## Une majoration de la fonction de Hilbert et ses conséquences pour l’interpolation algébrique. (Majorization of the Hilbert function and its consequences for algebraic interpolations).(French)Zbl 0709.13007

Let k be a field and I a homogeneous ideal in $$k[X_ 0,...,X_ n]$$. The main result in this paper is the following: If I is equi-dimensional and geometrically reduced (i.e. $$k[X_ 0,...,X_ n]/I$$ a separable k- algebra), then the following inequality for the Hilbert function $$H_ I(n)=\dim_ k(k[X_ 0,...,X_ n]/I)_ n$$ is valid: $$H_ I(n)\leq \left( \begin{matrix} n+D\\ D\end{matrix} \right)\deg (I)$$, where D is the dimension of I.
The equidimensionality is not essential to get a bound. Some consequences about the existence of regular sequences in ideals are derived. An example: If P is a geometrically reduced homogeneous ideal of codimension 2 in $$k[X_ 0,...,X_ n]$$, then there exist two polynomials $$p_ 1$$ and $$p_ 2$$ in P, without common factor, with $$\deg (p_ 1)\cdot \deg (p_ 2)\leq n(n-1)\deg (P)$$.
Reviewer: R.Fröberg

### MSC:

 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 14A05 Relevant commutative algebra

### Keywords:

homogeneous ideal; Hilbert function; regular sequences
Full Text:

### References:

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