We establish a new extremal property of the classical Chebyshev polynomials in the context of best rank-one approximation of tensors. We also give some necessary conditions for a tensor to be a minimizer of the ratio of spectral and Frobenius norms.

We study the average condition number for polynomial eigenvalues of collections of matrices drawn from various random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with Gaussian entries are very well-conditioned on the average. 1. Polynomial Root Finding: I is the set of univariate real polynomials of degree d, O = R and V = {(f, ζ) : f (ζ) = 0}. 2. Polynomial System Solving, which we can see as the homogeneous multivariate version of Polynomial Root Finding: I is the projective space of (dense or structured) systems of n real homogeneous polynomials of degrees d 1 , . . . , d n in variables x 0 , . . . , x n , O = RP n and V = {(f, ζ) : f (ζ) = 0}. 3. EigenValue Problem: I = R n×n , O = R and V = {(A, λ) : det(A − λ Id) = 0}.