On Computing Multilinear Polynomials Using Multi- <i>r</i> -ic Depth Four Circuits

In this article, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits which polynomial computed at every node has a bound on individual degree r ≥ 1 with respect to all its variables (referred as multi- -ic circuits). The goal study is make progress towards proving superpolynomial lower bounds for general polynomials, by better value increases. Recently, Kayal, Saha and Tavenas (Theory Computing, 2018) showed that any arithmetic circuit bounded an explicit n O (1) d must have size least ( / 1.1 ) Ω(√ . This bound, however, deteriorates It natural question ask if can prove does not deteriorate increases, or holds larger regime increasing values , albeit specific instance = but wider range Formally, large enough integers small constant η, show there exists Θ (log 2 such ≤ η exp(Ω(log )). improvement obtained suitably adapting measure Kayal et al. 2018). adaptation inspired used (SIAM J. 2017).