The Grobner basis of the ideal of vanishing polynomials

摘要

We construct an explicit minimal strong Crfibner 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 GrSbner basis is independent of the monomial order and that the set of leading terms of the constructed GrObner 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.