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The Hilbert series of algebras of the Veronese type. (English) Zbl 1107.13003

Fixed a positive integer \(d\) and an integer vector \( a=(a_1,a_2, \ldots, a_n)\), \((1 \leq a_1 \leq a_2 \leq \ldots \leq a_n\) and \( \sum_{i=1}^n a_i > d)\), the Veronese type algebra \(V(a;d)\) is the \(k\)-subalgebra of the polynomial ring \(k[x_1,x_2, \ldots, x_n]\) spanned by the monomials \(\prod_{i=1}^n x_i^{b_i}\) such that \(\sum_{i=1}^n b_i = d_i\) and \(b_i \leq a_i\) for all index \(i\).
In this paper, the author provides an explicit expression for the \(h\)-vector of the algebra \(V(a;d)\) which allows to obtain the multiplicity of \(V(a;d)\) and an upper bound for its \(a\)-invariant.

MSC:

13A02 Graded rings
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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References:

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