Slightly improved lower bounds for homogeneous formulas of bounded depth and bounded individual degree.

•A finer product decomposition for Homogeneous Formulas of small depth.•A finer product decomposition for small-depth homogeneous multi-r-ic formulas.•We give superpolynomial lower bounds for homogeneous multi-r-ic formulas of small-depth.•For all Δ∈[ω(1),o(log⁡nlog⁡r)] our lower bound is asymptotically better than Kayal et al. [STOC 2014].