## Slightly Improved Lower Bounds for Homogeneous Multi-r-ic Formulas of Small-Depth

### Authors

Suryajith Chillara

## Abstract

Kayal, Saha and Tavenas (Theory of Computing, 2018), showed that any bounded-depth homogeneous formula of bounded individual degree (bounded by $r$) that computes an explicit polynomial over $n$ variables must have size $\exp\left(\Omega\left(\frac{1}{r}\left(\frac{n}{4}\right)^{1/\Delta}\right)\right)$ for all depths $\Delta \leq O\left({\frac{\log n}{\log r+ \log\log n}}\right)$. In this article we show an improved size lower bound of $\exp\left(\Omega\left(\frac{\Delta}{r}\left({\frac{nr}{2}}\right)^{1/\Delta}\right)\right)$ for all depths $\Delta \leq O\left({\frac{\log n}{\log r}}\right)$, and for the same explicit polynomial.

In comparison to Kayal, Saha and Tavenas (Theory of Computing, 2018)

1. our results give superpolynomial lower bounds in a wider regime of depth $\Delta$, and
2. for all $\Delta \in \left[\omega(1), o\left(\frac{\log n}{\log r}\right)\right]$ our lower bound is asymptotically better.

This improvement is due to a finer product decomposition of general homogeneous formulas of bounded-depth. This follows from an adaptation of a new \emph{product decomposition} for bounded-depth multilinear formulas shown by Chillara, Limaye and Srinivasan (SIAM Journal of Computing, 2019).