Geodesic
On sets with prescribed number of power invariants
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 4, Tome 261 (1999), pp. 40-42 
V. V. Makeev. On sets with prescribed number of power invariants. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 4, Tome 261 (1999), pp. 40-42. http://geodesic.mathdoc.fr/item/ZNSL_1999_261_a2/
@article{ZNSL_1999_261_a2,
     author = {V. V. Makeev},
     title = {On sets with prescribed number of power invariants},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {40--42},
     year = {1999},
     volume = {261},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_261_a2/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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 $\{\lambda_1,\dots,\lambda_n\}$}.