Applications of lunacy to finding small solutions of two variable congruences

par Ted Chinburg (University of Pennsylvania)

jeudi 28 mai 2026, 17:00 → 18:00 Europe/Paris

Salle Pellos (1R2)

In 2000, Boneh and Durfee showed that if a certain heuristic hypothesis is true, one can find all small solutions of certain two variable congruences using the LLL algorithm for finding small non-zero elements of lattices.   The truth of the heuristic hypothesis has been an open question since 2000, however.  In this talk I will discuss how one can use capacity theory on curves to show that the heuristic hypothesis actually fails in a positive proportion of cases, and that it also holds in a positive proportion of cases.  One step in the proof involves the potential theory of regions in the complex plane that are the complement in a large closed disk of a smaller open disk.  Such regions are called lunes because they look like a waxing or waning moon.  This is joint work with Stephen Miller.