The Chinese remainder problem, multivariate interpolation, and Gröbner bases

Let K[X] be a multivariate polynomial ring over a field K, I],..., Im ideals in K[X], U ~ X. Using a single Gr6bner basis in an extension ring of KIX1, we solve the followir.w . .. problems effectively. Given jl,. . . . im G K[X], put Af : np=l(h + fk). (1) Decide whether Af n K[U] # 0 and if so, construct some element of Af n K[U]. (2) For given g c K[U], decide whether g 6 Af. (3) Construct all elements of Af n K[U]. Taking for 1~ a suitable vanishing ideal of some parametrized hypersurface in Kn (1 ~ k < m), this solves a generalized Hermite and spline interpolation problem. It is well-known and easy to see that the classical Lagrange interpolation formula is an instance of the Chinese remainder theorem in the PID K[X], where the residue classes whose intersection is computed are of the form T-i+(X – Ci ). The aim of this paper is to show that with the use of Grobner bases, this situation carries over to multivariat e polynomials to a surpringly large extent. We first discuss the computation of the intersection of finitely many residue classes, which is precisely what the Chinese remainder theorem does. This can then be used to interpolate point data by considering residue classes of the form r~ + ((XI – c~l), . . . . (Xn – c~n)). Instead of just prescribing values at points, one can actually prescribe “value functions” on parametrized hypersurfaces. Furthermore, one can, to some extent, prescribe derivatives too. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 0 199’1 ACM 0-89791-437-6191 /0006 /0064... S 1.50 64 1. A Chinese remainder theorem Let K be a field. For arbitrary finite set V of indeterminates we denote by K[V] the polynomial ring over K in the indeterminate in V and by T’(V) the set of all terms in K[V]. For an introduction to the theory of Grobner bases and further references, we refer the reader to [2]. Nowlet X= {X1,..., X.} (n ~ 1) and U ~ X. For given 2 ~ m G IN, we let Y1, . . . . Ym be new indeterminate, and we set K[X, Y] = K[X U Y]. We fix an admissible term order < on T(X U Y) such that t < Y for all t E T’(U) and Y c (X U Y) \ U; in other words, the only requirement on the term order < is that U << Y U (X \ U). With every m-tuple (11, . . . . lm) of ideals in K[X] and every rn-tuple f = (~1, . . . . fm) of polynomials in K[X] we associate the ideal 1’ generated by YII1 U . . .uYm Imu{Yl+. ..+ Y1}l} in K[X, Y], the element