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.

Abstract—We present a polynomial-time deterministic algorithm for testing whether constant-read multilinear arithmetic formulae are identically zero. In such a formula each variable occurs only a constant number of times and each subformula computes a multilinear polynomial. Before our work no subexponential-time deterministic algorithm was known for this class of formulae. We also present a deterministic algorithm that works in a blackbox fashion and runs in quasi-polynomial time in general, and polynomial time for constant depth. Finally, we extend our results and allow the inputs to be replaced with sparse polynomials. Our results encompass recent deterministic identity tests for sums of a constant number of read-once formulae, and for multilinear depth-four circuits. Keywords-arithmetic circuit; bounded-depth circuit; de-randomization; polynomial identity testing; I.

A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic black-box identity testing algorithm for (high-degree) univariate polynomials would imply a lower bound on the arithmetic complexity of the permanent. The lower bounds that are known to follow from derandomization of (low-degree) multivariate identity testing are weaker. To obtain a lower bound for the permanent it would be sufficient to derandomize identity testing for polynomials of a very specific norm: sums of products of sparse polynomials with sparse coefficients. This observation leads to new versions of the Shub-Smale τ-conjecture on integer roots of univariate polynomials. In particular, we show that a lower bound for the permanent would follow if one could give a polynomial upper bound on the number of real roots of sums of products of sparse polynomials (Descartes ’ rule of signs gives such a bound for sparse polynomials and products thereof). In fact the same lower bound would follow even if one could only prove a slightly superpolynomial upper bound on the number of real roots. This is a consequence of a new result on reduction to depth 4 for arithmetic circuits which we establish in a companion paper. We also show that an even weaker bound on the number of real roots would suffice to obtain a lower bound on the size of depth 4 circuits computing the permanent. These results suggest the intriguing possibility that tools from real analysis might be brought to bear on a longstanding open problem: what is the arithmetic complexity of the permanent polynomial?

Polynomial identity testing and arithmetic circuit lower bounds are two central questions in algebraic complexity theory. It is an intriguing fact that these questions are actually related. One of the authors of the present paper has recently proposed a “real τ-conjecture” which is inspired by this connection. The real τ-conjecture states that the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded. It implies a superpolynomial lower bound on the size of arithmetic circuits computing the permanent polynomial. In this paper we show that the real τ-conjecture holds true for a restricted class of sums of products of sparse polynomials. This result yields lower bounds for a restricted class of depth-4 circuits: we show that polynomial size circuits from this class cannot compute the permanent, and we also give a deterministic polynomial identity testing algorithm for the same class of circuits.

Polynomial identity testing (PIT) problem is known to be challenging even for constant depth arithmetic circuits. In this work, we study the complexity of two special but natural cases of identity testing- first is a case of depth-3 PIT, the other of depth-4 PIT. Our first problem is a vast generalization of: Verify whether a bounded top fanin depth-3 circuit equals a sparse polynomial (given as a sum of monomial terms). Formally, given a depth-3 circuit C, having constant many general product gates and arbitrarily many semidiagonal product gates, test if the output of C is identically zero. A semidiagonal product gate in C computes a product of the form m · ∏b, where m is a i=1 ℓei i monomial, ℓi is an affine linear polynomial and b is a constant. We give a deterministic polynomial time test, along with the computation of leading monomials of semidiagonal circuits over local rings. The second problem is on verifying a given sparse polynomial factorization, which is a classical question (von zur Gathen, FOCS 1983): Given multivariate sparse polynomials f, g1,..., gt explicitly, check if f = ∏ t i=1 gi. For the special case when every gi is a sum of univariate polynomials, we give a deterministic polynomial time test. We characterize the factors of such gi’s and even show how to test the divisibility of f by the powers of such polynomials. The common tools used are Chinese remaindering and dual representation. The dual representation of polynomials (Saxena, ICALP 2008) is a technique to express a product-of-sums of univariates as a sum-ofproducts of univariates. We generalize this technique by combining it with a generalized Chinese remaindering to solve these two problems (over any field).

