Signature-based Criteria for Möller's Algorithm for Computing Gröbner Bases over Principal Ideal Domains

A dynamical solution of Kronecker's problem

In this paper, I present a new decision procedure for the ideal membership problem for polynomial rings over principal domains using discrete valuation domains. As a particular case, I solve a fundamental algorithmic question in the theory of multivariate polynomials over the integers called “Kronecker’s problem”, that is the problem of finding a decision procedure for the ideal membership prob...

Signature-Based Gröbner Basis Algorithms - Extended MMM Algorithm for computing Gröbner bases

Signature-based algorithms is a popular kind of algorithms for computing Gröbner bases, and many related papers have been published recently. In this paper, no new signature-based algorithms and no new proofs are presented. Instead, a view of signature-based algorithms is given, that is, signature-based algorithms can be regarded as an extended version of the famous MMM algorithm. By this view,...

Revisiting strong Gröbner bases over Euclidean domains

Buchberger and Kandri−Rody and Kapur defined a strong Gröbner basis for a polynomial ideal over a Euclidean domain in a way that gives rise to canonical reductions. This retains what is perhaps the most important property of Gröbner bases over fields. A difficulty is that these can be substantially harder to compute than their field counterparts. We extend their results for computing these base...

A Monomial-Oriented GVW for Computing Gröbner Bases

The GVW algorithm, presented by Gao et al., is a signature-based algorithm for computing Gröbner bases. In this paper, a variant of GVW is presented. This new algorithm is called a monomial-oriented GVW algorithm or mo-GVW algorithm for short. The moGVW algorithm presents a new frame of GVW and regards labeled monomials instead of labeled polynomials as basic elements of the algorithm. Being di...

SAGBI and SAGBI-Gröbner Bases over Principal Ideal Domains