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

Bugeaud, Yann; Györy, Kálmán

Acta Arith. 74, No. 3, 273-292 (1996).

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 \(\kappa_3= (2m)^{-2(n-\sigma)/ \sigma} A^{-(n^2+1) (nm+2)/ \sigma} \exp (-R_M/ \sigma)\) , \(\tau_3= (3^{n+26} n^{15n+20} R_M \log^* R_M)^{-1}\).

Reviewer: N.Tzanakis (Iraklion)

MSC:

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

Keywords:

linear forms with algebraic coefficients; explicit bounds; Thue equations; binary form of degree \(\geq 3\); norm form diophantine equations; Thue-Mahler equations