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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
##### Keywords:
homogeneous ideal; Hilbert function; regular sequences
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##### References:
  BOURBAKI (N.) . - Algèbre Ch. 4 à 7 . - Masson, 1981 . Zbl 0498.12001 · Zbl 0498.12001  BOURBAKI (N.) . - Algèbre Commutative Ch. 1 à 9 (en trois volumes). - Masson, 1983 et 1985 . · Zbl 0579.13001  DEMAZURE (M.) . - Fonctions d’Hilbert Samuel , d’après Macaulay, Stanley et Bayer, Notes informelles de calcul formel, prépublications du Centre de Mathématiques de l’École Polytechnique, 1984 .  FULTON (W.) . - Intersection Theory . - Springer-Verlag, 1984 . MR 85k:14004 | Zbl 0541.14005 · Zbl 0541.14005  HARTSHORNE (R.) . - Algebraic Geometry . - Springer-Verlag, 1977 . MR 57 #3116 | Zbl 0367.14001 · Zbl 0367.14001  NESTERENKO (Y.) . - Estimates for the characteristic function of a prime ideal, Translation of the American Mathematical Society , 1985 , from the Math. USSR Sb., 51. Zbl 0579.10030 · Zbl 0579.10030  NORTHCOTT (D.G.) . - Lessons on Rings Modules and Multiplicities . Cambridge University Press, 1968 . MR 38 #144 | Zbl 0159.33001 · Zbl 0159.33001  MUMFORD (D.) . - Algebraic geometry I . Complex Projective Varieties. - Springer-Verlag, 1976 . MR 56 #11992 | Zbl 0356.14002 · Zbl 0356.14002  SERRE (J.-P.) . - Algèbre locale et multiplicités , Springer-Verlag Lecture Notes in Mathematics, no 11, 1965 . MR 34 #1352 | Zbl 0142.28603 · Zbl 0142.28603  TEICHMÜLLER (O.) . - Differentialrechnung bei Charakteristik p , J. reine angew. Math., t. 175, 1936 , p. 89-99. Zbl 0014.00401 | JFM 62.0114.01 · Zbl 0014.00401  et [Z-S 2] ZARISKI (O.) et SAMUEL (P.) . - Commutative Algebra , vol. 1 et vol. 2. - Springer-Verlag, 1960 . MR 22 #11006 · Zbl 0121.27801
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