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The Hilbert Nullstellensatz, Inequalities for Polynomials, and Algebraic Independence

Part of the Lecture Notes in Mathematics book series (LNM,volume 1752)

Abstract

Given a set of polynomials, P 1(x),⋯, P m(x), one of the most basic questions one can ask about them is whether they have common zeros or not. Hilbert gave a comprehensive answer to this sort of question in his famous theorem. To state it, we will use the following notation:

NOTATION. Let % be the polynomial ideal generated in R := k[x 1,⋯, x n] by the polynomials P 1,⋯P m of respective degrees D 2(=: D) ≥ D 3D mD 1.

Keywords

  • Polynomial Ideal
  • Common Zero
  • Diophantine Approximation
  • Homogeneous Ideal
  • Algebraic Independence

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

[Hil] D. Hilbert. Über die vollen Invariantensysteme. Math. Ann. 42, (1883), 313-373

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© 2001 Springer-Verlag Berlin Heidelberg

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(2001). The Hilbert Nullstellensatz, Inequalities for Polynomials, and Algebraic Independence. In: Nesterenko, Y.V., Philippon, P. (eds) Introduction to Algebraic Independence Theory. Lecture Notes in Mathematics, vol 1752. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44550-1_16

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  • DOI: https://doi.org/10.1007/3-540-44550-1_16

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41496-4

  • Online ISBN: 978-3-540-44550-0

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