# Seminars & Colloquia Calendar

Number Theory Seminar

## Bounds for the least solutions of quadratic inequalities

#### Thomas Hille, Yale

Location:  Room 425
Date & time: Tuesday, 10 March 2020 at 2:00PM - 3:00PM

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 $$d \geq 3$$ 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 $$d \geq 5$$. We will review historical, as well as recent results in this direction and 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.

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