Mathematical Research Letters

Volume 27 (2020)

Number 1

The smallest root of a polynomial congruence

Pages: 43 – 66



Vlad Crişan (Mathematisches Institut der Universität Göttingen, Germany)

Paul Pollack (Department of Mathematics, University of Georgia, Athens, Ga., U.S.A.)


Fix $f(t) \in \mathbb{Z}[t]$ having degree at least $2$ and no multiple roots. We prove that as k ranges over those integers for which the congruence $f(t) \equiv 0 (\operatorname{mod} k)$ is solvable, the least nonnegative solution is almost always smaller than $k / (\operatorname{log} k)^{c_f}$. Here $c_f$ is a positive constant depending on $f$. The proof uses a method of Hooley originally devised to show that the roots of $f$ are equidistributed modulo $k$ as $k$ varies.

The second author (P.P.) is supported by NSF award DMS-1402268.

Received 18 February 2018

Accepted 13 February 2019

Published 8 April 2020