ACM Communications in Computer Algebra, Vol. 42, No. 4, December 2008 Translation On a Combinatorial Theorem of Macaulay and its Applications to the Theory of Polynomial Ideals By Emanuel Sperner F. S. Macaulay has found a purely combinatorial theorem (see 2), with which he has been able to more simply derive the Hilbert characteristic function and some new results on polynomial ideals1 . However, his proof of this theorem is very complicated [2, p. 537: Note]. The aim of this paper is to produce a considerably simpler and shorter proof of this interesting theorem ( 2-3). For the sake of completeness and for the comfort of the reader, I also add the most important applications of this theorem ( 4-6), partly in new form, from which it will be shown that the Hilbert Basis Theorem is also a simple consequence of Macaulay s Theorem. We consider power products of degree l in n variables x1 , x2 , . . . , xn . Their quantity will be denoted by (l)n .2 It is well known that n+l 1 n+l 1 (l)n = = . l n 1 We set (0)n = 1, ( 1)n = 0.
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On a combinatorial theorem of Macaulay and its applications to the theory of polynomial ideals
Sperner, Emanuel
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, Volume 42 (4)
Association for Computing Machinery – Feb 6, 2009
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