Distribution of zeros of nondegenerate functions on short cuttings

Abstract

In this paper, we obtain estimates from above and from below the number of zeros of functions of a special kind, as well as an estimate of the measure of the set of points in which such functions take small values. Let f₁ (x), ..., f_n (x) function defined on an interval I, n + 1 times differentiable and Wronskian of derivatives almost everywhere (in the sense of Lebesgue measure) on I different from 0. Such functions are called nondegenerate. The problem of distributing zeros of F (x) = a_nf_n (x) + ... + a₁f₁ (x) + a₀, a_j ∈ Z, 1 ≤ j ≤ n is a generalization of many problems about the distribution of zeros of polynomials is important in the metric theory of Diophantine approximations. An interesting fact is that there is a lot in common in the distribution of roots of the function F (x) and the distribution of zeros of polynomials. For example, the number of zeros of F (x) on a fixed interval does not exceed n, as well as for polynomials — the number of zeros does not exceed the polynomial degree.

Three theorems were proved: on the evaluation of the number of zeros from above, on the evaluation of the number of zeros from below, as well as an auxiliary metric theorem, which is necessary to obtain estimates from below. While obtaining lower bounds method was used for major and minor fields, who introduced V. G. Sprindzuk.

Let Q > 1 be a sufficiently large integer, and the interval I has the length Q^−γ, 0 ≤ γ < 1. Produced estimates on the top and bottom for the number of zeros of the function F (x) on the interval I, with |a_j| ≤ Q, 0 ≤ γ < 1, and also indicate the dependence of this quantity from the interval I. When γ = 0 similar results are available from A. S. Pyartli, V. G. Sprindzhuk, V. I. Bernik, V. V. Beresnevich, N. V. Budarina.