We prove super-polynomial lower bounds for low-depth arithmetic circuits using the shifted partials measure [Gupta-Kamath-Kayal-Saptharishi, CCC 2013], [Kayal, ECCC 2012] and the affine projections of partials measure [Garg-Kayal-Saha, FOCS 2020], [Kayal-Nair-Saha, STACS 2016]. The recent breakthrough work of Limaye, Srinivasan and Tavenas [FOCS 2021] proved these lower bounds by proving ... more >>>

We study the polynomial equivalence problem for orbits of read-once arithmetic formulas (ROFs). Read-once formulas have received considerable attention in both algebraic and Boolean complexity and have served as a testbed for developing effective tools and techniques for analyzing circuits. Two n-variate polynomials f, g \in \mathbb{F}[\mathbf{x}] are equivalent, denoted ... more >>>

The orbit of an n-variate polynomial f(\mathbf{x}) over a field \mathbb{F} is the set \mathrm{orb}(f) := \{f(A\mathbf{x}+\mathbf{b}) : A \in \mathrm{GL}(n,\mathbb{F}) \ \mathrm{and} \ \mathbf{b} \in \mathbb{F}^n\}. This paper studies explicit hitting sets for the orbits of polynomials computable by certain well-studied circuit classes. This version of the hitting set ... more >>>

We show an \widetilde{\Omega}(n^{2.5}) lower bound for general depth four arithmetic circuits computing an explicit n-variate degree \Theta(n) multilinear polynomial over any field of characteristic zero. To our knowledge, and as stated in the survey by Shpilka and Yehudayoff (FnT-TCS, 2010), no super-quadratic lower bound was known for depth four ... more >>>