math.NA — Numerical Analysis
Quartic Fekete points on every real simplex
We prove that on every real simplex Bos's quartic interpolation nodes uniquely maximize the absolute Vandermonde determinant, up to ordering, among all configurations of the prescribed size. In dimension four the squared Euclidean norm of the cardinal vector exceeds one at the centroid. We bound its excess by products of the cardinal polynomials for tetrahedral face centers. These products also measure correlations between columns of the evaluation matrix, which give the required loss in the determinant. An elementary consequence of the same corrected squared-norm estimate is Bos's fourth-power bound in every dimension, with equality exactly at the nodes. The corrected estimate follows from cubic perturbations, reduction to at most two distinct positive coordinate values, and coefficient monotonicity in their multiplicities. All polynomial bounds are given explicitly. The common base case is the tetrahedral Fej\'er bound previously stated by Panda and Pal in equivalent optimal-design form. We give an independent proof of that bound and determine all its equality points.