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Bounds for the solutions of Thue-Mahler equations and norm form equations. (English) Zbl 0861.11024

The purpose of this paper is to improve on earlier results of the second author, which deal with explicit bounds for the size of norm form diophantine equations – in particular, with Thue and Thue-Mahler equations. Below we state three typical results.
Let \(F(X,Y) \in \mathbb{Z} [X,Y]\) be an irreducible binary form of degree \(n\geq 3\), \(b\) a nonsquare rational integer and put \(B= \max \{\log |b|,e\}\). Let also \(M= \mathbb{Q} (\alpha)\) for some root \(\alpha\) of \(F(X,1)\) and denote by \(R_M\), \(h_M,r\) the regulator, class-number and unit rank of \(M\). Further, let \(H\geq 3\) be an upper bound for the height of \(F\). Finally, define \(\log^* (x)= \max \{\log x,1\}\) for any positive real number \(x\).
Theorem 1: All integer solutions \((x,y)\) of \(F(x,y)= b\) satisfy \[ \max \bigl\{|x|, |y|\bigr\} < \exp \biggl \{c_3 R_M (\log^*R_M) \bigl(R_M+ \log (HB) \bigr) \biggr\} \] and \[ \max \bigl \{|x|, |y |\bigr\} < \exp \bigl\{c_4 H^{2n-2} (\log H)^{2n-1} \log B \bigr\}, \] where \(c_3= 3^{r+27} (r+ 1)^{7r+19} n^{2n+ 6r+ 14}\), \(c_4= 3^{3(n+9)} n^{18(n+1)}\).
Theorem 2: Let \(p_1, \dots, p_s\) be distinct rational primes not exceeding \(P\) and \(b,B\) as in the previous theorem. Then, all solutions \((x,y,z_1, \dots, z_s)\) of the Thue-Mahler equation \(F(x,y) = bp_1^{z_1} \dots p_s^{z_s}\) satisfy \[ \begin{split} \max \bigl\{|x|, |y|, p_1^{z_1}, \dots, p_s^{z_s} \bigr\} < \\ \exp \biggl\{c_5P^N (\log^*P)^{ns+2} R_Mh_M \bigl(\log^* (R_Mh_M) \bigr)^2 \bigl( R_M + s h_M+ \log (HB) \bigr)\biggr\} \end{split} \] and \[ \max \bigl\{|x |, |y |, p_1^{z_1}, \dots, p_s^{z_s} \bigr\} < \exp \bigl\{c_6P^N (\log^* P)^{ns+2} H^{2n+2} (\log h)^{2n-1} \log B \bigr\}, \] where \(N=n (n-1)(n-2)\) and \(c_5= 3^{n(2s+1) +27}n^{2n(7s+13) +13} (s+1)^{5n (s+1) +15}\), \(c_6=2^{5n} n^{3n}c_5\).
In the following theorem \(|z|\) denotes the distance of the complex number \(z\) from the nearest rational integer.
Theorem 3: Let \(\alpha_1, \dots, \alpha_m\) be algebraic numbers with absolute heights at most \(A\) \((\geq e)\) such that, for every \(i=1, \dots, m-1\), \(\alpha_{i+1}\) is of degree \(\geq 3\) over \(\mathbb{Q} (\alpha_0, \dots, \alpha_i)\) (where \(\alpha_0 =1)\). Let \(M= \mathbb{Q} (\alpha_1, \dots, \alpha_m)\) be of degree \(n\). Denote by \(R_M\) the regulator of \(M\) and put \(\sigma =1\) or 2, according as \(M\) is real or not. Then, for every nonzero \(m\)-tuple of integers \((x_1, \dots, x_m)\) we have \[ |x_1 \alpha_1+ \cdots+ x_m \alpha_m |> \kappa_3 X^{-(n- \sigma- \tau_3)/ \sigma}, \quad X= \max \bigl(|x_1 |, \dots, |x_m |\bigr), \] where \(\kappa_3= (2m)^{-2(n-\sigma)/ \sigma} A^{-(n^2+1) (nm+2)/ \sigma} \exp (-R_M/ \sigma)\) , \(\tau_3= (3^{n+26} n^{15n+20} R_M \log^* R_M)^{-1}\).

MSC:

11D57 Multiplicative and norm form equations
11J13 Simultaneous homogeneous approximation, linear forms
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