Sharper Upper Bounds for Roots of Polynomials Generated by Positive Sequences

Abstract

Finding sharp and easily computable upper bounds for the moduli of the roots of polynomials with real coefficients is a long-standing problem with applications in numerical analysis, control theory, and the study of linear recurrence relations. The classical bounds of Cauchy and Lagrange, despite their age, remain the most frequently used estimates because of their extreme simplicity. This paper introduces a new family of upper bounds specifically designed for polynomials whose coefficients are the initial terms of a positive real sequence a_n that does not grow too rapidly. For each such polynomial we construct an explicit number by taking the two largest values appearing among the (i+1)-th roots of the successive absolute differences of the sequence together with the simple quantity a_1+1, and adding them. We prove that the resulting value rigorously bounds the modulus of every root. A companion bound based on second differences is obtained as an immediate corollary. Extensive numerical tests on constant, arithmetic, harmonic, and exponential sequences show that the new estimates are often several times tighter than Cauchy’s bound and, in many cases, also outperform recently published refinements. The contribution is twofold: (i) a new, fully explicit bound using first differences, and (ii) an even sharper variant using second differences presented as a corollary.