The Gröbner basis of the ideal of vanishing polynomials

Abstract

We construct an explicit minimal strong Gröbner 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 Gröbner basis is independent of the monomial order and that the set of leading terms of the constructed Gröbner 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 Gröbner basis in Z/m[x1, x2, . . . , xn] 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.

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“…But, in recent years, several works with different coefficient rings have accelerated. For example, when proving correctness of data paths in system on chip design, usage of Gröbner basis in polynomial rings over has led to emergence of many works in this direction (Greuel et al 2011), (Greuel et al 2008), (Shekhar et al 2005), (Wienand et al 2008…”

“…An explicit minimal strong Gröbner basis has been obtained for vanishing ideal ofR[x 1 ,x 2 ,...,x n ] where R Zm = (m ≥ 2) in(Greuel et al 2011). In this paper, we determine some vanishing polynomials in R[x 1 ,x 2 ,...,x n ] where…”

Reduced Gröbner bases and Macaulay–Buchberger Basis Theorem over Noetherian rings

“…However, these approaches only looked at extending basic definitions and concepts over rings, and validity of many important results were not explored (Greuel et al, 2011). Recently, there has been renewed interest in polynomial rings over rings (Greuel et al, 2011). For instance, certain residue class rings over Z[x] called ideal lattices (Micciancio, 2002) have shown to be isomorphic to integer lattices, an important cryptographic primitive (Ajtai, 1996) and certain cyclic lattices in Z [x] have been used in NTRU cryptographic schemes (Hoffstein et al, 1998).…”

“…For a good exposition on Gröbner bases over rings one can refer to (Adams & Loustaunau, 1994). However, these approaches only looked at extending basic definitions and concepts over rings, and validity of many important results were not explored (Greuel et al, 2011). Recently, there has been renewed interest in polynomial rings over rings (Greuel et al, 2011).…”

“…Boolean polynomial rings over a boolean ring have been used to solve Sudoku and other combinatorial puzzles (Sato et al, 2011). Further, polynomial rings over Z/2 k have been used to prove the correctness of data paths in system-on-chip design (Greuel et al, 2011). Also, the widely used degree truncated polynomial rings are actually the quotient rings of the form, Z[x 1 , .…”