Tag · 3 problems · 2 fields
Open problems tagged polynomials
3 problems in the Math Lab index touch polynomials, spanning Number Theory, Dynamics & Analysis. Each carries a precise statement, an honest status, and a computational line of attack.
Casas-Alvero conjectureNumber Theoryuntouched
If a complex polynomial of degree n shares a root with each of its derivatives P′, P″, …, P^(n−1), then it must be c(x − a)^n — a power of a single linear factor.
Lehmer's Mahler measure problemNumber Theoryuntouched
Is there a gap above 1 in Mahler measures of integer polynomials? Lehmer's degree-10 polynomial has measure ≈ 1.176280818; the question is whether any non-cyclotomic integer polynomial does better.
a.k.a. Lehmer's conjecture
Sendov's conjectureDynamics & Analysisuntouched
If all roots of a polynomial lie in the closed unit disk, then within distance 1 of each root lies a critical point (root of the derivative).