Estimate of complete sums of Dirichlet characters of polynomials (Q1584105)

The paper is devoted to estimate complete sums of Dirichlet characters modulo the power of a prime number for polynomials of one or several variables. Such estimates were first obtained in the 1980s by D. I. Ismoilov, who successfully applied the method of Hua Loo-Keng for the estimation of complete rational trigonometric sums and the formula of A. G. Postnikov for a primitive Dirichlet character. In this paper the author establishes estimates of sums of primitive characters modulo \(p^k\) for polynomials of the form \(f(x)= a_0 + p^\alpha h(x)\), where the coefficients of the polynomial \(h(x)\) are relatively prime with \(p\) in common and \(h(0)= 0\). For \(\alpha= 0\) estimates of such sums were obtained by D. I. Ismoilov. Estimates of complete sums of characters for polynomials of several variables are also established in the paper. Here the author follows the approach proposed by V. N. Chubarikov in estimating complete multiple rational trigonometric sums.