Ideal Projectors of Type Partial Derivative and Their Perturbations

In this paper, we verify Carl de Boor's conjecture on ideal projectors for real ideal projectors of type partial derivative by proving that there exists a positive $\eta\in \mathbb{R}$ such that a real ideal projector of type partial derivative $P$ is the pointwise limit of a sequence of Lagrange projectors which are perturbed from $P$ up to $\eta$ in magnitude. Furthermore, we present an algorithm for computing the value of such $\eta$ when the range of the Lagrange projectors is spanned by the Gr\"{o}bner \'{e}scalier of their kernels w.r.t. lexicographic order.