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Philippon, P.

Effective Nullstellensatz and elimination. (Théorème des zéros effectif et élimination.) (French) Zbl 0743.11036

Sémin. Théor. Nombres Bordx., Sér. II 1, No. 1, 137-155 (1989).
This interesting article consists of two distinct parts. In each of them a prominent role is played by elimination theory, to whose recent development the author has contributed much. The first part reflects a talk given in the Seminar on the various effective versions of the Nullstellensatz developed by Brownawell, Kollár, and Berenstein and Yger. This last version depends critically on Philippon’s estimate for a denominator.
The second part gives a quick development of a novel approach in characteristic zero, joint with F. Amoroso, to the multiplicity of a prime ideal at a point as a sort of order of vanishing of the generators of the Chow ideal. Six equivalent criteria involving eliminant forms and differentiation are derived. Then this notion is extended to general ideals and related to the more usual definition of multiplicity in the local ring of the point.
A few explanatory words (for example just mentioning the product rule and Euler’s expression of a homogeneous polynomial in terms of its partial derivatives) on the equivalence of the two definitions at the top of p.145 would have been useful, especially since the underlying ideas are invoked again later. The reader who survives the necessarily heavy notation and surprising number of misprints in this part will be rewarded.
Reviewer: W.D.Brownawell (University Park)

Cited in 2 Documents

MSC:

11J95 Results involving abelian varieties
12D99 Real and complex fields
13B25 Polynomials over commutative rings

Keywords:

elimination theory; effective versions of the Nullstellensatz; multiplicity of a prime ideal; Chow ideal
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References:

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