On the number of solutions of a polynomial congruence modulo \(p^ \alpha\)
Let \(p\) be a prime number, \(a\) and \(\alpha\geq 1\) be integers, and \(f(x)= a_n x^n+ \dots+ a_1 x+ a_0\) \((n\geq 2)\) be a polynomial with integral coefficients such that \((a_n, \dots, a_0) =1\). Also we take \(V_a (f; p^\alpha)= \{x\bmod p^\alpha\mid f(x)\equiv a\pmod {p^\alpha} \}\) and \(N_a (f; p^\alpha)= \text{Card } V_a (f; p^\alpha)\). The object of this paper is to improve the estimation of the superior bound of \(N_a (f; p^\alpha)\) by using the estimate of the complete rational trigonometric sum \(S(F; p^\alpha)= \sum_x e(F (x) p^{-\alpha})\), where \(F(x)\in \mathbb{Z} [x]\) and the summation is over a complete residue system modulo \(p^\alpha\). Two sharper estimates for \(N_a (f; p^\alpha)\) are obtained in the present paper. The reviewer believes that by using the reviewer's results [Sci. Sin., Ser. A 28, 561-578 (1985; Zbl 0599.10030)], the author's conclusion may be improved when \(p>n\).
number of solutions of a polynomial congruence modulo \(p^ \alpha\)
scientific article; zbMATH DE number 815655