The Eisenstein constant. (English) Zbl 0770.11051

Let \(f(x)=\sum^ \infty_{i=0}\alpha_ ix^ i\) be an algebraic power series, i.e. such that \(P(x,f)=0\) for some polynomial \(P\) of \(\mathbb{Z}[x,y]\). Then there are natural numbers \(a_ 0\) and \(a\) such that \(a_ 0a^ i\alpha_ i\) are algebraic integers. This paper is devoted to bound \(a\) by means of the degree \(n\) in \(y\) and the height of the coefficients of \(P\). The authors obtain \[ a\leq\lambda H^{2n- 1}\exp[n(\alpha n+\beta\log^ 2(n)+\gamma\log(n))] \] for explicit contants \(\lambda,\alpha,\beta\) and \(\gamma\). It seems to be the best bound known at the moment.
For the proof, one considers the differential equation satisfied by the Vandermonde matrix constructed with the powers of the roots of \(P\). Then a weak transfer theorem for \(p\)-adic differential equations, gives, for each prime \(p\), a lower bound for the radius of convergence of \(f\). From that, one infers an upper bound for the \(p\)-adic absolute value of \(a\).
Reviewer: G.Christol (Paris)


11S05 Polynomials
12H25 \(p\)-adic differential equations
Full Text: DOI


[1] E. Artin, Algebraic numbers and algebraic functions , Gordon and Breach Science Publishers, New York, 1967. · Zbl 0194.35301
[2] G. Christol, Un théorème de transfert pour les disques singuliers réguliers , Astérisque (1984), no. 119-120, 5, 151-168. · Zbl 0553.12014
[3] G. Christol and B. Dwork, Effective \(p\)-adic bounds at regular singular points , Duke Math. J. 62 (1991), no. 3, 689-720. · Zbl 0762.12004
[4] G. Christol and B. Dwork, Differential Modules of Bounded Spectral Norm , Contemp. Math., · Zbl 0765.12003
[5] B. Dwork and P. Robba, On natural radii of \(p\)-adic convergence , Trans. Amer. Math. Soc. 256 (1979), 199-213. JSTOR: · Zbl 0426.12013
[6] P. Dienes, The Taylor series: an introduction to the theory of functions of a complex variable , Dover Publications Inc., New York, 1957. · Zbl 0078.05901
[7] 1 G. Eisenstein, Über eine allgemeine Eigenschaft der Reihen-Entwicklungen aller algebraischen Funktionen , Bericht Königl. Preuß. Akad. Wiss., Berlin, 1852.
[8] 2 G. Eisenstein, Mathematische Werke. Band II , Chelsea Publishing Co., New York, 1975. · Zbl 0339.01018
[9] H. P. Epp, Eliminating wild ramification , Invent. Math. 19 (1973), 235-249. · Zbl 0254.13008
[10] J. Rosser and L. Schoenfeld, Sharper bounds for the Chebyshev functions \(\theta (x)\) and \(\psi (x)\) , Math. Comp. 29 (1975), 243-269. JSTOR: · Zbl 0295.10036
[11] O. F. G. Schilling, The Theory of Valuations , Mathematical Surveys, No. 4, American Mathematical Society, New York, N. Y., 1950. · Zbl 0037.30702
[12] W. M. Schmidt, Eisenstein’s theorem on power series expansions of algebraic functions , Acta Arith. 56 (1990), no. 2, 161-179. · Zbl 0659.12003
[13] H. Shapiro, Introduction to the theory of numbers , Pure and Applied Mathematics, John Wiley & Sons Inc., New York, 1983. · Zbl 0515.10001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.