Quadratic forms which have only large zeros

We give lower bounds for zeros of quadratic forms. For example, givenn2d>0 there are infinitely many nonsingular quadratic formsF with integral coefficients inn variables which vanish on ad-dimensional rational subspace such that anyn linearly independent integral zerosx ₁,...,x _n satisfy |x ₁|...|x _n |F(ⁿ²/^2d)–d, whereF is the maximum modulus of the coefficients ofF. This complements a recent result of the authors in the opposite direction and shows that it is best possible.Written whileSchmidt had a Senior Scientist Award from the Alexander von Humboldt-Stiftung in the Federal Republic of Germany.