Sets with a Prescribed Number of Power Invariants

Abstract

Let A₁,...,A_n be points in $\mathbb{R}^d$, let ${O} \in {\mathbb{R}}^{d}$ be a fixed point, let p be a positive integer, and let λ₁,...,λ_n be positive real numbers. If the $s_p (M)= \sum_{i=1}^n \lambda_i |A_iM|^{2p}$ does not depend on the position of M on a sphere with center O, then one says that the point system {A₁,...,A_n} has an invariant of degree p with weight system {λ,...,λ_n}. It is proved that for arbitrary positive integers d and N there exists a finite point system $\left\{ {{A}_{1} , \ldots ,{A}_{n} } \right\} \subset {\mathbb{R}}^{d}$ having invariants of degrees p=1,...,N with common positive weight system {λ₁,...,λ_n}. Bibliography: 2 titles.