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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