Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1 Quadrature Rules and Orthogonal Polynomials
- 2 Cubature Rules: Basics
- 3 Orthogonal Polynomials of Several Variables
- 4 Gauss Cubature Rules
- 5 Lower Bounds for the Number of Nodes
- 6 First Minimal Cubature Rules
- 7 Further Minimal Cubature Rules
- 8 Discrete Fourier Transform and Cubature Rules
- 9 Cubature Rules and Polynomial Ideals
- 10 Epilogue: Two Open Problems
- 11 Addendum: Cubature Rules on the Triangle and Simplex
- References
- Index
9 - Cubature Rules and Polynomial Ideals
Published online by Cambridge University Press: 10 October 2025
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1 Quadrature Rules and Orthogonal Polynomials
- 2 Cubature Rules: Basics
- 3 Orthogonal Polynomials of Several Variables
- 4 Gauss Cubature Rules
- 5 Lower Bounds for the Number of Nodes
- 6 First Minimal Cubature Rules
- 7 Further Minimal Cubature Rules
- 8 Discrete Fourier Transform and Cubature Rules
- 9 Cubature Rules and Polynomial Ideals
- 10 Epilogue: Two Open Problems
- 11 Addendum: Cubature Rules on the Triangle and Simplex
- References
- Index
Summary
The chapter explains another angle of looking at minimal cubature rules, using the language of ideal and variety in algebraic geometry. In essence, the existence of a cubature rule of degree m amounts to the existence of a polynomial ideal generated by m-orthogonal polynomials with zero-dimensional and real variety, and the codimension of the ideal equals the cardinality of the variety. The abstract point of view pinpoints the root of the difficulty in understanding the minimal cubature rules and indicates a theoretical roadmap.
Information
- Type
- Chapter
- Information
- Minimal Cubature RulesTheory and Practice, pp. 213 - 221Publisher: Cambridge University PressPrint publication year: 2025
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