On the Location of Critical Points of Polynomials

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 { |α − v| : v ∈ Z(p) \ {α} } and τ(p, α) := min { |α − v| : v ∈ Z(p′) \ {α} } . 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, we prove that (1/n)ω(p) ≤ τ(p) ≤ ( 1/2 sin(π/n) ) ω(p), for any polynomial of degree n.