Proving explicit lower bounds on the size of algebraic formulas is a long-standing open problem in the area of algebraic complexity theory. Recent results in the area (e.g. a lower bound against constant-depth algebraic formulas due to Limaye, Srinivasan, and Tavenas (FOCS 2021)) have indicated a way forward for attacking this question: show that we can convert a general algebraic formula to a homogeneous algebraic formula with moderate blow-up in size, and prove strong lower bounds against the latter model. Here, a homogeneous algebraic formula F for a polynomial P is a formula in which all subformulas compute homogeneous polynomials. In particular, if P is homogeneous of degree d, F does not contain subformulas that compute polynomials of degree greater than d. We investigate the feasibility of the above strategy and prove a number of positive and negative results in this direction.

Classical results of Brent, Kuck and Maruyama (IEEE Trans.Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth Oplog sq with only a polynomial blow-up in size.In this paper, we consider a fine-grained version of this result depending on the degree of the polynomial computed by the algebraic formula.Given a homogeneous algebraic formula of size s computing a polynomial P of degree d, we show that P can also be computed by an (unbounded fan-in) algebraic formula of depth Oplog dq and size polypsq.Our proof shows that this result also holds in the highly restricted setting of monotone, non-commutative algebraic formulas.This improves on previous results in the regime when d is small (i.e., d " s op1q ).In particular, for the setting of d " Oplog sq, along with a result of Raz (STOC 2010, JACM 2013), our result implies the same depth reduction even for inhomogeneous formulas.This is particularly interesting in light of recent algebraic formula lower bounds, which work precisely in this "low-degree" and "low-depth" setting.We also show that these results cannot be improved in the monotone setting, even for commutative formulas.

Inspired by Nisan’s characterization of noncommutative complexity (Nisan 1991), we study different notions of nonnegative rank, associated complexity measures and their link with monotone computations. In particular we answer negatively an open question of Nisan asking whether nonnegative rank characterizes monotone noncommutative complexity for algebraic branching programs. We also prove a rather tight lower bound for the computation of elementary symmetric polynomials by algebraic branching programs in the monotone setting or, equivalently, in the homogeneous syntactically multilinear setting.

We study the arithmetic complexity of iterated matrix multiplication. We show that any multilinear homogeneous depth-4 arithmetic formula computing the product of $d$ generic matrices of size $n \times n$, $\mathrm{IMM}_{n,d}$, has size $n^{\Omega(\sqrt{d})}$ as long as $d = n^{O(1)}$. This improves the result of Nisan and Wigderson [Comput. Complexity, 6 (1997), pp. 217--234] for depth-4 set-multilinear formulas. We also study $\Sigma\Pi^{[O(d/t)]}\Sigma\Pi^{[t]}$ formulas, which are depth-4 formulas with the stated bounds on the fan-ins of the $\Pi$ gates. A recent depth reduction result of Tavenas [Lecture Notes in Comput. Sci. 8087, 2013, pp. 813--824] shows that any $n$-variate degree $d = n^{O(1)}$ polynomial computable by a circuit of size $\mathop{\mathrm{poly}}(n)$ can also be computed by a depth-4 $\Sigma\Pi^{[O(d/t)]}\Sigma\Pi^{[t]}$ formula of top fan-in $n^{O(d/t)}$. We show that any such formula computing $\mathrm{IMM}_{n,d}$ has top fan-in $n^{\Omega({d/t})}$, proving the optimality of Tavenas' result. This also strengthens a result of Kayal, Saha, and Saptharishi [Proceedings of STOC, 2014, pp. 146--153], which gives a similar lower bound for an explicit polynomial in VNP.

We study the arithmetic complexity of iterated matrix multiplication. We show that any multilinear homogeneous depth 4 arithmetic formula computing the product of d generic matrices of size n × n, IMMn,d, has size nΩ(√d) as long as d = nO(1). This improves the result of Nisan and Wigderson (Computational Complexity, 1997) for depth 4 set-multilinear formulas.

We consider the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of monomials. We show that these problems are complete for subclasses of the counting hierarchy which had few or no known natural complete problems. We also study these questions for circuits computing multilinear polynomials.

We show that proving lower bounds in algebraic models of computntion may not be easier than in the standard %ring machine model.For instance, a superpolynomial lower bound on the size of an algebraic circuit solving the real knapsack problem (or on the running time of a real 'Brring machine) would imply a separation of P from PSPACE.A more general result relates parallel complexity classes in boolean and real models of computation.We also propose a few problems in algebraic complexity and topological complexity,