The Limits of Depth Reduction for Arithmetic Formulas: It's All About the Top Fan-In

In recent years, a very exciting and promising method for proving lower bounds for arithmetic circuits has been proposed. This method combines the method of depth reduction developed in the works of Agrawal and Vinay [FOCS, IEEE, Piscataway, NJ, 2008, pp. 67--75], Koiran [Theoret. Comput. Sci., 448 (2012), pp. 56--65], and Tavenas [Inform. and Comput., 240 (2015), pp, 2--11], and the use of the shifted partial derivative complexity measure developed in the works of Kayal [Electronic Colloquium on Computational Complexity, TR12-081, 2012] and Gupta et al. [J. ACM, 61 (2014), 33]. These results inspired a flurry of other beautiful results and strong lower bounds for various classes of arithmetic circuits, in particular a recent work of Kayal, Saha, and Saptharishi [STOC, ACM, New York, 2014, pp. 146--153] showing superpolynomial lower bounds for regular arithmetic formulas via an improved depth reduction for these formulas. It was left as an intriguing question if these methods could prove superpolynomial l...