Let n>2 be a fixed integer. Lenstra conjectured that when monic integer polynomials of degree n are ordered by the size of their largest coefficient, the proportion of polynomials which correspond to maximal rings is 1/ζ(2), independent of n. In this talk, we will discuss the analogous question for the space of all (not necessarily monic) integer polynomials of degree n, and prove that a proportion of 1/ζ(2)ζ(3) of them correspond to maximal rings, obtaining also the Bertini (regularity) theorem for P1(ℤ). This is joint work with Manjul Bhargava and Xiaoheng Wang.
This video was produced by the International Centre for Mathematical Sciences, as part of the workshop Arithmetic, Algebra, and Algorithms.
