On an estimate of A. O. Gel'fond (Q1054781)

This paper deals with Gel'fond's inequality used in applications of number theory to numerical analysis. Suppose that \(a_1,\ldots, a_m\) are integers, \(p\) is prime or a power of a prime and \((a_i ,p) =1\) \((i=1,\ldots, s)\). Define \(\overline m = \max(1, \vert m\vert)\). Consider the congruence \[ m_0+ a_1m_1 + \cdots + a_sm_s \equiv 0\pmod p, \quad -p/2 <m_i\le p/2,\ i = 0,1, \ldots, s, \tag{1} \] and the system of congruences \[ a_1k_0\equiv k_1 \pmod p, \ldots, a_mk_0 \equiv k_s \pmod p,\quad - p/2 <k_i \le p/2,\ i = 0,1, \ldots, s. \tag{2} \] Let \(q_s(p) = \min \overline m_0\cdots \overline m_s\) and \(Q_k(P) = \min \vert k_0\vert \cdots \vert k_s\vert\), where the minima are taken over the solutions \((m_0, \ldots, m_s) \ne (0,\ldots, 0)\) of (1) and \((k_0 , \ldots, k_s) \ne (0, \ldots, 0)\) of (2), respectively. Theorem. The inequality \[ q_s(p) \ge (Q_s(p))^s/\bigl(C_s p^{s^2 - 1}\bigr) \] holds with \(C_s = (2s +3)^{s+1}\). This result generalizes Gel'fond's inequality [see the author, Russ. Math. Surv. 22, No. 3, 80--118 (1967; Zbl 0178.04603); translation from Usp. Mat. Nauk 22, No. 3(135), 83--118 (1967)].