## 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

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.

### 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

Zbl 0808.11023
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