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The range of all regularities for polynomial ideals with a given Hilbert function. (English) Zbl 07268500
Let $$A$$ be a polynomial ring over a field $$K$$ and let $$I$$ denote any homogeneous ideal of $$A$$. This paper presents a proof of the following statement suggested by Le Tuan Hoa in a private communication,
Theorem 1: Given an Hilbert function $$u$$, the set $$\{reg(I) : A/I \text{ has Hilbert function } u \}$$ is an interval of integers.
Since for every homogenous ideal $$I$$, there exists a strongly stable ideal $$J$$ such that $$reg(I)=reg(J)$$ and $$reg(J)$$ is the highest degree, denoted by $$\nu(J)$$, of a minimal generator of $$J$$, the theorem 1 is equivalent to the following one,
Theorem 2: Given an Hilbert function $$u$$, the set $$\{\nu(J) : J \text{ is strongly stable and }A/J \text{ has Hilbert function } u\}$$ is an interval of integers.
The proof is based on constructive arguments presented in the paper [F. Cioffi et al., Exp. Math. 24, No. 4, 424–437 (2015; Zbl 1333.13028)], mainly on a construction called expanded lifting, that allows, given a saturated homogeneous ideal $$I$$ with Hilbert function $$\preceq u$$, to create a homogenous ideal $$J$$ with Hilbert function $$u$$.
##### MSC:
 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 14Q99 Computational aspects in algebraic geometry 68W30 Symbolic computation and algebraic computation 11Y55 Calculation of integer sequences 13C99 Theory of modules and ideals in commutative rings
CoCoA
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