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, P ≤n, degx P ≤h and
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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