Bivariate Hermite interpolation and numerical curves
Here the authors return to their favorite theme of Hermite interpolation by bivariate algebraic polynomials of degree \(n\) given by a scheme \({\mathcal N}=\{m_1,\dots,m_s;n\}\) where \(m_1,\dots,m_s\) are non-negative integers, \(n\) is the degree of the polynomial and \(s\) is the length of \(\mathcal N\). The set of all schemes is denoted by \(\mathcal J\). The interpolation parameters are the values of a function and its partial derivative up to some order \(n_\nu-1\) at the nodes \(z_\nu=(x_\nu,y_\nu)\), \(\nu=1,\dots,s\), where \(n_\nu\) is the multiplicity of \(z_\nu\). A scheme \({\mathcal N}\in {\mathcal J}\) is called an interpolation scheme if \(\sum^s_{\nu=1} \overline n_\nu=\overline{(n+1)}\), where \(\overline n_\nu= {n_\nu(n_\nu+1)\over 2}\). A scheme \(\mathcal N\) is called regular if \(({\mathcal N},{\mathcal L})\) is correct for at least one node set \(\mathcal L\). It is called singular if it is not correct for any node set. Geometrically the singularity of \(\mathcal N\) means that for any node set \(\mathcal L\) there exists an algebraic curve of degree \(\leq n\) passing through \(\mathcal L\) with multiplicity \(m_\nu\) \((\nu=1,\dots,s)\) at \(z_\nu\). The authors introduce the idea of quadratic transformation and reduction of schemes. A quadratic transformation \(\mathcal N\) into \(N^*\) while reduction takes \(N\) into \(N^*\). The significance of quadratic transformation and reduction is illustrated by the following theorem which was proved in 1992 [\textit{A. A. Akopyan}, \textit{O. V. Gevorkyan}, and \textit{A. A. Saakyan}, Russ. Acad. Sci., Sb., Math. 76, No. 1, 211-223 (1993; Zbl 0782.41001)]: Theorem. Let \({\mathcal N}=\{m_1,\dots,m_s; n\}\in S\). Then (i) if \(n_1+n_2=n+r\geq n+1\), \(n_1\leq n\), \(n_2\leq n\), then the schemes \(N\) and \(N^*\) are both singular or both regular, (ii) if \(n_i+n_j\leq n\), \(1\leq i< j\leq3\), then the schemes \(N\) and \(N^*\) are both regular or both singular. The intersection (product) of two schemes \({\mathcal N}=\{m_1,\dots,m_s, n\}\), \(M=\{m_1,\dots,m_s; m\}\) is denoted by \(\langle N,M\rangle\), where \[ \langle{\mathcal N},M\rangle:=\sum^s_{\nu=1} m_\nu m_r-mn. \] Two schemes \(\mathcal N\) and \(M\) an equivalent (denoted by \({\mathcal N}\sim M\)) if the scheme \(\mathcal N\) can be obtained from \(\mathcal M\) by one or more of the following operations: quadratic transformation, rearrangement of numbers, adding zero numbers. Those schemes which are equivalent to the scheme \(\{1,1;1\}\) are called prime curves (PC). As an application of the various results obtained they prove the following numerical analogue of the well-known Bézout's theorem. Theorem 5.7. Let \(\mathcal N\), \(M\) be numerical curves with \(\langle{\mathcal N},M\rangle>0\). Then \(\mathcal N\) and \(M\) have a common prime curve in their canonical decomposition, i.e. there is an \({\mathcal A}\in \text{PC}\) such that \(\langle{\mathcal A},{\mathcal N}\rangle>0\), \(\langle{\mathcal A},M\rangle>0\). For more details the reader should read the paper.
Hermite interpolation by bivariate algebraic polynomials
scientific article; zbMATH DE number 894210