Positive values of indefinite quadratic forms

Abstract

Let f(x)= ∑ ⁿ _i=1 ∑ ⁿ _j=1 f_ijx_ix_j (f_ij= f_ji) (1) be a real quadratic form in n variables with integral coefficients (i.e., 2f_ijε Z,f_ij ε Z.) and determinant D ≠ O. A well-known theorem of Cassels [1] states that if the equation f = 0 is properly soluble in integers x₁ … , x_n then there is a solution satisfying 0<||x||=max|x_i≪ F^(n-1)/2, (2). ∏ⁿ_i=1||x|| ≪ F^{n(n-1)/2+(n-2)/2}, (3).