# Article

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Keywords:
Birkhoff interpolation; 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.
References:
[1] Barbeau E.J.: Pell's Equation. Springer, New York, 2003. MR 1949691 | Zbl 1030.11008
[2] Lorentz R.A.: Multivariate Birkhoff Interpolation. Lecture Notes in Mathematics, 1516, Springer, Berlin-Heidelberg, 1992. MR 1222648 | Zbl 0760.41002
[3] Gasca M., Maeztu J.I.: On Lagrange and Hermite interpolation in $\mathbb R^n$. Numer. Math. 39 (1982), 1--14. DOI 10.1007/BF01399308 | MR 0664533
[4] Stillwell J.: Elements of number theory. Undergraduate Texts in Mathematics, Springer, New York, 2003. MR 1944957 | Zbl 1112.11002

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