Unisolvence of Symmetric Node Patterns for Polynomial Spaces on the Simplex

Abstract Finite elements with polynomial basis functions on the simplex a symmetric distribution of nodes should have unique representation. Unisolvence not only requires that number equals independent polynomials spanning space given degree, but also Vandermonde matrix controlling their mapping to Lagrange interpolating can be inverted. Here, necessary condition for unisolvence is presented spaces non-decreasing degrees when going from edges and various faces interior simplex. It leads proof conjecture unisolvence, requiring node pattern same as regular