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1. The basic setting and two ternary cases We take a finite-dimensional vector space V over a field F and take also an F-bilinear symmetric form ϕ: V × V → F. We then put ϕ[x] =ϕ(x, x) for x ∈ V, thus using the same letter ϕ for the quadratic form and the corresponding symmetric form. By a quadratic Diophantine equation we mean an equation of the type (1) ϕ[x] =q with a given q ∈ F ×. In particular, in the classical case with F = Q and V = Q n, we usually assume that ϕ is Z-valued on Z n and q ∈ Z. The purpose of the present article is to present some new ideas on various arithmetical questions on such an equation. We start with some of our basic symbols and terminology. For a set X we denote by #X or #{X} the number ( ≤ ∞)ofelementsofX. For an associative ring R with identity element, we denote by R × the group of invertible elements of R and by Mn(R) the ring of all square matrices of size n with entries in R. We then put GLn(R) =Mn(R) × and denote by 1n the identity element of Mn(R). Fo

Year: 2011

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oai:CiteSeerX.psu:10.1.1.192.589

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