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Multiplicity estimates for solutions of algebraic differential equations

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

Abstract

Let f 1(z),⋯, (z) be a set of functions analytic at the point 0 and n, h natural numbers. It’s easy to see that there exist a nonzero polynomial P(z,x 1,⋯,x m) ε C[z,x 1,⋯, x m], such that degz, Pn, degx Ph and

$$ ord_{z = 0} P\left( {z,f_1 \left( z \right), \ldots ,f_m \left( z \right)} \right) \geqslant \left( {m!} \right)^{ - 1} nh^m . $$

The upper bounds for this order of zero in terms of n and h depends on individual properties of functions f l, ⋯, f m. For example, if functions are algebraically dependent over C(z), and P is a polynomial which realise the dependence we have ordz=o P = ∞. In this article we are interested in upper bounds. Of course, instead of the point z = 0 we can ask the same question at any point z = ε

Keywords

  • Prime Ideal
  • Linear Differential Equation
  • Polynomial Ideal
  • Algebraic Function
  • Irreducible Polynomial

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.

Chapter’s author : Yuri V. Nesterenko.

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© 2001 Springer-Verlag Berlin Heidelberg

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(2001). Multiplicity estimates for solutions of algebraic differential equations. 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_10

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  • DOI: https://doi.org/10.1007/3-540-44550-1_10

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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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