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Báo cáo toán học: "Proof of the combinatorial nullstellensatz over integral domains, in the spirit of Kouba"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Proof of the combinatorial nullstellensatz over integral domains, in the spirit of Kouba. | Proof of the combinatorial nullstellensatz over integral domains in the spirit of Kouba Peter Heinig Lehr- und Forschungseinheit M9 fur Angewandte Geometrie und Diskrete Mathematik Zentrum Mathematik Technische Universitat Munchen Boltzmannstrafie 3 D-85748 Garching bei Munchen Germany heinig@ma.tum.de Submitted Jan 4 2010 Accepted Feb 11 2010 Published Feb 22 2010 Mathematics Subject Classification 2010 13G05 15A06 Abstract It is shown that by eliminating duality theory of vector spaces from a recent proof of Kouba A duality based proof of the Combinatorial Nullstellensatz Electron. J. Combin. 16 2009 N9 one obtains a direct proof of the nonvanishing-version of Alon s Combinatorial Nullstellensatz for polynomials over an arbitrary integral domain. The proof relies on Cramer s rule and Vandermonde s determinant to explicitly describe a map used by Kouba in terms of cofactors of a certain matrix. That the Combinatorial Nullstellensatz is true over integral domains is a well-known fact which is already contained in Alon s work and emphasized in recent articles of Michalek and Schauz the sole purpose of the present note is to point out that not only is it not necessary to invoke duality of vector spaces but by not doing so one easily obtains a more general result. 1 Introduction The Combinatorial Nullstellensatz is a very useful theorem see 1 about multivariate polynomials over an integral domain which bears some resemblance to the classical Nullstellensatz of Hilbert. Theorem 1 Alon Combinatorial Nullstellensatz ideal-containment-version Theorem 1.1 in 1 . Let K be a field R c K a subring f E R x1 . xn S1 . Sn arbitrary nonempty subsets of K and g ỊỊ seS. Xi s for every 1 i n. If f s1 . sn 0 The author was supported by a scholarship from the Max Weber-Programm Bayern and by the ENB graduate program TopMath. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 N14 1 for every s1 . Sn G S1 X X Sn then there exist polynomials h. G R x1 . xn with the property that deg h. deg