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Regularity of lex-segment ideals: Some closed formulas and applications. (English) Zbl 1036.13014
The authors provide a closed formula for the (Castelnuovo-Mumford) regularity of a lex-segment ideal which depends only on its Hilbert function. Hence it bounds the regularity of any ideal with the same Hilbert function. As a consequence they are able to characterize those polynomials which are Hilbert polynomials of some projective scheme. Moreover, they give an explicit bound for the regularity of a projective scheme in terms of its Hilbert polynomial. Further applications include a (necessarily huge) bound for the maximal degree of an element of a reduced Gröbner basis of a polynomial ideal in terms of the degrees of the generating polynomials.

MSC:
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13D45 Local cohomology and commutative rings
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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