Advertisement

Multiplicity estimates for solutions of algebraic differential equations

Chapter
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Personalised recommendations