On Ascertaining Inductively the Dimension of the Joint Kernel of Certain Commuting Linear Operators

Given an index set X, a collection IB of subsets of X (all of the same cardinality), and a collection f`xgx2X of commuting linear maps on some linear space, the family of linear operators whose joint kernel K = K(IB) is sought consists of all `A :=Qa2A `a with A any subset of X which intersects every B 2 IB. The goal is to establish conditions, on IB and `, which ensure that dimK(IB) = X B2IBdimK(fBg); or, at least, one or the other of the two inequalities contained in this equality. Concrete instances of this problem arise in box spline theory, and speci c conditions on ` were given by Dahmen and Micchelli for the case that IB consists of the bases of a matroid. We give a new approach to this problem, and establish the inequalities and the equality under various rather weak conditions on IB and `. These conditions involve the solvability of certain linear systems of the form `b? = b, b 2 B, with B 2 IB, and the existence of `placeable' elements of X, i.e., of x 2 X for which every B 2 IB not containing x has all but one element in common with some B0 2 IB containing x. AMS (MOS) Subject Classi cations: primary: 47A50; secondary: 05B353, 41A63, 35G05, 47D03.