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Given two polynomials $f$ and $g$, let $f_i = \partial_i f$ and $g_j = \partial_j g$ (directional partial derivatives). Consider the following $\star$-product acting on quotient rings/algebras given by \begin{align} \mathbb{C} \{ z_1, \dots, z_n \} / J \star \mathbb{C} \{z_1, \dots, z_m \} / K \mapsto \mathbb{C} \{ z_{1} \dots, z_{nm} \} / L, \end{align} where the ideals $J = \langle f_1, \dots, f_n \rangle $, $K = \langle g_1, \dots, g_m \rangle $ and $L = \langle (f \boxtimes g)_1, \dots, (f \boxtimes g)_{nm} \rangle$, where $f \boxtimes g$ is defined as the corresponding polynomial whose exponent matrix is the Kronecker product of the exponent matrices of $f$ and $g$. For example, if $f = z_1^a + z_2^b$ and $g = z_1^c z_2^d + z_2^e$, then \begin{align} E_f = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}, \, E_{g} = \begin{pmatrix} c & d \\ 0 & e \end{pmatrix}, \quad \text{and} \quad E_{f \, \boxtimes \, g} = E_{f} \otimes E_{g} = \begin{pmatrix} ac & ad & 0 & 0 \\ 0 & ae & 0 & 0 \\ 0 & 0 & bc & bd \\ 0 & 0 & 0 & be \end{pmatrix} \end{align} and, therefore, $f \boxtimes g = z_1^{ac} z_2^{ad} + z_2^{ae} + z_3^{bc} z_4^{bd} + z_4^{be}$. Here, $\mathbb{C}\{ \cdot \}$ denotes the polynomial ring of convergent power series (about the origin). It seems to me the operation $\star$ is related to the tensor product, and I am somehow reminded of the Segre embedding of projective spaces with this operation. What is the $\star$-product?

If $A_f$ and $A_g$ denote the algebras above with, then the algebras admit the filtration $A_f = \bigoplus_i A_{f,i}$ and $A_g = \bigoplus_j A_{g,j}$ by degree. What is the structure of the direct sum decomposition of $A_f \star A_g$?

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It is unclear to me how $L$ is generally a subset of ${\bf C}[z_i\otimes w_j]$. –  anon Feb 5 '13 at 2:05
    
My edit should clarify things a bit. –  user02138 Feb 5 '13 at 3:27

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