We study limitations of polynomials computed by depth two circuits built over read-once polynomials (ROPs) and depth three syntactically multi-linear formulas.
We prove an exponential lower bound for the size of the $\Sigma\Pi^{[N^{1/30}]}$ arithmetic circuits built over syntactically multi-linear $\Sigma\Pi\Sigma^{[N^{8/15}]}$ arithmetic circuits computing a product of variable disjoint linear forms on $N$ variables. We extend the result to the case of $\Sigma\Pi^{[N^{1/30}]}$ arithmetic circuits built over ROPs of unbounded depth, where the number of variables with $+$ gates as a parent in an proper sub formula is bounded by $N^{1/2+1/30}$. We show that the same lower bound holds for the permanent polynomial. Finally we obtain an exponential lower bound for the sum of ROPs computing a polynomial in ${ VP}$ defined by Raz and Yehudayoff~\cite{RY09}.


Our results demonstrate a class of formulas of unbounded depth with exponential size lower bound against the permanent and can be seen as an exponential improvement over the multilinear formula size lower bounds given by Raz~\cite{Raz04a} for a sub-class of multi-linear and non-multi-linear formulas.
Our proof techniques are built on the one developed by Raz~\cite{Raz04a} and later extended by Kumar et. al.~\cite{KMS13} and are based on non-trivial analysis of ROPs under random partitions. Further, our results exhibit strengths and limitations of the lower bound techniques introduced by Raz~\cite{Raz04a}.

Errors in Proofs of Lemma 7 and 15 of the previous version have been fixed in this version

We study limitations of polynomials computed by depth two circuits built over read-once polynomials (ROPs) and depth three syntactically multi-linear formulas.
We prove an exponential lower bound for the size of the $\Sigma\Pi^{[N^{1/30}]}$ arithmetic circuits built over syntactically multi-linear $\Sigma\Pi\Sigma^{[N^{8/15}]}$ arithmetic circuits computing a product of variable disjoint linear forms on $N$ variables. We extend the result to the case of $\Sigma\Pi^{[N^{1/30}]}$ arithmetic circuits built over ROPs of unbounded depth, where the number of variables with $+$ gates as a parent in an proper sub formula is bounded by $N^{1/2+1/30}$. We show that the same lower bound holds for the permanent polynomial. Finally we obtain an exponential lower bound for the sum of ROPs computing a polynomial in ${ VP}$ defined by Raz and Yehudayoff~\cite{RY09}.


Our results demonstrate a class of formulas of unbounded depth with exponential size lower bound against the permanent and can be seen as an exponential improvement over the multilinear formula size lower bounds given by Raz~\cite{Raz04a} for a sub-class of multi-linear and non-multi-linear formulas.
Our proof techniques are built on the one developed by Raz~\cite{Raz04a} and later extended by Kumar et. al.~\cite{KMS13} and are based on non-trivial analysis of ROPs under random partitions. Further, our results exhibit strengths and limitations of the lower bound techniques introduced by Raz~\cite{Raz04a}.