Sharp estimates for the arithmetic Nullstellensatz

Details Sharp estimates for the arithmetic Nullstellensatz Sharp estimates for the arithmetic Nullstellensatz

1999-11-14

arxiv.org/abs/math/9911094v1

Mathematics

Algebraic Geometry

55 pages, LaTeX2e

Scientific paper

We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over $\Z$. The result improves previous work of Philippon, Berenstein-Yger and Krick-Pardo. We also present degree and height estimates of intrinsic type, which depend mainly on the degree and the height of the input polynomial system. As an application, we derive an effective arithmetic Nullstellensatz for sparse polynomial systems. The proof of these results relies heavily on the notion of local height of an affine variety defined over a number field. We introduce this notion and study its basic properties.

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