Fekete problem

In mathematics, the Fekete problem is, given a natural number N and a real s ≥ 0, to find the points x₁,...,x_N on the 2-sphere for which the s-energy, defined by

: $\sum_{1 \leq i < j \leq N} \|x_i - x_j \|^{-s}$

for s > 0 and by

: $\sum_{1 \leq i < j \leq N} \log \|x_i - x_j \|^{-1}$

for s = 0, is minimal. For s > 0, such points are called s-Fekete points, and for s = 0, logarithmic Fekete points (see {{harvtxt|Saff|Kuijlaars|1997}}).

More generally, one can consider the same problem on the d-dimensional sphere, or on a Riemannian manifold (in which case ||x_i −x_j|| is replaced with the Riemannian distance between x_i and x_j).

The problem originated in the paper by {{harvs|txt|first=Michael|last=Fekete||authorlink=Michael Fekete|year=1923}} who considered the one-dimensional, s = 0 case, answering a question of Issai Schur.

An algorithmic version of the Fekete problem is number 7 on the list of problems discussed by {{harvtxt|Smale|1998}}.

References

{{Citation | last1=Bendito | first1=E. | last2=Carmona | first2=A. | last3=Encinas | first3=A. M. | last4=Gesto | first4=J. M. | last5=Gómez | first5=A. | last6=Mouriño | first6=C. | last7=Sánchez | first7=M. T. | title=Computational cost of the Fekete problem. I. The forces method on the 2-sphere | doi=10.1016/j.jcp.2009.01.021 | mr=2513833 | year=2009 | journal=Journal of Computational Physics | issn=0021-9991 | volume=228 | issue=9 | pages=3288–3306| bibcode=2009JCoPh.228.3288B }}
{{Citation | last1=Fekete | first1=M. | author-link=Michael Fekete|title=Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten | doi=10.1007/BF01504345 | mr=1544613 | year=1923 | journal=Mathematische Zeitschrift | issn=0025-5874 | volume=17 | issue=1 | pages=228–249 | s2cid=186223729 |url=http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002367300| url-access=subscription }}
{{cite journal|mr=1439152|last1=Saff|first1=E. B.|author-link=Edward B. Saff|last2=Kuijlaars|first2=A. B. J.|authorlink2=Arno Kuijlaars|title=Distributing many points on a sphere|journal=Math. Intelligencer|volume=19|year=1997|issue=1|pages=5–11|doi=10.1007/BF03024331|s2cid=122562170}}
{{Citation | last1=Smale | first1=Stephen | author-link = Stephen Smale | title=Mathematical problems for the next century | doi=10.1007/BF03025291 | mr=1631413 | year=1998 | journal=The Mathematical Intelligencer | issn=0343-6993 | volume=20 | issue=2 | pages=7–15| s2cid=1331144 }}

Category:Mathematical analysis

Category:Approximation theory