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Eisenstein’s theorem on power series expansions of algebraic functions. (English) Zbl 0659.12003

A theorem of Eisenstein asserts that if a formal series \(y=\alpha_ 0+\alpha_ 1x+\alpha_ 2x^ 2+...\) satisfies a non-trivial polynomial equation \(F(x,y)=0\) with algebraic coefficients, then \(\alpha_ 0,\alpha_ 1,..\). lie in an algebraic number field, and there are natural numbers \(a_ 0, a\) such that \(a_ 0a^ j\alpha_ j\) \((j=0,1,...)\) are algebraic integers. The present work gives estimates on the size of \(a_ 0, a\). Suppose the coefficients of F lie in an algebraic number field of degree \( d,\) suppose F is of total degree N and has a suitably defined height H. Then we may take \(a_ 0=a^ N\) and \(a<c(N,d)H^{8N^ 3d^ 2}.\)
A reformulation of the theorem in terms of absolute values in number fields is given. To deal with the non-archimedean absolute values, p-adic analysis is employed. In particular, a theorem of Dwork and Robba on p- adic radii of convergence of algebraic functions, and a theorem of Clark on p-adic radii of convergence of solutions of linear differential equations are used.
Reviewer: W.M.Schmidt

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11R58 Arithmetic theory of algebraic function fields
11S99 Algebraic number theory: local fields
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