Pisot, Charles Familles normales de fractions rationnelles et ensembles fermés de nombres algébriques. (French) Zbl 0207.36201 Algèbre Théorie Nombres, Sém. P. Dubreil, M.-L. Dubreil-Jacotin, L. Lesieur et C. Pisot 16 (1962/63), Exp. No. 14, 12 p. (1967). Let \(\mathbb Q_p\) denote the completion of the rationals with respect to the \(p\)-adic absolute value defined by \(\vert p\vert_p= p^{-1}\) and \(\vert p'\vert_p = 1\) if \(p'\) is a prime other than \(p\). Let \(\Omega_p\) be the algebraic closure of \(\mathbb Q_p\). Main theorem: Let \(k\), \(q\) be integers, \(k\ge 0\), and let \(0 < \delta< 1\). Let \(\mathcal F\) be the family of rational functions \(\varphi = P/Q\), where \(P\) and \(Q\) are polynomials whose coefficients are rational integers, \(\vert\varphi(z)\vert\le 1\) for \(\vert z\vert = 1\), \(Q(0) = q\), and \(Q'(0)\) is relatively prime to \(q\). Let \(Q\) have, in \(\vert z\vert < 1\), at most \(k\) zeros, all contained in the annulus \(\delta\le\vert z\vert < 1\). Suppose also that \(\vert P(0)\vert_p > \vert q\vert_p\) for every prime \(p\) that divides \(q\). Then \(\mathcal F\) is a normal family in \(\vert z\vert < 1\). Applications: 1. Let \(q\) be a nonzero integer. Let \(S_q\) be the set of real algebraic numbers \(\theta >1\) such that: \(\theta^{-1}\) is a root of a polynomial \(Q\) with integer coefficients, and the other roots of \(Q\) are in \(\vert z\vert > 1\); \(Q(0) = q\) and \(Q'(0)\) is prime to \(q\); and there exists a polynomial \(P\) with integer coefficients such that \(\vert P(z)\vert \le \vert Q(z)\vert\) for \(\vert z\vert = 1\), \(P(1/\theta)\ne 0\), \(\vert P(0)\vert \ge\vert q\vert\), and \(\vert P(0)\vert_p > \vert q\vert_p\) for every prime \(p\) that divides \(q\). Then \(S_q\) is closed. (Raphael Salem proved this result in the case \(q=1\).)2. The following result of Chabauty is also obtained: Let \(\mathcal C\) be the set of numbers \(\alpha\in\mathbb Q_p\), algebraic over \(\mathbb Q\), for which there exists an integer \(h\ge 1\) such that \(p^h\alpha\) is an algebraic integer over \(\mathbb Q\) and the irreducible equation in \(\mathbb Q\) having \(\alpha\) for zero has all its roots in \(\vert z\vert < 1\), and in \(\Omega_p\) has all its roots other than \(\alpha\) in \(\vert z\vert_p \le 1\), whereas \(\vert \alpha\vert_p > 1\). Then \(\mathcal C\) is closed in \(\mathbb Q_p\). [For the entire collection see Zbl 0189.01501.] Reviewer: O. C. McGehee Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 2 Documents MSC: 11R04 Algebraic numbers; rings of algebraic integers 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11R09 Polynomials (irreducibility, etc.) 11S05 Polynomials Keywords:normal family of rational fractions; closed set of algebraic numbers Citations:Zbl 0189.01501 × Cite Format Result Cite Review PDF Full Text: Numdam EuDML