On a conjecture of E. Thomas concerning parametrized Thue equations (Q2717636)

The author considers parametrized Thue equations of the form NEWLINE\[NEWLINE F_a(X,Y)=\prod_{i=1}^n \bigl( X-p_i(a) Y\bigr) -Y^n = \pm 1, \quad n\geq 3, \leqno (1) NEWLINE\]NEWLINE where \(p_1\), \dots, \(p_n\) are polynomials with integer coefficients. It is clear that \(\pm (p_i(a),1)\) and \(\pm(1,0)\) are solutions, called trivial. E. Thomas conjectured that if \(p_1=0\) and \(0<\deg p_2<\ldots<\deg p_n\) are monic then (1) has only trivial solutions for large \(a\). In the present paper, the author proves such a result under suitable hypotheses on the \(p_i\)'s. The proof proceeds in two steps: to show that for a nontrivial solution \(|y|\) must be very large (the proof of this ``gap principle'' is the main difficulty) and to use upper bounds on \(|y|\) (Bugeaud-Győry estimates) to get a contradiction when \(a\) is large enough.