Normal bivariate Birkhoff interpolation schemes and Pell equation

summary:Finding the normal Birkhoff interpolation schemes where the interpolation space and the set of derivatives both have a given regular ``shape'' often amounts to number-theoretic equations. In this paper we discuss the relevance of the Pell equation to the normality of bivariate schemes for different types of ``shapes''. In particular, when looking at triangular shapes, we will see that the conjecture in Lorentz R.A., {\it Multivariate Birkhoff Interpolation\/}, Lecture Notes in Mathematics, 1516, Springer, Berlin-Heidelberg, 1992, is not satisfied, and, at the same time, we will describe the complete solution

UR Birkhoff interpolation with rectangular sets of derivatives

summary:In this paper we characterize the regular UR Birkhoff interpolation schemes ($U=$ uniform, $R=$ rectangular sets of nodes) with rectangular sets of derivatives, and beyond