A p-Adic Approach to the Computation of Gröbner Bases

A method for the p-adic lifting of a Grobner basis is presented. If F is a finite vector of polynomials in @?[x"1,...,x"1] and p is a lucky prime for F (it turns out that there are only finitely many unlucky primes) then in a first step the normalized reduced Grobner basis G^(^0^) for F modulo p is computed, together with matrices Y^(^0^) and R^(^0^) such that Y^(^0^), G^(^0^)=F (mod p) and R^(^0^). G^(^0^)=0(mod p), where the rows of R^(^0^) are the syzygies of G^(^0^) derived from the reduction of the S-polynomials of G^(^0^) to 0. These congruences can be lifted to congruences modulo p^i, for any natural number i, finally leading to the normalized reduced Grobner basis for F in @?[x"1,...,x"1].

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