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We consider the problem of interpolating an unknown multivariate polynomial with coefficients taken from a finite field or as numerical approximations of complex numbers. Building on the recent work of Garg and Schost, we improve on the best-known algorithm for inter-polation over large finite fields by presenting a Las Vegas randomized algorithm that uses fewer black box evaluations. Using related tech-niques, we also address numerical interpolation of sparse polynomials with complex coefficients, and provide the first provably stable algo-rithm (in the sense of relative error) for this problem, at the cost of modestly more evaluations. A key new technique is a randomization which makes all coefficients of the unknown polynomial distinguish-able, producing what we call a diverse polynomial. Another departure from most previous approaches is that our algorithms do not rely on root finding as a subroutine. We show how these improvements affect the practical performance with trial implementations. 1

According to the real τ-conjecture, the number of real roots of a sum of products of sparse univariate polynomials should be polynomi-ally bounded in the size of such an expression. It is known that this conjecture implies a superpolynomial lower bound on the arithmetic circuit complexity of the permanent. In this paper, we use the Wronksian determinant to give an upper bound on the number of real roots of sums of products of sparse poly-nomials of a special form. We focus on the case where the number of distinct sparse polynomials is small, but each polynomial may be re-peated several times. We also give a deterministic polynomial identity testing algorithm for the same class of polynomials. Our proof techniques are quite versatile; they can in particular be applied to some sparse geometric problems that do not originate from arithmetic circuit complexity. The paper should therefore be of interest to researchers from these two communities (complexity theory and sparse polynomial systems).

In this paper we study algebraic branching programs (ABPs) with restrictions on the order and the number of reads of variables in the program. An ABP is given by a layered directed acyclic graph with source s and sink t, whose edges are labeled by variables taken from the set {x1, x2,..., xn} or field constants. It computes the sum of weights of all paths from s to t, where the weight of a path is defined as the product of edge-labels on the path. Given a permutation pi of the n variables, for a pi-ordered ABP (pi-OABP), for any directed path p from s to t, a variable can appear at most once on p, and the order in which variables appear on p must respect pi. One can think of OABPs as being the arithmetic analogue of ordered binary decision diagrams (OBDDs). We say an ABP A is of read r, if any variable appears at most r times in A. Our main result pertains to the polynomial identity testing problem, i.e. the problem of deciding whether a given n-variate polynomial is identical to the zero polynomial or not. We prove that over any field F, and in the black-box model, i.e. given only query access to the polynomial, read r pi-OABP computable polynomials can be tested in DTIME[2O(r log r·log2 n log logn)]. In case F is a finite field, the above time bound holds provided the identity testing algorithm is allowed

We associate to each Boolean language complexity class C the algebraic class a·C consisting of families of polynomials {fn} for which the evaluation problem over Z is in C. We prove the following lower bound and randomness-to-hardness results: 1. If polynomial identity testing (PIT) is in NSUBEXP then a·NEXP does not have poly size constant-free arithmetic circuits. 2. a·NEXP RP does not have poly size constant-free arithmetic circuits. 3. For every fixed k, a·MA does not have arithmetic circuits of size nk. Items 1 and 2 strengthen two results due to Kabanets and Impagliazzo [7]. The third item improves a lower bound due to Santhanam [11]. We consider the special case low-PIT of identity testing for (constant-free) arithmetic circuits with low formal degree, and give improved hardness-to-randomness trade-offs that apply to this case. Combining our results for both directions of the hardness-randomness connection, we demonstrate a case where derandomization of PIT and proving lower bounds are equivalent. Namely, we show that low-PIT ∈ i.o-NTIME[2no(1)]/no(1) if and only if there exists a family of multilinear polynomials in a·NE/lin that requires constant-free arithmetic circuits of super-polynomial size and formal degree.