Algebra/Number Theory Seminar

Near injectivity of polynomial functions

Alexander Carney, U Rochester

Wednesday, September 11th, 2019
12:00 PM - 1:00 PM
Hylan 1106A

A polynomial in Q[X] induces a map Q—>Q. We show that this map is always at most 6-to-1 over all but finitely many values. Analogous bounds hold over every number field K, depending only on the number of roots of unity zeta such that zeta+zeta^-1 is in K. If we interpret Mazur and Merel’s theorems on rational torsion of elliptic curves as bounding the rational N-to-1 behavior for morphisms between genus one curves, then our result can be seen as a parallel for the affine line. We formulate a conjecture about morphisms between arbitrary varieties which implies both our result and the uniform boundedness conjecture for rational torsion on abelian varieties, and discuss possible techniques and obstructions to proving additional cases.

Event contact: dinesh dot thakur at rochester dot edu