samples/texts/102047/page_4.md

Assume now that $X$ is unimodular, i.e., the columns of $X$ are from $\mathbb{Z}^s\setminus0$ and every $B \in \mathbb{B}(X)$ has determinant $\pm1$. For such unimodular $X$, the observations made in [R; §4] (especially before the proof of Theorem 4.1 and in the proof of Corollary 4.2) confirm the existence, for each $h \in \mathbb{H}(X)$, of consecutive integers $c_{h,j}$ so that $\nu_X$ is contained in the union of the hyperplanes

$$⟨h^{\perp },\cdot ⟩=c_{h,j},j=1,\dots ,\mathrm{\#}X\mathrm{\backslash }h\mathrm{.}\backslash langle\; h^\{\backslash perp\},\; \backslash cdot\; \backslash rangle\; =\; c\_\{h,j\},\; \backslash quad\; j\; =\; 1,\; \backslash dots,\; \backslash \#X\backslash backslash\; h.$$

Indeed, fixing $h \in \mathbb{H}(X)$ and taking the normal $h^\perp$ to be a relatively prime integer vector implies that $\langle h^\perp, x \rangle = 1$ for every $x \in X\backslash h$ since $X$ is unimodular. By choosing the signs of the columns of $X$ appropriately (which amounts to a shift in $M_X$), we can achieve that $\langle h^\perp, x \rangle > 0$ for all $x \in X\backslash h$; hence

$$c_h:=\sum _{x\in X\mathrm{\backslash }h}⟨h^{\perp },x⟩=\mathrm{\#}X\mathrm{\backslash }h\mathrm{.}c\_h\; :=\; \backslash sum\_\{x\; \backslash in\; X\; \backslash backslash\; h\}\; \backslash langle\; h^\{\backslash perp\},\; x\; \backslash rangle\; =\; \backslash \#X\; \backslash backslash\; h.$$

Consequently, with $z$ chosen so as to satisfy $c_h - 1 < \langle h^\perp, z \rangle < c_h$, $\nu_X(z)$ must lie in the union of the hyperplanes

$$⟨h^{\perp },\cdot ⟩=j,j=0,\dots ,c_h-1.\backslash langle\; h^\{\backslash perp\},\; \backslash cdot\; \backslash rangle\; =\; j,\; \backslash quad\; j\; =\; 0,\; \backslash dots,\; c\_h\; -\; 1.$$

Moreover, $#\nu_X = \dim \mathcal{P}(X)$, because of (2.17) Corollary and the following.

(3.11) RESULT [DM2]. If $X$ is unimodular, then

$$\mathrm{\#}{\nu }_X=b(X)\mathrm{.}\backslash \#\backslash nu\_X\; =\; b(X).$$

This establishes the following theorem.

(3.12) THEOREM. If $X$ is unimodular, then $(\exp_{\nu_X})_\downarrow = \mathcal{P}(X)$. In particular, $\nu_X$ is correct for $\mathcal{P}(X)$, and $\mathcal{P}(X)$ is of least degree among all polynomial spaces for which $\nu_X$ is correct.

Note that only the inequality $#\nu_X \ge b(X)$ was needed in the proof of (3.12) Theorem. As a matter of fact, the converse inequality is a consequence of the theorem.

4. The duality between $\mathcal{H}(X)$ and $\mathcal{P}(X)$. The ideal $I_X$ and its kernel $\mathcal{P}(X)$ are intimately related to another ideal $I_X$ and its kernel $\mathcal{H}(X)$, which play a fundamental role in the theory of box splines. In the following, we review some of the basics about $I_X$ and $\mathcal{H}(X)$ and draw several connections between the two settings.