Intersections of sequences of ideals generated by polynomials
The main result of the paper gives the reduced Gröbner basis with respect to an admissible term of order type \(\omega\) of the intersection of a descending chain of polynomial ideals. It is also shown that the result does not hold for any admissible order. Using this and previous results of \textit{P. Gianni, B. Trager} and \textit{G. Zacharias} [J. Symb. Comput. 6, No. 2/3, 149-167 (1988; Zbl 0667.13008)], one can compute a generating set for arbitrary intersections. The authors apply their main theorem to Lagrange interpolation on algebraic sets, obtaining the ideal \(I\) of an algebraic set as the intersection of zero-dimensional ideals whose affine Hilbert functions converge towards the affine Hilbert function of \(I\).
admissible term order of order type \(\omega\)
Lagrange interpolation on algebraic sets
scientific article; zbMATH DE number 1308223
scientific article