Christoffel transformations for multivariate orthogonal polynomials

Polynomial perturbations of real multivariate measures are discussed and corresponding Christoffel type formulas are found. The 1D Christoffel formula is extended to the multidimensional realm: multivariate orthogonal polynomials are expressed in terms of last quasi-determinants and sample matrices. The coefficients of these matrices are the original orthogonal polynomials evaluated at a set of nodes, which is supposed to be poised. A discussion for the existence of poised sets is given in terms of algebraic hypersurfaces in the complex affine space. Two examples of irreducible perturbations of total degree 1 and 2, for the bivariate product Legendre orthogonal polynomials, are discussed in detail.