## 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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