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Abstract

Mourrain \cite{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.