(cf. [BR2, §3]). From $I_H$, we obtain its homogeneous counterpart $I_{H\uparrow}$ as
where $p_\uparrow$ is the leading term of the polynomial $p$, namely the homogeneous polynomial satisfying
The result (3.6) is of interest here since it is easy to identify elements of $I_{H\uparrow}$ in case $H = \exp_\Theta$: If $(p_j)$ are linear homogeneous polynomials for which the union of the corresponding hyperplanes ${x \in \mathbb{R}^s : p_j(x) = c_j}$ (for suitable choices of the constants $c_j$) contains $\Theta$, then $p := \prod_j (p_j - c_j) \in I_H$; hence $\prod_j p_j \in (I_{H\uparrow})$. If we obtain enough of these $p$ to generate all of $I_{H\uparrow}$, then we know by (3.6) Result that $H_↓$ is the joint kernel of all the corresponding differential operators $p(D)$. In fact, since we know from (3.3) that $\dim H_↓ = \dim H$, we can already reach this conclusion when we only know that the $p$ so constructed generate an ideal $J$ of codimension $\le \dim H$.
(3.9) RESULT [BR2]. If the ideal $J$ generated from the leading terms of some polynomials in $I_{H\uparrow}$ has codimension $\le \dim H$, then $J = I_{H\uparrow}$; therefore
In our case, we have identified (in (2.7) Theorem) $\mathcal{P}(X)$ as the joint kernel of the differential operators $(D_h)^{#{X\setminus h}}$ (with $h$ running over $\mathbb{H}(X)$), hence are entitled to conclude that $(\exp_\Theta)↓ = \mathcal{P}(X)$ whenever we can find, for each such $h$, constants $c{j,h}$ so that
\prod_{j=1}^{#\{X\setminus h\}} ((h^\perp, \cdot) - c_{j,h})
vanishes on $\Theta$, and know additionally that $# \Theta \ge \dim \mathcal{P}(X)$.
Just such a pointset is, under certain assumptions on $X$, provided by
with $z \in \mathbb{R}^s \setminus \bigcup_{h \in \mathbb{H}(X)} (h + \mathbb{Z}^s)$. The set $-\nu_X$ comprises the integer points in the support of the shifted box spline $M_X(\cdot + z)$ (cf. [DM2]).