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Hilbert series of a quotient algebra of polynomials. (Série de Hilbert d’une algèbre de polynômes quotient.) (French) Zbl 0836.13011
Let \(R = k[x_1, \dots, x_r]/(F_1, \dots, F_n)\), where \(F_i\) are homogeneous of degree \(d_i = \deg F_i\) (and \(\deg x_i = 1\) for all \(i)\). Then \(R\) is graded, \(R = \bigoplus R_i\), and the Hilbert series of \(R\) is \(\text{Hilb} R(t) = \sum \dim_k R_i t^i\). It has been conjectured that \(\text{Hilb} R(t) \geq |(1 - t)^r \prod^n_{k = 1} (1 - t^{d_k}) |\), where \(|\sum a_i t^i |= \sum c_i t^i\) with \(c_i = a_i\) if \(a_j \geq 0\) for \(j \leq i\), and \(c_i = 0\) otherwise, and where \(\sum a_i t^i \geq \sum b_i t^i\) if \(a_i \geq b_i\) for all \(i\). The conjecture also predicts equality if the \(F_i\)’s are generic. The conjecture is proved for \(r \leq 3\) and for \(n \leq r + 1\). It has also been proved [by M. Hochster and D. Laksov, Commun. Algebra 15, 227-239 (1987; Zbl 0619.13007)] that, for any \(r\) and \(n\), the inequality holds for the first nontrivial degree, i.e. in degree \(1 + \min \{d_k\}\). The author extends the result of Hochster and Laksov to a wide range of degrees.

MSC:
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13A02 Graded rings
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