Multiplicity estimates for solutions of algebraic differential equations

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


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


Prime Ideal Linear Differential Equation Polynomial Ideal Algebraic Function Irreducible Polynomial 
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© Springer-Verlag Berlin Heidelberg 2001

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