15 September 2001 Sharp estimates for the arithmetic Nullstellensatz
Teresa Krick, Luis Miguel Pardo, Martín Sombra
Duke Math. J. 109(3): 521-598 (15 September 2001). DOI: 10.1215/S0012-7094-01-10934-4

Abstract

We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring ℤ. The result improves previous work of P. Philippon, C. Berenstein and A. Yger, and T. Krick and L. M. 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.

Citation

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Teresa Krick. Luis Miguel Pardo. Martín Sombra. "Sharp estimates for the arithmetic Nullstellensatz." Duke Math. J. 109 (3) 521 - 598, 15 September 2001. https://doi.org/10.1215/S0012-7094-01-10934-4

Information

Published: 15 September 2001
First available in Project Euclid: 5 August 2004

zbMATH: 1010.11035
MathSciNet: MR1853355
Digital Object Identifier: 10.1215/S0012-7094-01-10934-4

Subjects:
Primary: 11G50
Secondary: 11G35 , 13P10 , 14Q20

Rights: Copyright © 2001 Duke University Press

Vol.109 • No. 3 • 15 September 2001
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