EVENT DETAILS AND ABSTRACT

Title: Bounds for the Least Solution of Quadratic Inequalities
Speaker: Thomas Hille
Speaker Info: Northwestern U.
Brief Description: Part of Midwest Dynamics Conference
Special Note: Part of Midwest Dynamics Conference
Abstract:

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]|< epsilon for any positive epsilon if d ≥5. We will show how to obtain explicit bounds that are polynomial in \epsilon^{-1}$, with exponents depending only on the signature of Q or if applicable, the Diophantine properties of Q. This talk is based on joint work with P. Buterus, F. Götze and G. Margulis.