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\notag
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$
Useful tricks
Identities
Newton identity
Newton identity provides a bridge between $\Sym_k$ and $\Pow_k$. For our interest in arithmetic circuits, Newton identity tells us that $\Sym_d$ has a $\Sigma\Pi\Sigma\Lambda$ circuit of size $2^{O(\sqrt{d})}\cdot\poly(n)$. Note that the resulting circuit is homogeneous . Recall that there is $n^{\Omega(d)}$ lower bound for $\Sigma\Lambda\Sigma$ circuits computing $\Sym_d$.
For any $k\in[n]$, we have
\begin{eqnarray*}
\Sym_k &=\frac{1}{k!}
\begin{vmatrix}
\Pow_1 & 1 & 0 & \cdots & 0 \\
\Pow_2 & \Pow_1 & 2 & \cdots & 0 \\
\vdots & \vdots & \ddots & \cdots & 0 \\
\Pow_{k-1} & \Pow_{k-2} & \cdots & \Pow_1 & k-1 \\
\Pow_k & \Pow_{k-1} & \cdots & \Pow_2 & \Pow_1
\end{vmatrix}.
\end{eqnarray*}
Specifically, one can write $x_1x_2\cdots x_k=\sum_{\mathbf{a}:\ \sum_iia_i=k}\alpha_{\mathbf{a}}(\Pow_1)^{a_1}(\Pow_2)^{a_2}\cdots(\Pow_k)^{a_k}$.
Applications:
Any non-homogeneous $\Sigma\Pi\Sigma$ circuit can be converted to homogeneous $\Sigma\Pi\Sigma\Lambda\Sigma$ circuit with $2^{O(\sqrt{d})}\cdot\poly(n)$ blow-up.
Ryser-Fischer identity
Ryser-Fischer identity expresses multilinear monomial into sums of the power of sum. To our interest in arithmetic circuits, it implies that one can replace monomial $x_1x_2\cdots c_d$ with a size $2^d$ $\Sigma\Lambda\Sigma$ circuit. Note that the circuit is homogeneous .
For any $d\in\N$, we have
\begin{equation}
x_1x_2\cdots x_d = \sum_{S\subseteq[d]}(-1)^{d-\card{S}}(\sum_{i\in S}x_i)^d.
\end{equation}
The proof is based on simple inclusion-exclusion principle so here we omit the details.
Duality trick
The duality trick converts $\Lambda\Sigma\Lambda$ circuit to $\Sigma\Pi\Sigma$ circuit.
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