## Strong Gröbner bases for polynomials over a principal ideal ring

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.

### History

#### School

- Science

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- Computer Science