  Polynomial Identity Testing (PIT) is a fundamental computational problem. The
famous depth-$4$ reduction result by Agrawal and Vinay (FOCS 2008) has made PIT
for depth-$4$ circuits an enticing pursuit. A restricted depth-4 circuit
computing a $n$-variate degree-$d$ polynomial of the form $\sum_{i = 1}^{k}
\prod_{j} g_{ij}$, where $\deg g_{ij} \leq \delta$ is called
$\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}$ circuit. On further restricting $g_{ij}$
to be sum of univariates we obtain $\Sigma^{[k]}\Pi\Sigma\wedge$ circuits. The
largely open, special-cases of $\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}$ for
constant $k$ and $\delta$, and $\Sigma^{[k]}\Pi\Sigma\wedge$ have been a source
of many great ideas in the last two decades. For eg. depth-$3$ ideas of Dvir
and Shpilka (STOC 2005), Kayal and Saxena (CCC 2006), and Saxena and Seshadhri
(FOCS 2010 and STOC 2011). Further, depth-$4$ ideas of Beecken, Mittmann and
Saxena (ICALP 2011), Saha, Saxena and Saptharishi (Comput.Compl. 2013), Forbes
(FOCS 2015), and Kumar and Saraf (CCC 2016). Additionally, geometric
Sylvester-Gallai ideas of Kayal and Saraf (FOCS 2009), Shpilka (STOC 2019), and
Peleg and Shpilka (CCC 2020, STOC 2021). Very recently, a subexponential-time
blackbox PIT algorithm for constant-depth circuits was obtained via lower bound
breakthrough of Limaye, Srinivasan, Tavenas (FOCS 2021). We solve two of the
basic underlying open problems in this work.
  We give the first polynomial-time PIT for $\Sigma^{[k]}\Pi\Sigma\wedge$. We
also give the first quasipolynomial time blackbox PIT for both
$\Sigma^{[k]}\Pi\Sigma\wedge$ and $\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}$. A key
technical ingredient in all the three algorithms is how the logarithmic
derivative, and its power-series, modify the top $\Pi$-gate to $\wedge$.
