Bounds for the Least Integral Solutions of Inequalities Involving Diophantine Quadratic Forms.

Let $Q$ be a non-degenerate indefinite Diophantine quadratic form in d variables. In the mid 80’s, Margulis proved the Oppenheim conjecture, which states that if $d \geq 3$ and $Q$ is not proportional to a rational form then $Q$ takes values arbitrarily close to zero at integral points. In this talk we will discuss the problem of obtaining bounds for the least integral solution of the Diophantine inequality $|Q[x]|< \epsilon$ for any positive $\epsilon$ if $d \geq 5$ if in addition $Q$ satisfies a certain Diophantine condition. We will show how to obtain explicit bounds that are polynomial in $\epsilon$, with exponents depending only on the Diophantine properties of Q.