Multivariate Lagrange and Hermite interpolation are examples of ideal interpolation. More generally an ideal interpolation problem is defined by a set of linear forms, on the polynomial ring, whose kernels intersect into an ideal.

The article addresses multivariate interpolation in the presence of symmetry. Interpolation is a prime tool in algebraic computation while symmetry is a qualitative feature that can be more relevant to a mathematical model than the numerical accuracy of the parameters. The article shows how to exactly preserve symmetry in multivariate interpolation while exploiting it to alleviate the computational cost. We revisit minimal degree and least interpolation with symmetry adapted bases, rather than monomial bases. This allows to construct bases of invariant interpolation spaces in blocks, capturing the inherent redundancy in the computations. We show that the so constructed symmetry adapted interpolation bases alleviate the computational cost of any interpolation problem and automatically preserve any equivariance of this interpolation problem might have.