Indefinite quadratic polynomials of small signature

Abstract

Let F(X)=Q(X)+L(X) be a real quadratic polynomial with no constant term. Suppose that the quadratic part Q(X) is indefinite of type (r, n-r). For an integer k≥4 we show that if min (r, n-r) ≥k there exists a function f(n, k)=–1/2+3/(4k+2)+O_k (1/n) with the following property. For any η>0 and all large enough X there is an integer vector χ≠0 such that |χ| ≤X and |F(X)|«X^ƒ(n,k)+n.