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Automating elementary numbertheoretic proofs using Gröbner bases
"... Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates ..."
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Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates such as ‘divisible by’, ‘congruent ’ and ‘coprime’; one notable example in this class is the Chinese Remainder Theorem (for a specific number of moduli). The method is based on a reduction to ideal membership assertions that are then solved using Gröbner bases. As well as illustrating the usefulness of the procedure on examples, and considering some extensions, we prove a limited form of completeness for properties that hold in all rings. 1
Computational ideal theory in finitely generated extension rings
, 1998
"... One of the most general extensions of Buchberger's theory of Gröbner bases is the concept of graded structures due to Robbiano and Mora. But in order to obtain algorithmic solutions for the computation of Gröbner bases it needs additional computability assumptions. In this paper we introduce na ..."
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One of the most general extensions of Buchberger's theory of Gröbner bases is the concept of graded structures due to Robbiano and Mora. But in order to obtain algorithmic solutions for the computation of Gröbner bases it needs additional computability assumptions. In this paper we introduce natural graded structures of finitely generated extension rings and present subclasses of such structures which allow uniform algorithmic solutions of the basic problems in the associated graded ring and, hence, of the computation of Gröbner bases with respect to the graded structure. Among the considered rings there are many of the known generalizations. But, in addition, a wide class of rings appears first time in the context of algorithmic Gröbner basis computations. Finally, we discuss which conditions could be changed in order to find further effective Gröbner structures and it will turn out that the most interesting constructive instances of graded structures are covered by our results.
Effective Gröbner structures
, 1997
"... Since Buchberger introduced the theory of Gröbner bases in 1965 it has become one of the most important tools in constructive algebra and, nowadays, it is the kernel of many algorithms in the theory of polynomial ideals and algebraic geometry. Motivated by the results in polynomial rings there have ..."
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Since Buchberger introduced the theory of Gröbner bases in 1965 it has become one of the most important tools in constructive algebra and, nowadays, it is the kernel of many algorithms in the theory of polynomial ideals and algebraic geometry. Motivated by the results in polynomial rings there have been investigated a lot of possibilities to generalise Buchberger's ideas to other types of rings. The perhaps most general concept, though it does not cover all extensions reported in the literature, is the extension to graded structures due to Robbiano and Mora. But in order to obtain algorithmic solutions for the computation of Gröbner bases it needs additional computability assumptions. The subject of this paper is the presentation of some classes of effective graded structures.
A Tutte
"... We dene a polynomial W on graphs with colours on the edges, by generalizing the spanning tree expansion of the Tutte polynomial as far as possible: we give necessary and sucient conditions on the edge weights for this expansion not to depend on the order used. We give a contractiondeletion formula ..."
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We dene a polynomial W on graphs with colours on the edges, by generalizing the spanning tree expansion of the Tutte polynomial as far as possible: we give necessary and sucient conditions on the edge weights for this expansion not to depend on the order used. We give a contractiondeletion formula for W analogous to that for the Tutte polynomial, and show that any coloured graph invariant satisfying such a formula can be obtained from W. In particular, we show that generalizations of the Tutte polynomial obtained from its rank generating function formulation, or from a random cluster model, can be obtained from W. Finally, we nd the most general conditions under which W gives rise to a link invariant, and give as examples the onevariable Jones polynomial, and an invariant taking values in Z=22Z. 1.