Abstract
In this paper, we deeply research Lagrange interpolation by polynomial in several variables and give an application of Cayley–Bacharach theorem for it. Particularly on the sufficiently intersected algebraic manifold (or SIAM, for short), we introduce a general method of constructing properly posed set of nodes (or PPSN, for short) for Lagrange interpolation, namely the superposition interpolation process. Then we give an equivalent condition about a PPSN along a SIAM. Further we introduce a relation between the sufficiently intersected algebraic hypersurfaces and H-basis. At the end of this paper, we use the extended Cayley–Bacharach theorem to resolve some problems of Lagrange interpolation along the zero-dimensional and one-dimensional algebraic manifolds.
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