4 research outputs found
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
Normalization of Rings
We present a new algorithm to compute the integral closure of a reduced
Noetherian ring in its total ring of fractions. A modification, applicable in
positive characteristic, where actually all computations are over the original
ring, is also described. The new algorithm of this paper has been implemented
in Singular, for localizations of affine rings with respect to arbitrary
monomial orderings. Benchmark tests show that it is in general much faster than
any other implementation of normalization algorithms known to us.Comment: Final version, to be published in JSC 201