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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
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