Abstract
The value of the Vandermonde determinant is optimized over various surfaces, including the sphere, ellipsoid and torus. Lagrange multipliers are used to find a system of polynomial equations which give the local extreme points in its solutions. Using Gröbner basis and other techniques the extreme points are given either explicitly or as roots of polynomials in one variable. The behavior of the Vandermonde determinant is also presented visually in some interesting cases.
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Lundengård, K., Österberg, J. & Silvestrov, S. Optimization of the Determinant of the Vandermonde Matrix and Related Matrices. Methodol Comput Appl Probab 20, 1417–1428 (2018). https://doi.org/10.1007/s11009-017-9595-y
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DOI: https://doi.org/10.1007/s11009-017-9595-y
Keywords
- Vandermonde determinant
- Optimization
- Gröbner basis
- Orthogonal polynomials
- Ellipsoid
- Optimal experiment design
- Homogeneous polynomials