Points on polynomial curves in small boxes modulo an integer

TL;DR

This paper investigates the number of solutions to polynomial congruences within small boxes modulo an integer, employing advanced number theory techniques including the Vinogradov mean value theorem and lattice point counting.

Contribution

It introduces new bounds on solutions to polynomial congruences in small regions using innovative methods from the Geometry of Numbers and recent advances in the Vinogradov mean value theorem.

Findings

Derived new bounds for solutions to polynomial congruences

Applied transference principles from Geometry of Numbers

Utilized recent breakthroughs in Vinogradov mean value theorem

Abstract

Given an integer $q$ and a polynomial $f\in \mathbb Z_{q}[X]$ of degree $d$ with coefficients in the residue ring $\mathbb Z_q=\mathbb Z/q\mathbb Z,$ we obtain new results concerning the number of solutions to congruences of the form $$y\equiv f(x) \pmod{q},$$ with integer variables lying in some cube $\mathcal B$ of side length $H$. Our argument uses ideas of Cilleruelo, Garaev, Ostafe and Shparlinski which reduces the problem to the Vinogradov mean value theorem and a lattice point counting problem. We treat the lattice point problem differently using transference principles from the Geometry of Numbers. We also use a variant of the main conjecture for the Vinogradov mean value theorem of Bourgain, Demeter and Guth and of Wooley which allows one to deal with rather sparse sets.