Abstract — We study ΣΠΣ(k) circuits, i.e., depth three arithmetic circuits with top fanin k. We give the first deterministic polynomial time blackbox identity test for ΣΠΣ(k) circuits over the field Q of rational numbers, thus resolving a question posed by Klivans and Spielman (STOC 2001). Our main technical result is a structural theorem for ΣΠΣ(k) circuits that compute the zero polynomial. In particular we show that if a ΣΠΣ(k) circuit C = ∑ i∈[k] Ai

An arithmetic read-once formula (ROF for short) is a formula (a circuit in which the fanout of every gate is at most 1) in which the operations are {+, ×} and such that every input variable labels at most one leaf. In this paper we study the problems of identity testing and reconstruction of read-once formulas. the following are some of the results that we obtain. 1. Given k ROFs in n variables, over a field F, we give a deterministic (non black-box) algorithm that checks whether they sum to zero or not. The running time of the algorithm is n O(k2). 2. We give an n O(d+k2) time deterministic algorithm for checking whether a black box holding the sum of k depth d ROFs in n variables computes the zero polynomial. In other words, we provide a hitting set of size n O(d+k2) for the sum of k depth d ROFs. If |F | is too small then we make queries from a polynomial size extension field. This implies a deterministic algorithm that runs in time n O(d) for the reconstruction of depth d ROFs. 3. We give a hitting set of size exp ( Õ( √ n + k 2)) for the sum of k ROFs (without depth restrictions). In particular this implies a sub-exponential time deterministic algorithm for

In this paper we give the first deterministic polynomial time algorithm for testing whether a diagonal depth-3 circuit C(x1,..., xn) (i.e. C is a sum of powers of linear functions) is zero. We also prove an exponential lower bound showing that such a circuit will compute determinant or permanent only if there are exponentially many linear functions. Our techniques generalize to the following new results: 1. Suppose we are given a depth-3 circuit (over any field F) of the form: C(x1,..., xn):= k� i=1 ℓ ei,1 i,1 · · · ℓei,s i,s where, the ℓi,j’s are linear functions living in F[x1,..., xn]. We can test whether C is zero deterministically in poly (nk, max{(1 + ei,1) · · · (1 + ei,s) | 1 � i � k}) field operations. This immediately gives a deterministic poly(nk2 d) time identity test for general depth-3 circuits of degree d. 2. We prove that if the above circuit C(x1,..., xn) computes the determinant � (or permanent) of an m × m formal matrix with a “small ” s = o then � m log m k = 2 Ω(m). Our lower bounds work for all fields F. (Previous exponential lower bounds for depth-3 only work for nonzero characteristic.) 3. We present applications of our ideas to depth-4 circuits and an exponentially faster identity test for homogeneous diagonal circuits (deterministically in poly(n k log(d)) field operations over finite fields).

In this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits. More formally, if there exists an explicit polynomial f(x1,..., xm) that cannot be computed by a depth d arithmetic circuit of small size then there exists an efficient deterministic algorithm to test whether a given depth d − 8 circuit is identically zero or not (assuming the individual degrees of the tested circuit are not too high). In particular, if we are guaranteed that the tested circuit computes a multilinear polynomial then we can perform the identity test efficiently. To the best of our knowledge this is the first hardness-randomness tradeoff for bounded depth arithmetic circuits. The above results are obtained using the the arithmetic Nisan-Wigderson generator of [KI04] together with a new theorem on bounded depth circuits, which is the main technical contribution of our work. This theorem deals with polynomial equations of the form P (x1,..., xn, y) ≡ 0 and shows that if P has a circuit of depth d and size s and if the polynomial f(x1,..., xn) satisfies P (x1,..., xn, f(x1,..., xn)) ≡ 0 then f has a circuit of depth d + 3 and size O(s · r + m r), where m is the total degree of f and r is the degree of y in P.

In this paper we consider the problem of determining whether an unknown arithmetic circuit, for which we have oracle access, computes the identically zero polynomial. This problem is known as the black-box polynomial identity testing (PIT) problem. Our focus is on polynomials that can be written in the form f(¯x) = ∑ k i=1 hi(¯x) · gi(¯x), where each hi is a polynomial that depends on only ρ linear functions, and each gi is a product of linear functions (when hi = 1, for each i, then we get the class of depth-3 circuits with k multiplication gates, also known as ΣΠΣ(k) circuits, but the general case is much richer). When maxi(deg(hi · gi)) = d we say that f is computable by a ΣΠΣ(k, d, ρ) circuit. We obtain the following results. 1. A deterministic black-box identity testing algorithm for ΣΠΣ(k, d, ρ) circuits that runs in quasi-polynomial time (for ρ = polylog(n + d)). In particular this gives the first black-box quasi-polynomial time PIT algorithm for depth-3 circuits with k multiplication gates. 2. A deterministic black-box identity testing algorithm for read-k ΣΠΣ circuits (depth-3 circuits where each variable appears at most k times) that runs in time n 2O(k2). In particular

In this paper we consider the problem of constructing a small arithmetic circuit for a polynomial for which we have oracle access. Our focus is on n-variate polynomials, over a finite field F, that have depth-3 arithmetic circuits with two multiplication gates of degree d. We obtain the following results: 1. Multilinear case: When the circuit is multilinear (multiplication gates compute multilinear polynomials) we give an algorithm that outputs, with probability 1 − o(1), all the depth-3 circuits with two multiplication gates computing the same polynomial. The running time of the algorithm is poly(n, |F|). 2. General case: When the circuit is not multilinear we give a quasi-polynomial (in n, d, |F|) time algorithm that outputs, with probability 1 − o(1), a succinct representation of the polynomial. In particular, if the depth-3 circuit for the polynomial is not of small depth-3 rank (namely, after removing the g.c.d. of the two multiplication gates, the remaining linear functions span a not too small linear space) then we output the depth-3 circuit itself. In case that the rank is small we output a depth-3 circuit with a quasi-polynomial number of multiplication gates. Our proof technique is new and relies on the factorization algorithm for multivariate black-box polynomials, on lower bounds on the length of linear locally decodable codes with 2 queries, and on a theorem regarding the structure of identically zero depth-3 circuits with four multiplication gates.

