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1 results

  • 9 - Cubature Rules and Polynomial Ideals

  • Yuan Xu, University of Oregon
  • Book:
    Minimal Cubature Rules
    Published online:
    10 October 2025
    Print publication:
    30 October 2025, pp 213-221

  • 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.