Problems
Everything in the archive: the problem as it was posed, what has been attempted against it, and who checked each attempt. The mark down the left of the list says who has looked — a person, a machine, or nobody yet. Human reviews and machine checks are counted separately and are never added together.
8 problems
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Let p be a complex polynomial of degree n ≥ 2 whose zeros all lie in the closed unit disk. Then for every zero a of p, there exists a critical point ζ of p such that |ζ-a| ≤ 1. This is the standard Sendov statement and exactly matches the…
Crouzeix conjectured in 2004 that for every square complex matrix A and every polynomial p, lVert p(A)rVert ≤ 2 max_z ∈ W(A) |p(z)|, where W(A) is the numerical range of A - that is, the numerical range is a 2-spectral set. Crouzeix proved…
For which lattice parameters does a totally positive window function generate a Gabor frame? Gröchenig and Stöckler initiated the program in 2013; this paper gives the complete characterization, together with a Kadets-type theorem for…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
For unit-modulus complex numbers z_i, let p_n(z)=∏_i≤ n(z-z_i) and M_n=max_|z|=1|p_n(z)|. Erdős's prize question: is there c>0 with ∑_k≤ n M_k > n^1+c?
For f(z) = ∏_i=1^n (z - z_i) with all |z_i| ≤ 1, let ρ(f) be the radius of the largest disc contained in z : |f(z)| < 1. Is ρ(f) ≫ 1/n? The worst case is now known to be Θ(1/n), with the explicit bound ρ(f) ≥ (log 2)/n.
Let p be a complex polynomial of degree n≥2 whose zeros all lie in the closed unit disk. For every zero a of p, there is a critical point ζ satisfying |ζ-a|<1, except when |a|=1 and p is a nonzero scalar multiple of z^n-a^n.