Small solutions of polynomial congruences
Let \(\delta>0\) be fixed. If \(f(x)\in{\mathbb Z}[x]\) is a polynomial of degree \(\geq p^\delta\) and \(f(x)\equiv 0\pmod p\) has a solution, the author asks whether there is a solution \(x_0\) with \(|x_0|<p^{1-\varepsilon} \) for some \(\varepsilon=\varepsilon(\delta)\). For \(f(x)=x^q-a\), \(q\mid p-1\), which admits a solution modulo \(p\), he estimates the smallest solution \(x_0\) with \(0<x_0<p\) as \(x_0\ll p^{\frac32}q^{-1}\log p\) and by applying the Burgess character sum estimates [\textit{D. A. Burgess}, Proc. Lond. Math. Soc. (3) 13, 524--536 (1963; Zbl 0123.04404)], and estimates of certain exponential sums due to \textit{J. Bourgain, A. A. Glibichuk} and \textit{J. V. Konyagin} [J. Lond. Math. Soc., II. Ser. 73, No. 2, 380--398 (2006; Zbl 1093.11057)] derives refinements of this result.
scientific article; zbMATH DE number 5734646