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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)

MSC:

11S05 Polynomials
12H25 \(p\)-adic differential equations
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References:

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