The Number of Roots of Polynomials of Large Degree in a Prime Field

We establish asymptotic upper bounds on the number of roots modulo p of some polynomials with rational coefficients, with p an arbitrarily large prime numbers, using a variant of Stepanov's method. The polynomials we consider have degree of size close to p and are obtained by truncating power series of polylogarithms, polyexponentials, and Bessel functions. This answers a question of Heath-Brown.