Mahler, Kurt

Reprint: Zur Approximation algebraischer Zahlen. I: Über den größten Primteiler binärer Formen (1933)

Doc. Math. Extra Vol. Mahler Selecta, 333-366 (2019)
DOI: 10.25537/dm.2019.SB-333-366

Summary

In 1908, Thue showed that, if \(\zeta\) is a real algebraic number of degree \(n\) and \(\Theta\) is a positive number, the inequality \[\left|\frac{p}{q}-\zeta\right|\le q^{-\left(\frac{n}{2}+1+\Theta\right)}\] has only finitely many rational solutions \(p/q\). Siegel, in 1920, showed that one can replace the exponent \(\frac{n}{2}+1+\Theta\) by \[\beta=\min_{1\le s\le n-1}\left(\frac{n}{s+1}+s+\Theta\right). \] \par In this article, Mahler establishes the following \(p\)-adic extension of Siegel's result. Let \(f(x)\) be an irreducible polynomial with rational integer coefficients of degree \(n\ge 3\), let \(P_1,P_2,\ldots,P_t\) be finitely many prime numbers and let \(\zeta,\zeta_1,\zeta_2,\ldots,\zeta_t\) be real zero of \(f(x)\), a \(P_1\)-adic zero of \(f(x)\), a \(P_2\)-adic zero of \(f(x), \ldots \), and a \(P_t\)-adic zero of \(f(x)\), respectively. Mahler proves that, if \(\beta\) is Siegel's exponent and \(k\ge 1\) is fixed, the inequality \[\min\left\{1,\left|\tfrac{p}{q}-\zeta\right|\right\} \prod_{\tau=1}^t \min\{1,|p-q\zeta_\tau|_{P_\tau}\}\le k\, \max\{|p|,|q|\}^{-\beta}\] has finitely many solutions in reduced rational numbers \(p/q\). \par Reprint of the author's paper [Math. Ann. 107, 691--730 (1933; Zbl 0006.10502; JFM 59.0220.01)]. For Part II see [Zbl 1465.11013].

Mathematics Subject Classification

11-03, 11J68

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