Abstract
Studying Hilbert functions of concrete examples of normal toric rings, it is demonstrated that for each 1 ≤ s ≤ 5 , an O-sequence ( h 0 , h 1 , … , h 2 s − 1 ) ∈ Z ≥ 0 2 s satisfying the properties that (i) h 0 ≤ h 1 ≤ ⋯ ≤ h s − 1 , (ii) h 2 s − 1 = h 0 , h 2 s − 2 = h 1 and (iii) h 2 s − 1 − i = h i + ( − 1 ) i , 2 ≤ i ≤ s − 1 , can be the h-vector of a Cohen-Macaulay standard G-domain.
Highlights
In the paper [1] published in 1989, several conjectures on Hilbert functions of Cohen-Macaulay integral domains are studied
A classical result ([3], Chapter 5, Section 13) says that H ( A, n) is a polynomial for n sufficiently large and its degree is d − 1. It follows that the sequence h( A) = (h0, h1, h2, . . .), called the h-vector of A, defined by the formula
The toric ring of C2s+1 is the subring K [C2s+1 ] of S which is generated by those squarefree monomials (∏i∈W xi )y for which W ⊂ [2s + 1] is stable in C2s+1
Summary
In the paper [1] published in 1989, several conjectures on Hilbert functions of Cohen-Macaulay integral domains are studied. A classical result ([3], Chapter 5, Section 13) says that H ( A, n) is a polynomial for n sufficiently large and its degree is d − 1 It follows that the sequence h( A) = Hs ) of non-negative integers is said to be an O-sequence if there exists an order ideal M of monomials in Y, . Mathematics 2020, 8, 22 the h-vector of a Cohen-Macaulay standard G-algebra if and only if A standard G-domain is a standard G-algebra which is an integral domain It was conjectured ([1], Conjecture 1.4) that the h-vector of a Cohen-Macaulay standard G-domain is flawless. In general, the h-vector of a Cohen-Macaulay standard. O-sequence, which is non-flawless for each of s = 4 and s = 5, can be the h-vector of a Cohen-Macaulay standard G-domain
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