Ideal Interpolation: Mourrain’s Condition Vs D-invariance

Mourrain [Mo] characterizes those linear projectors on a finite-dimensional polynomial space that can be extended to an ideal projector, i.e., a projector on polynomials whose kernel is an ideal. This is important in the construction of normal form algorithms for a polynomial ideal. Mourrain’s characterization requires the polynomial space to be ‘connected to 1’, a condition that is implied by D-invariance in case the polynomial space is spanned by monomials. We give examples to show that, for more general polynomial spaces, D-invariance and being ‘connected at 1’ are unrelated, and that Mourrain’s characterization need not hold when his condition is replaced by D-invariance. By definition (see [Bi]), ideal interpolation is provided by a linear projector whose kernel is an ideal in the ring Π of polynomials (in d real (IF = IR) or complex (IF = C) variables). The standard example is Lagrange interpolation; the most general example has been called ‘Hermite interpolation’ (in [M] and [Bo]) since that is what it reduces to in the univariate case. Ideal projectors also occur in computer algebra, as the maps that associate a polynomial with its normal form with respect to an ideal; see, e.g., [CLO]. It is in this latter context that Mourrain [Mo] poses and solves the following problem. Among all linear projectors N on