Arithmetic Circuits with Locally Low Algebraic Rank

In recent years there has been a flurry of activity proving lower bounds for homogeneous depth-4 arithmetic circuits [14, 11, 18, 27], which has brought us very close to statements that are known to imply VP 4 VNP. It is a big question to go beyond homogeneity, and in this paper we make progress towards this by considering depth-4 circuits of low algebraic rank, which are a natural extension of homogeneous depth-4 arithmetic circuits. A depth-4 circuit is a representation of an N-variate, degree n polynomial P as P = Qii Qi2 Qit where the C2,3 are given by their monomial expansion. Homogeneity adds the constraint that for every i E [7], E3 degree(Q,3) = n. We study an extension where, for every i E [T], the algebraic rank of the set of polynomials {Qit, Qi2,., Qit} is at most some parameter k. We call this the class of EE(') EH circuits. Already for k = n, these circuits are a strong generalization of the class of homogeneous depth-4 circuits, where in particular t < n (and hence k < n). We study lower bounds and polynomial identity tests for such circuits and prove the following results. 1. Lower bounds: We give an explicit family of polynomials {Pri} of degree n in N = n (1) variables in VNP, such that any Ell(n) Ell circuit computing P. has size at least exp (S2 (\ log N)). This strengthens and unifies two lines of work: it generalizes the recent exponential lower bounds for homogeneous depth-4 circuits [18, 27] as well as the Jacobian based lower bounds of Agrawal et al. [2] which worked for EII(k)Ell circuits in the restricted setting where T.k < n. 2. Hitting sets: Let EH(k)EH[d] be the class of EH(k)EH circuits with bottom fan-in at most d. We show that if d and k are at most poly(log N), then there is an explicit hitting set for EH(k)EH[d] circuits of size quasipolynomial in N and the size of the circuit. This strengthens a result of Forbes [8] which showed such quasipolynomial sized hitting sets in the setting where d and t are at most poly(log N). A key technical ingredient of the proofs is a result which states that over any field of characteristic zero (or sufficiently large characteristic), up to a translation, every polynomial in a set of algebraically dependent polynomials can be written as a function of the polynomials in the transcendence basis. We believe this may be of independent interest. We combine this with shifted partial derivative based methods to obtain our final results.