Abstract—We show that the rank of a depth-3 circuit (over any field) that is simple, minimal and zero is at most O(k 3 log d). The previous best rank bound known was 2 O(k2) (log d) k−2 by Dvir and Shpilka (STOC 2005). This almost resolves the rank question first posed by Dvir and Shpilka (as we also provide a simple and minimal identity of rank Ω(k log d)). Our rank bound significantly improves (dependence on k exponentially reduced) the best known deterministic blackbox identity tests for depth-3 circuits by Karnin and Shpilka (CCC 2008). Our techniques also shed light on the factorization pattern of nonzero depth-3 circuits, most strikingly: the rank of linear factors of a simple, minimal and nonzero depth-3 circuit (over any field) is at most O(k 3 log d). The novel feature of this work is a new notion of maps between sets of linear forms, called ideal matchings, used to study depth-3 circuits. We prove interesting structural results about depth-3 identities using these techniques. We believe that these can lead to the goal of a deterministic polynomial time identity test for these circuits. I.

Abstract. We study the problem of identity testing for depth-3 circuits of top fanin k and degree d (called ΣΠΣ(k, d) identities). We give a new structure theorem for such identities. A direct application of our theorem improves the known deterministic d kO(k) time black-box identity test over rationals (Kayal & Saraf, FOCS 2009) to one that takes d O(k2)-time. Our structure theorem essentially says that the number of independent variables in a real depth-3 identity is very small. This theorem settles affirmatively the stronger rank conjectures posed by Dvir & Shpilka (STOC 2005) and Kayal & Saraf (FOCS 2009). Our techniques provide a unified framework that actually beats all known rank bounds and hence gives the best running time (for every field) for black-box identity tests. Our main theorem (almost optimally) pins down the relation between higher dimensional Sylvester-Gallai theorems and the rank of depth-3 identities in a very transparent manner. The existence of this was hinted at by Dvir & Shpilka (STOC 2005), but first proven, for reals, by Kayal & Saraf (FOCS 2009). We introduce the concept of Sylvester-Gallai rank bounds for any field, and show the intimate connection between this and depth-3 identity rank bounds. We also prove the first ever theorem about high dimensional Sylvester-Gallai configurations over any field. Our proofs and techniques are very different from previous results and devise a very interesting ensemble of combinatorics and algebra. The latter concepts are ideal theoretic and involve a new Chinese remainder theorem. Our proof methods explain the structure of any depth-3 identity C: there is a nucleus of C that forms a low rank identity, while the remainder is a high dimensional Sylvester-Gallai configuration. 1.

In this paper we give reconstruction algorithms for depth-3 arithmetic circuits with k multiplication gates (also known as ΣΠΣ(k) circuits), where k = O(1). Namely, we give an algorithm that when given a black box holding a ΣΠΣ(k) circuit C over a field F as input, makes queries to the black box (possibly over a polynomial sized extension field of F) and outputs a circuit C ′ computing the same polynomial as C. In particular we obtain the following results. 1. When C is a multilinear ΣΠΣ(k) circuit (i.e. each of its multiplication gates computes a multilinear polynomial) then our algorithm runs in polynomial time (when k is a constant) and outputs a multilinear ΣΠΣ(k) circuits computing the same polynomial. 2. In the general case, our algorithm runs in quasi polynomial time and outputs a generalized depth-3 circuit (a notion that is defined in the paper) with k multiplication gates. In fact, this algorithm works in the slightly more general case where the black box holds a generalized depth-3 circuits. Prior to this work there were reconstruction algorithms for several different models of bounded depth circuits: the well studied class of depth-2 arithmetic circuits (that compute sparse polynomials)

Using ideas from automata theory we design a new efficient (deterministic) identity test for the noncommutative polynomial identity testing problem (first introduced and studied in [RS05, BW05]). More precisely, given as input a noncommutative circuit C(x1, · · · , xn) computing a polynomial in F{x1, · · · , xn} of degree d with at most t monomials, where the variables xi are noncommuting, we give a deterministic polynomial identity test that checks if C ≡ 0 and runs in time polynomial in d, n, |C|, and t. The same methods works in a black-box setting: Given a noncommuting black-box polynomial f ∈ F{x1, · · · , xn} of degree d with t monomials we can, in fact, reconstruct the entire polynomial f in time polynomial in n, d and t. Indeed, we apply this idea to the reconstruction of black-box noncommuting algebraic branching programs (the ABPs considered by Nisan in [N91] and Raz-Shpilka in [RS05]). Assuming that the black-box model allows us to query the ABP for the output at any given gate then we can reconstruct an (equivalent) ABP in deterministic polynomial time. Finally, we turn to commutative identity testing and explore the complexity of the problem when the coefficients of the input polynomial come from an arbitrary finite commutative ring with unity whose elements are uniformly encoded as strings and the ring operations are given by an oracle. We show that several algorithmic results for polynomial identity testing over fields also hold when the coefficients come from such finite rings. 1