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Borel-fixed ideals and reduction number

Le Tuan Hoa·Ngo Viet Trung
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Abstract

The aim of this paper is to study the relationship between reduction numbers and Borel-fixed ideals in all characteristics. By definition, Borel-fixed ideals are closed under certain specializations which is similar to the strong stability. We will estimate the number of monomials which can be specialized to a given monomial. As a consequence, we obtain a combinatorial version of the well-known Eakin-Sathaye's theorem which bounds the reduction number in terms of the Hilbert function. Furthermore, we show that the bound of Eakin-Sathaye's theorem is attained by the reduction number of a lex-segment monomial ideal. This result answers a question of Conca in the affirmative. We will also show that the reduction number of the lex-segment ideal is bounded exponentially by the reduction number of the given ideal.

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Le Tuan Hoa, Ngo Viet Trung. 2003-11-18. Borel-fixed ideals and reduction number. https://arxiv.org/abs/math/0311300

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