On sets with prescribed number of power invariants

V. V. Makeev

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 4, Tome 261 (1999), pp. 40-42

Résumé

Let $A_1,\dots,A_n$ be points in $\mathbb R^d$, $O\in\mathbb R^d$ the fixed point, $p$ the positive integer and $\lambda_1,\dots,\lambda_n$ positive numbers. If the sum $s_p(M)=\sum^n_{i=1}\lambda_i|A_iM|^{2p}$ does not depend on the position of $M$ on the sphere with center at point $O$, then the point system $\{A_1,\dots,A_n\}$ has an invariant of degree $p$ with weight system $\{\lambda,\dots,\lambda_n\}$. Theorem. {\it For given positive integers $d$ and $N$ there exists a point system $\{A_1,\dots,A_n\}\subset\mathbb R^d$ with invariants of degree $p\le N$ with some common weight system $\{\lambda_1,\dots,\lambda_n\}$}.