The Groebner basis of the ideal of vanishing polynomials

We construct an explicit minimal strong Groebner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m>=2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Groebner basis is independent of the monomial order and that the set of leading terms of the constructed Groebner basis is unique, up to multiplication by units. We also present a fast algorithm to compute reduced normal forms, and furthermore, we give a recursive algorithm for building a Groebner basis in Z/m[x_1,x_2,...,x_n] along the prime factorization of m. The obtained results are not only of mathematical interest but have immediate applications in formal verification of data paths for microelectronic systems-on-chip.Comment: 15 pages, 1 table, 2 algorithms (corrected version with new Prop. 3.8 and proof); Journal of Symbolic Computation 46 (2011

On Border Basis and Groebner Basis Schemes

Hilbert schemes of zero-dimensional ideals in a polynomial ring can be covered with suitable affine open subschemes whose construction is achieved using border bases. Moreover, border bases have proved to be an excellent tool for describing zero-dimensional ideals when the coefficients are inexact. And in this situation they show a clear advantage with respect to Groebner bases which, nevertheless, can also be used in the study of Hilbert schemes, since they provide tools for constructing suitable stratifications. In this paper we compare Groebner basis schemes with border basis schemes. It is shown that Groebner basis schemes and their associated universal families can be viewed as weighted projective schemes. A first consequence of our approach is the proof that all the ideals which define a Groebner basis scheme and are obtained using Buchberger's Algorithm, are equal. Another result is that if the origin (i.e. the point corresponding to the unique monomial ideal) in the Groebner basis scheme is smooth, then the scheme itself is isomorphic to an affine space. This fact represents a remarkable difference between border basis and Groebner basis schemes. Since it is natural to look for situations where a Groebner basis scheme and the corresponding border basis scheme are equal, we address the issue, provide an answer, and exhibit some consequences. Open problems are discussed at the end of the paper.Comment: Some typos fixed, some small corrections done. The final version of the paper will be published on "Collectanea Mathematica

The Symmetric Algebra for Certain Monomial Curves

In this article we compute a minimal Groebner basis for the symmetric algebra for certain affine Monomial Curves, as an R-module. Keywords: Monomial Curves, Groebner Basis, Symmetric Algebra. Mathematics Subject Classification 2000: 13P10, 13A30