Revisiting Lower Bounds for Multi-r-ic Depth Four Circuits

9 Multi-r-ic arithmetic circuits are arithmetic circuits where the individual degree of every variable, in 10 polynomials computed at every node, is restricted to be at most r. This is a natural generalization 11 of the multilinear restriction. 12 Raz and Yehudayoff (CC, 2008) introduced a "full rank polynomial" and proved strong lower 13 bounds against the multilinear circuits (of small depth) and formulas computing it, using the method 14 of partial derivatives (cf. Nisan-Wigderson (CC, 1997), Raz (ToC 2006)). It is a natural question to 15 ask if these techniques and polynomial constructions be extended beyond the multilinear setting, to 16 multi-r-ic circuit models and thus prove lower bounds against multi-r-ic circuits and in particular, 17 depth four multi-r-ic circuits. 18 In this paper, we prove a superpolynomial yet quasipolynomial lower bound on the size of depth 19 four multi-r-ic circuits computing an explicit "full rank polynomial" by just using the method of 20 partial derivatives. 21 Recently Kayal, Saha and Tavenas (ToC 2018) proved an exponential lower bound on the size of 22 depth four multi-r-ic circuits computing explicit polynomials using a stronger complexity measure. 23 Our proof however borrows no elements from theirs and is for a completely different polynomial. 24 The proof strategy is inspired by Saptharishi’s proof of an exponential lower bound for depth three 25 multi-r-ic circuits (cf. Chapter 14, Ramprasad’s survey, 2019). 26 The point of this paper is to retrospectively extend our understanding of the power of the method 27 of Partial Derivatives and thus achieve superpolynomial lower bounds, and its limits to realize the 28 need to use stronger complexity measures to prove exponential bounds. 29 2012 ACM Subject Classification Theory of Computation → Algebraic Complexity Theory 30