally, we give a complete and explicit description of all the border bases for ideals IM in case M is a 2-dimensional lattice contained in Z 2 .Comment: 25 pages, 3 figures. Comments welcome!, MEGA'2015 (Special Issue), Jun 2015, Trento, Ital

Scienza e fede: dio e l’infinito – Considerazioni di un matematico

Elementary examples show that mathematical infinity is actually present in the applications of mathematics to reality. This leads to remarks on Cantor’s defini- tive introduction of actual infinity in mathematics, through his theory of sets, which is part of the current formalization of mathematics. However, mathematics goes beyond its formalization in a somewhat “mysterious” way. The equally “mysterious” effectiveness of mathematics in the universe ultimately suggests a dynamic understanding of creation, an open-ended process where infinite and finite are intertwinedEsempi elementari mostrano che l’infinito matematico è attualmente presente nelle applicazioni della matematica alla realtà. Ciò porta a osservazioni sull’introduzione definitiva a opera di Cantor dell’infinito attuale in matematica, attraverso la sua teoria degli insiemi, che fa parte della corrente formalizzazione della matematica. Tuttavia la matematica va oltre la sua formalizzazione in un modo un po’ “misterioso”. L’efficacia altrettanto “misteriosa” della matematica nell’universo suggerisce in ultima analisi una comprensione dinamica della creazione, un processo aperto in cui infinito e finito sono intrecciat

Gröbner Bases Related to 3-Dimensional Transportation Problems

This paper illustrates some work in progress on 3-dimensional transportation problems,of format rst say. Following Conti and Traverso, a suitable Grobner basis is sought for, which is hard to be calculated by means of Buchberger algorithm. A different approach involving graph theory makes the calculation tractable when r = s = t = 3 (and in fact whenever 3 \in {r, s, t})

Computing Gröbner bases of pure binomial ideals via submodules of Zn

2A binomial ideal is an ideal of the polynomial ring which is gener- ated by binomials. In a previous paper, we gave a correspondence between pure saturated binomial ideals of K [x1 , . . . , xn ] and sub- modules of Zn and we showed that it is possible to construct a the- ory of Gröbner bases for submodules of Zn . As a consequence, it is possible to follow alternative strategies for the computation of Gröbner bases of submodules of Zn (and hence of binomial ideals) which avoid the use of Buchberger algorithm. In the present pa- per, we show that a Gröbner basis of a Z-module M ⊆ Zn of rank m lies into a finite set of cones of Zm which cover a half-space of Zm . More precisely, in each of these cones C , we can find a suitable subset Y (C ) which has the structure of a finite abelian group and such that a Gröbner basis of the module M (and hence of the pure saturated binomial ideal represented by M) is described using the elements of the groups Y (C ) together with the generators of the cones.nonemixedBoffi G.; Logar A.Boffi, G.; Logar, Alessandr

A Littlewood-Richardson filtration at roots of 1 for quantum deformations of skew Schur modules

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