Bivariate truncated powers: Complete intersection decompositions and the spline representation

Cone polynomials, also known as volume polynomials and/or spline polynomials, are the polynomials that appear in the local structure of the truncated powers, hence in the local structure of any derived construction such as box splines, simplex splines, character formulæ and moment maps. We provide a fresh look at bivariate cone polynomials. Two main principles underlie our approach here. The first is that understanding the truncated powers does not necessarily require us to analyse as a whole the ideal of differential operators that defines the cone polynomials. Instead, we create a host of much simpler ideals, and analyse each of them separately, with each of them making a contribution of a single cone polynomial. This concept is formalized under the notion of Complete Intersection Decomposition (CID) of ideals. The second observation is that the coefficients of the cone polynomial in a suitable monomial representation are piecewise-analytic. In a sense, one can say that not only cone polynomials underlie the structure of truncated powers, but truncated powers underlie the structure of cone polynomials, too. This surprising cycle must go beyond piecewise-polynomials: the coefficients of cone polynomials are rarely piecewise-polynomial: in general they are only piecewise-exponential. AMS (MOS) Subject Classifications: 41A15, 41A63, 52C35, 13A02, 13C40, 13F20, 13N10, 13P25, 05B20, 05B35, 39A12, 47F05, 47L20, 52B40.