On minima of rational indefinite quadratic forms

Abstract

Let f( x) be an indefinite quadratic form with real coefficients in n variables with nonzero determinant d. The collection of all values of v(f) = |d| −1 n inf |f(x)| , where infimum is taken over x ∈ Z n such that f( x) ≠ 0 ( x ≠ 0) is called the spectrum of nonzero minima (spectrum of minima) of such forms. The spectrum is said to be discrete if for every δ > 0, there are only finitely many values of v( f) > δ. It is proved that for rational quadratic forms in n ≥ 3 variables and real quadratic forms in n ≥ 21 variables the spectra of nonzero minima are discrete. Also the spectra of minima of indefinite ternary and quaternary rational quadratic forms are discrete.