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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
A Solution to Kronecker's Problem
, 1994
"... : Consider Z[x 1 , : : :, x n ], the multivariate polynomial ring over integers involving n variables. For a fixed n, we show that the ideal membership problem as well as the associated representation problem for Z[x 1 , : : :, x n ] are primitive recursive. The precise complexity bounds are easil ..."
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: Consider Z[x 1 , : : :, x n ], the multivariate polynomial ring over integers involving n variables. For a fixed n, we show that the ideal membership problem as well as the associated representation problem for Z[x 1 , : : :, x n ] are primitive recursive. The precise complexity bounds are easily expressible by functions in the Wainer hierarchy. Thus, we solve a fundamental algorithmic question in the theory of multivariate polynomials over the integers. As a direct consequence, we also obtain a solution to certain foundational problems intrinsic to Kronecker's programme for constructive mathematics and provide an effective version of Hilbert's basis theorem. Our original interest in this area was aroused by Edwards' historical account of the Kronecker's problem in the context of Kronecker's version of constructive mathematics. Key Words: Ascending chain condition, Ebases, Grobner bases, Ideal membership problem, Rapidly growing functions, Ring of polynomials over the integers, ...
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.
Computational ideal theory in finitely generated extension rings
, 1998
"... One of the most general extensions of Buchberger's theory of Grobner bases is the concept of graded structures due to Robbiano and Mora. But in order to obtain algorithmic solutions for the computation of Grobner bases it needs additional computability assumptions. In this paper we introduce natural ..."
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One of the most general extensions of Buchberger's theory of Grobner bases is the concept of graded structures due to Robbiano and Mora. But in order to obtain algorithmic solutions for the computation of Grobner bases it needs additional computability assumptions. In this paper we introduce natural graded structures of nitely 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 Grobner 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 rst time in the context of algorithmic Grobner basis computations. Finally, we discuss which conditions could be changed in order to nd further e ective Grobner structures and it will turn out that the most interesting constructive instances of graded structures are covered by our results. Key words: ideal membership problem, e ective graded structure, Grobner basis, Buchberger's algorithm 1