Timothy Browning: Random Diophantine equations (NTWS 010)
Channel: Number Theory Web Seminar
Date: 2023-04-11

Abstract: I’ll survey some of the key challenges around the solubility of polynomial Diophantine equations over the integers.
While studying individual equations is often extraordinarily difficult, the situation is more accessible if we merely ask what happens on average and if we restrict to the so-called Fano range, where the number of variables exceeds the degree of the polynomial.  Indeed, about 20 years ago, it was conjectured by Poonen and Voloch that random Fano hypersurfaces satisfy the Hasse principle, which is the simplest necessary condition for solubility.  After discussing related results I’ll report on joint work with Pierre Le Boudec and Will Sawin where we establish this conjecture for all Fano hypersurfaces, except cubic surfaces.

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Thursday, June 4, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time)
Friday, June 5, 2020 (1am AEST, 3am NZST)