TAILIEUCHUNG - On cauchy’s bound for zeros of a polynomial
In this note, we improve upon Cauchy’s classical bound, and upon some recent bounds for the moduli of the zeros of a polynomial. Be a polynomial of degree n, with complex coefficients. Then, according to Cauchy’s classical result, we have the following theorem. | Turk J Math 30 (2006) , 95 – 100. ¨ ITAK ˙ c TUB On Cauchy’s Bound for Zeros of a Polynomial V. K. Jain Abstract In this note, we improve upon Cauchy’s classical bound, and upon some recent bounds for the moduli of the zeros of a polynomial. Key Words: Zeros, polynomials, upper bound, moduli, refinement. 1. Introduction and Statement of Results Let f(z) = z n + an−1 z n−1 + an−2 z n−2 + . . . + a1 z + a0 , ai 6= 0, for at least one i ∈ I, I = {0, 1, 2, . . ., n − 1}, be a polynomial of degree n, with complex coefficients. Then, according to Cauchy’s classical result [1], we have the following theorem. Theorem A Z[f(z)] ⊂ B(η) ⊂ B(1 + a), where η is the unique positive root of the equation Q(x) = 0, Q(x) = xn −
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