Existence of Solutions to Systems of Underdetermined Equations and Spherical Designs

  title={Existence of Solutions to Systems of Underdetermined Equations and Spherical Designs},
  author={Xiaojun Chen and Robert S. Womersley},
  journal={SIAM J. Numer. Anal.},
This paper is concerned with proving the existence of solutions to an underdetermined system of equations and with the application to existence of spherical $t$-designs with $(t+1)^2$ points on the unit sphere $S^2$ in $R^3$. We show that the construction of spherical designs is equivalent to solution of underdetermined equations. A new verification method for underdetermined equations is derived using Brouwer’s fixed point theorem. Application of the method provides spherical $t$-designs which… 

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