  We prove super-polynomial lower bounds for low-depth arithmetic circuits
using the shifted partials measure [Gupta-Kamath-Kayal-Saptharishi, CCC 2013],
[Kayal, ECCC 2012] and the affine projections of partials measure
[Garg-Kayal-Saha, FOCS 2020], [Kayal-Nair-Saha, STACS 2016]. The recent
breakthrough work of Limaye, Srinivasan and Tavenas [FOCS 2021] proved these
lower bounds by proving lower bounds for low-depth set-multilinear circuits. An
interesting aspect of our proof is that it does not require conversion of a
circuit to a set-multilinear circuit, nor does it involve a random restriction.
We are able to upper bound the measures for homogeneous formulas directly,
without going via set-multilinearity. Our lower bounds hold for the iterated
matrix multiplication as well as the Nisan-Wigderson design polynomials. We
also define a subclass of homogeneous formulas which we call unique parse tree
(UPT) formulas, and prove superpolynomial lower bounds for these. This
generalizes the superpolynomial lower bounds for regular formulas in
[Kayal-Saha-Saptharishi, STOC 2014], [Fournier-Limaye-Malod-Srinivasan, STOC
2014].
