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Distribution of solutions of diophantine equations \(f_ 1 (x_ 1) f_ 2 (x_ 2)= f_ 3 (x_ 3)\), where \(f_ i\) are polynomials. II. (English) Zbl 0855.11014

The investigations of Part I [Rend. Semin. Mat. Univ. Padova 87, 39-68 (1992; Zbl 0762.11011)] are continued for the distribution of the solutions of diophantine equations \((*)\) \(f_1 (x_1) f_2 (x_2)= f_3 (x_3)\). Let \(x>0\) and denote \(N(x)\) the number of solutions \(x_1, x_2, x_3\in \mathbb{Z}\) of \((*)\) such that \(|x_3 |\leq x\). Assuming that all three polynomials are quadratic with leading coefficient 1, moreover that the discriminants of \(f_1\) and \(f_2\) are 16 while the discriminant of \(f_3\) is 16 times a square of an integer, the author proves the asymptotic formula \[ N(x)= 2\sqrt {x}+ O(x^{1/3}). \] The main tool in the proof is a detailed description of possibilities for when the equation \((p_1^2- \delta_1) (p_2^2- \delta_2)= (p_3^2- \delta_3)\) has infinitely many solutions in polynomials \(p_i\in k[ t]\), where \(k\) denotes a field and \(\delta_1, \delta_2, \delta_3\) are elements of \(k\) such that \(\delta_1 \delta_2 \neq 0\).

MSC:

11D41 Higher degree equations; Fermat’s equation
11D25 Cubic and quartic Diophantine equations

Citations:

Zbl 0762.11011
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References:

[1] A. Schinzel - U. Zannier , Distribution of solutions of diophantine equations f1 (x1) f2 (X2) = f3 (x3), where fi are polynomials , Rend. Sem. Mat. Univ. Padova , 87 ( 1992 ), pp. 39 - 68 . Numdam | MR 1183901 | Zbl 0762.11011 · Zbl 0762.11011
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