We can regard interpolation as another way to compute the normal form of a polynomial modulo an ideal, recalling that the remainder of a polynomial f takes the same values as f on the zeros of the ideal.

Polynomial interpolation is also a very important symbolic-numeric algorithm to create formulas. Instead of expanding the determinant of a symbolic matrix algebraically, via row expansion, we could sample the determinant numerically and reconstruct or evaluate the formula via interpolation.

Interpolation in several variables is not as well posed as univariate interpolation, see the html version of a Maple worksheet.

For some recent applications of the Cayley-Bacharach

• Xue-Zhang Liang, Chun-Mei Lu, and Ren-Zhong Feng: "Properly posed sets of nodes for multivariate Lagrange interpolation in C^s". SIAM J. Numer. Anal. 39(2):587-595, 2001.
• Xue-Zhang Liang, Li-Hong Cui, and Jie-Lin Zhang: "The application of Cayley-Bacharach theorem to bivariate Lagrange interpolation". Journal of Computational and Applied Mathematics Volume 163(1): 177-187, 2004.