A number is called y-smooth if all of its prime factors are bounded above by y. The set of y-smooth numbers below x forms a sparse subset of the integers below x as soon as x is sufficiently large in terms of y. If f1, …, fr \in ℤ[x1,…,xs] is a system of pairwise ℚ-linearly independent linear forms, one may ask how often these forms simultaneously take values in the set of y-smooth numbers when the variables xi are all restricted to be of size about x. I will discuss asymptotic results on this counting problem.
This talk is based on joint work with Mengdi Wang.
This video is part of the Institute for Advanced Study‘s Special year research seminar.
