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On Osgood’s effective Thue theorem for algebraic functions. (English) Zbl 0333.10019

Let \(K\) be the field of formal series \(\alpha = a_\rho t^\rho +a_{\rho-1} t^{\rho-1} + \cdots\), with \(\rho\) an integer and with coefficients in a field \(k\) of characteristic zero. Let \(\vert\alpha\vert\) be the non-archimedean absolute value with \(\vert\alpha\vert = e^\rho\) if \(\alpha_\rho \ne 0\). Recently, it was shown by C. F. Osgood [Nederl. Akad. Wet., Proc., Ser. A 78, 105–119 (1975; Zbl 0302.10034)] that if \(\alpha\) in \(K\) is algebraic of degree \(\nu\) over the rational function field \(k(t)\), and if \(\mu\) is an integer with \(\tfrac12 \nu < \mu < \nu\), then \[ \vert \alpha - p(t)/q(t) \vert \ge c\,\vert q(t)\vert^{-\mu}, \] for every rational function \(p(t)/q(t)\); here \(c=c(\alpha,\mu) >0\) is effectively computable. The author gives an explicit value of \(c\) in terms of \(\nu\), \(\mu\) and the height of \(\alpha\). Consider a “Thue equation” \(F(p,q)=m\) where \(m\ne 0\) is a polynomial in \(t\) and \(F\) is a form of degree \(> 2\) without multiple factors and coefficients which are polynomials in \(t\). The height \(\mathfrak H(F)\) is the maximum absolute value of the coefficients of \(F\). It is shown that, for every polynomial solution \(p = p(t)\), \(q = q(t)\) of the equation, \[ \vert p \vert, \vert q \vert \le \vert m \vert\, \mathfrak H(F)^{80}. \]

MSC:

11J61 Approximation in non-Archimedean valuations
11J68 Approximation to algebraic numbers
11D59 Thue-Mahler equations

Citations:

Zbl 0302.10034
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Full Text: DOI

References:

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