samples/texts/102047/page_19.md

i.e., equal to the number of equations. Here, the sums are over all $l \in H(X)$ which contain $k'$, which implies that $X \setminus k'$ is the disjoint union of the sets $(X \cap l) \setminus k'$ and so justifies the second equality.

Now organize the unknowns by $l$ and, within $l$, by $i=0, \dots, \mu(l)$, and order the equations by $j=0, \dots, A$. Then the matrix consists, more precisely, of one block of columns for each $l$, with the $i$th column (in the block for $l$) containing the value at $\beta = \beta_i$ of the $i$th derivative of all the polynomials $B_j^A$, $j=0, \dots, A$, $i=0, \dots, \mu(l)$. Hence our matrix is the transpose of the matrix which occurs in the linear system for the determination of the Bernstein form of the polynomial in $\Pi_A$ which agrees with some function $(\mu(l)+1)$-fold at $\beta = \beta_i$, all $l$. Since such univariate Hermite interpolation is correct (since $\beta_i \neq \beta_{i'}$ for $l \neq l'$), the invertibility of our matrix follows. $\square$

We note that (2.7) Theorem now allows us to conclude that all inequalities appearing in its proof must be equalities. This implies, e.g., that

$$I^{X^{\mathrm{\prime }}}\subset I^X\text{ whenever }{X}^{\mathrm{\prime }}\subset X,I^\{X\text{'}\}\; \backslash subset\; I^X\; \backslash text\{\; whenever\; \}\; X\text{'}\; \backslash subset\; X,$$

and that $p_{h,X'}(D)(I^X \perp) = I^{h \cup X} \perp$. Another immediate consequence is the following

(2.17) COROLLARY [DM3], [DR1].

$$\dim \mathcal{P}(X)=b(X)\mathrm{.}\backslash dim\; \backslash mathcal\{P\}(X)\; =\; b(X).$$

Furthermore,

$$(2.18)d(X):=\min \{\mathrm{\#}(X\setminus h):h\in \mathbb{H}(X)\}(2.18)\; \backslash qquad\; d(X)\; :=\; \backslash min\backslash \{\backslash \#(X\backslash setminus\; h)\; :\; h\; \backslash in\; \backslash mathbb\{H\}(X)\backslash \}$$

is the least degree of the generators of $I^X$; hence, since $I^X \perp = \mathcal{P}(X)$, we have the following.

(2.19) COROLLARY [DR1]. With $d(X)$ as in (2.18),

but

$${\mathrm{\Pi }}_d(X)\subset ̸\mathcal{P}(X)\mathrm{.}\backslash Pi\_d(X)\; \backslash not\backslash subset\; \backslash mathcal\{P\}(X).$$

By its definition, $I^X \perp$ is the set of all polynomials $p \in \Pi$ that satisfy the following condition:

(2.20) Condition. For every $h \in H(X)$ and $v \in \mathbb{R}^s$, $p|{v+h\perp} \subset \Pi{#(X\setminus h)-1}$ (with $h\perp$ the subspace orthogonal to $h$).