Positive values of inhomogeneous quaternary ouadratic forms, II

Abstract

In a previous paper [4] we showed that Γ3,1 = 16/. For the definition of Γr, s for an indefinite quadratic form in n = r + s variables of the type (r, s) see the above paper. Here we shall show that Γ2,2 = 16. More precisely we prove: Theorem. Let Q (x, y, z, t) be an indefinite quaternary quadratic form with determinant D > 0 and signature (2, 2). Then given any real numbers x0, y0, z0, t0 we can find integers x, y, z, t such thatEquality is necessary if and only if either where ρ ≠ 0. For Q1 equality occurs if and only if