Signature-based Criteria for Möller's Algorithm for Computing Gröbner Bases over Principal Ideal Domains
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Signature-based algorithms have become a standard approach for Gröbner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this paper, we present a signature-based algorithm for computing Gröbner bases over principal ideal domains (e.g. the ring of integers or the ring of univariate polynomials over a field). It is adapted from Möller's algorithm (1988) which considers reductions by multiple polynomials at each step. This ensures that, in our signature-based adaptation, signature drops are not a problem, and it allows us to implement classic signature-based criteria to eliminate some redundant reductions. A toy implementation in Magma confirms that the signature-based algorithm is more efficient in terms of the number of S-polynomials computed. Early experimental results suggest that the algorithm might even work for polynomials over more general rings, such as unique factorization domains (e.g. the ring of multivariate polynomials over a field or a PID).
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