Physical Sciences and Mathematics Commons^™

On The Location Of Critical Points Of Polynomials, Branko Ćurgus, Vania Mascioni

Mathematics Faculty Publications

Given a polynomial p of degree n ≥ 2 and with at least two distinct roots let Z(p) = { z: p(z) = 0}. For a fixed root α ∈ Z(p) we define the quantities ω(p, α) := min (formula) and (formula). We also define ω (p) and τ (p) to be the corresponding minima of ω (p,α) and τ (p,α) as α runs over Z(p). Our main results show that the ratios τ (p,α)/ω (p,α) and τ (p)/ω (p) are bounded above and below by constants that only depend on the degree of p. In particular, …