Macaulay Dual Space

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For $\alpha \in \bN^n$ and $y\in\bC^n$, define $\p^\alpha = \frac{1}{\alpha!}\frac{\p^{|\alpha|}}{\p x^\alpha}$ and $\p_y^\alpha: R \rightarrow \bC$ as

(1)
\begin{align} \p^\alpha_y(g) = (\p^\alpha g)(y). \end{align}

Let $D_y = \<\partial^\alpha_y~|~\alpha \in \bN^n \>_{\bC}$
be the vector space of differential functionals at $y$ and let $D_y^{(d)}\subset D_y$ be the (finite-dimensional) subset of functionals of order at most $d$.

Define the (Macaulay) {\em dual space}} (a.k.a. {\em inverse system})
of differential functionals that vanish at $y$
for an ideal $I\subset\bC[x]=\bC[x_1,\dots,x_N]$ as

(2)
\begin{align} D_y[I] = \{q\in D_y~|~q(g)=0\hbox{~for all~}g\in I\}. \end{align}

(a) We call $D_y^{(d)}[I] = D_y[I]\cap D_y^{(d)}$ the truncated dual space of order $d$.
Show that

(3)
\begin{align} D_0^{k}[I] \cong \ker M^{(d)}_0, \end{align}

where $M^{(d)}_0$ is the Macaulay matrix of order $d$.

(b) Compute $D_0^{(d)}[I]$ for

(4)
\begin{align} I = \< x_1^3+x_1 x_2^2, x_1 x_2^2 + x_2^3, x_1^2x_2+x_1x_2^2\> \end{align}

and all $d\in\bN$.

(c) For any ideal $I$, show that $D_0[I]$ is closed under

(5)
\begin{align} \der_i:D_0 \to D_0,\ \ \ \der_i \p^\alpha = \p^\alpha/\p_i, \end{align}

the derivations with respect to $\p_i$; prove that

(6)
\begin{align} D_0[I:x_i] = \der_i D_0[I], \end{align}

for $i=1,\ldots,n$.