Introduction to Algebraic Independence Theory pp 149-165 | Cite as

# Multiplicity estimates for solutions of algebraic differential equations

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## Abstract

Let The upper bounds for this order of zero in terms of n and h depends on individual properties of functions

*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 deg_{z},*P*≤*n*, deg_{x}*P*≤*h*and$$
ord_{z = 0} P\left( {z,f_1 \left( z \right), \ldots ,f_m \left( z \right)} \right) \geqslant \left( {m!} \right)^{ - 1} nh^m .
$$

*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 ord_{z}=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.

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© Springer-Verlag Berlin Heidelberg 2001