Optimal density for values of generic polynomial maps

Abstract

We establish that the optimal bound for the size of the smallest integral solution of the Oppenheim Diophantine approximation problem $|Q(x)-\xi|<\epsilon$ for a generic ternary form $Q$ is $|x|\ll\epsilon^{-1}$. We also establish an optimal rate of density for the values of polynomials maps in a number of other natural problems, including the values of linear forms restricted to suitable quadratic surfaces, and the values of the polynomial map defined by the generators of the ring of conjugation-invariant polynomials on $M_3(\Bbb{C})$ Show more