RESEARCH IN RESIDENCE — Explicit local solubility and applications

The question of whether or not a collection of equations has a solution in the integers is notoriously challenging, with the answer depending heavily on the geometry of the constituent equations. An important reduction is localization – looking at the equations modulo a prime or over the real numbers. Having solutions locally is a key necessary condition for the equations to have an integral solution and much more tractable to determine explicitly. The proposed research involves determining explicit and exact expressions for how often certain families of equations have local solutions, which can sometimes yield explicit results for how often an equation has an integral solution. In particular, we propose an approach to determine how often a degree 3 polynomial in n+1 variables has an integral zero, by reducing to the local probabilities (via recent work of Browning, Le Boudec, and Sawin) and determining them by a careful recursive argument.