Simultaneous Representation of Integers by a Pair of Ternary Quadratic Forms—With an Application to Index Form Equations in Quartic Number Fields
✍ Scribed by István Gaál; Attila Pethő; Michael Pohst
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 411 KB
- Volume
- 57
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
✦ Synopsis
Let Q 1 , Q 2 # Z[X, Y, Z] be two ternary quadratic forms and u 1 , u 2 # Z. In this paper we consider the problem of solving the system of equations
(1) Q 2 (x, y, z)=u 2 in x, y, z # Z with gcd(x, y, z)=1.
According to Mordell [12] the coprime solutions of
can be presented 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 # 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 irreducible 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).