Effective bounds for the least solutions of homogeneous quadratic Diophantine inequalities

Let Q be a non-degenerate indefinite quadratic form in d variables. In the mid 80's, Margulis proved the Oppenheim conjecture, which states that if d ≥ 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]|< ε for any positive ε if d ≥ 5. We will review historical, as well as recent results in this direction and show how to obtain explicit bounds that are polynomial in ε-1, with exponents depending only on the signature of Q or if applicable, the Diophantine properties of Q.