Indefinite quadratic polynomials of small signature (Q790876)

Let Q(x) be an indefinite quadratic form in n variables, then Q may be expressed as a sum of squares of real linear forms. Suppose that there are r positive signs and n-r negative signs with \(\min(r,n-r)\geq k.\) Theorem. Let \(F(x)=Q(x)+L(x)\) be a polynomial having no constant term. For any \(\eta>0\) and all large X there is an integer vector \(x\neq 0\) satisfying \(| x| \leq X\) and \(| F(x)|<<X^{f(n,k)+\eta}\) where \[ f(n,k)=-1/2+3/(4k+2)+O_ k(1/n)\quad \to \quad -1/2\quad as\quad n\to \infty,\quad k\to \infty. \] The analogous results for Hermitian forms and polynomials in the order \(x+y\sqrt{d}\), \(d<0\) square- free, are also obtained.