Existence of Solutions to Systems of Underdetermined Equations and Spherical Designs

Abstract

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 are close to extremal (maximum determinant) points and have the optimal order $O(t^2)$ for the number of points. An error bound for the computed spherical designs is provided.