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Strong Gröbner bases for polynomials over a principal ideal ring

2001; Cambridge University Press; Volume: 64; Issue: 3 Linguagem: Inglês

10.1017/s0004972700019973

1755-1633

Graham H. Norton, Ana Sălăgean,

Coding theory and cryptography

Gröbner bases have been generalised to polynomials over a commutative ring A in several ways. Here we focus on strong Gröbner bases, also known as D-bases. Several authors have shown that strong Gröbner bases can be effectively constructed over a principal ideal domain. We show that this extends to any principal ideal ring. We characterise Gröbner bases and strong Gröbner bases when A is a principal ideal ring. We also give algorithms for computing Gröbner bases and strong Gröbner bases which generalise known algorithms to principal ideal rings. In particular, we give an algorithm for computing a strong Gröbner basis over a finite-chain ring, for example a Galois ring.