Simultaneous representation of integers by a pair of ternary quadratic forms – with an application to index form equations in quartic number fields. (English) Zbl 0853.11023

Gaál, István; Pethö, Attila; Pohst, Michael

J. Number Theory 57, No. 1, 90-104 (1996).

From the author’s abstract: “Let \(Q_1, Q_2\in \mathbb{Z} [X, Y, Z]\) be two ternary quadratic forms and \(u_1, u_2\in \mathbb{Z}\). In this paper we consider the problem of solving the system of equations \[ Q_1 (x, y, z)= u_1, \quad Q_2 (x, y, z)= u_2,\qquad x, y, z\in \mathbb{Z}, \quad \text{gcd} (x, y, z)=1. \tag{1} \] According to Mordell, the coprime solutions of \(Q_0 (x, y, z)= u_1 Q_2 (x, y, z)- u_2 Q_1 (x, y, z) =0\) can be represented by finitely many expressions of the form \(x= f_x (p, q)\), \(y= f_y (p, q)\), \(z= f_z (p, q)\), where \(f_x, f_y, f_z\in \mathbb{Z} [P, Q]\) are binary quadratic forms and \(p\), \(q\) are coprime integers. Substituting these expressions into one of the equations of (1), we obtain a quartic homogeneous equation in two variables. If it is reducible, it is a quartic Thue equation, otherwise it can be solved even easier. The finitely many solutions \(p\), \(q\) of that equation then yield all solutions \(x\), \(y\), \(z\) of (1).”
An application to the solution of index form equations in a quartic number field \(K\) is discussed. In an earlier paper, the same authors [J. Symb. Comput. 16, 563-584 (1993; Zbl 0808.11023)] have shown that the solution of such equations is reduced to the solution of a cubic equation \(F(u, v)= i\), where \(F\) is a cubic form, and a system like (1). Quoting again the authors: “In this case the application of the new ideas for the resolution of (1) leads to quartic Thue equations which split over the same quartic field \(K\).”
The second application, to the computation of all integral points on a Weierstrass model of an elliptic curve, will be discussed in a forthcoming paper. Many numerical examples of the first application and two examples of the second, without details, are given.

Reviewer: N.Tzanakis (Iraklion)

MSC:

11D25	Cubic and quartic Diophantine equations
11D57	Multiplicative and norm form equations
11R16	Cubic and quartic extensions
11E25	Sums of squares and representations by other particular quadratic forms

Keywords:

simultaneous representation of integers; ternary quadratic forms; quartic Thue equation; index form equations; quartic number field

Citations:

Zbl 0808.11023