Values of random polynomials in shrinking targets

Relying on the classical second moment formula of Rogers we give an effective asymptotic formula for the number of integer vectors $v$ in a ball of radius $t$, with value $Q(v)$ in a shrinking interval of size $t^{-\kappa}$, that is valid for almost all indefinite quadratic forms in $n$ variables for any $\kappa<n-2$. This implies in particular, the existence of such integer solutions establishing the prediction made by Ghosh Gorodnik and Nevo. We also obtain similar results for random polynomials of higher degree.