Values of Random Polynomials at Integer Points

Using classical results of Rogers bounding the $L^2$-norm of Siegel transforms, we give bounds on the heights of approximate integral solutions of quadratic equations and error terms in the quantiative Oppenheim theorem of Eskin-Margulis-Mozes for almost every quadratic form. Further applications yield quantitative information on the distribution of values of random polynomials at integral points.