A symbolic determinant under rank-one restriction computes a polynomial of the form det(A0 + A1y1 + … + Anyn), where A0, A1, …, An are square matrices over a field F and rank(Ai) = 1 for each i ∈ [n]. This class of polynomials has been studied extensively, since the work of Edmonds (1967), in the context of linear matroids, matching, matrix completion and polynomial identity testing. We study the following learning problem for this class: Given black-box access to an n-variate polynomial f = det(A0 + A1y1 + … + Anyn), where A0, A1, …, An are unknown square matrices over F and rank(Ai) = 1 for each i ∈ [n], find a square matrix B0 and rank-one square matrices B1, …, Bn over F such that f = det(B0 + B1y1 + … + Bnyn). In this work, we give a randomized poly(n) time algorithm to solve this problem; the algorithm can be derandomized in quasi-polynomial time. To our knowledge, this is the first efficient learning algorithm for this class. As the above-mentioned class is known to be equivalent to the class of read-once determinants (RODs), we will refer to the problem as learning RODs. An ROD computes the determinant of a matrix whose entries are field constants or variables and every variable appears at most once in the matrix. Thus, the class of RODs is a rare example of a well-studied class of polynomials that admits efficient proper learning.

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 ∈

An s-sparse polynomial has at most s monomials with nonzero coefficients. The Equivalence Testing problem for sparse polynomials (ETsparse) asks to decide if a given polynomial f is equivalent to (i.e., in the orbit of) some s-sparse polynomial. In other words, given f ∈ 𝔽[𝐱] and s ∈ ℕ, ETsparse asks to check if there exist A ∈ GL(|𝐱|, 𝔽) and 𝐛 ∈ 𝔽^|𝐱| such that f(A𝐱 + 𝐛) is s-sparse. We show that ETsparse is NP-hard over any field 𝔽, if f is given in the sparse representation, i.e., as a list of nonzero coefficients and exponent vectors. This answers a question posed by Gupta, Saha and Thankey (SODA 2023) and also, more explicitly, by Baraskar, Dewan and Saha (STACS 2024). The result implies that the Minimum Circuit Size Problem (MCSP) is NP-hard for a dense subclass of depth-3 arithmetic circuits if the input is given in sparse representation. We also show that approximating the smallest s₀ such that a given s-sparse polynomial f is in the orbit of some s₀-sparse polynomial to within a factor of s^{1/3 - ε} is NP-hard for any ε > 0; observe that s-factor approximation is trivial as the input is s-sparse. Finally, we show that for any constant σ ≥ 6, checking if a polynomial (given in sparse representation) is in the orbit of some support-σ polynomial is NP-hard. Support of a polynomial f is the maximum number of variables present in any monomial of f. These results are obtained via direct reductions from the 3-SAT problem.

The orbit of an n-variate polynomial f(𝐱) over a field 𝔽 is the set {f(A𝐱+𝐛) : A ∈ GL(n,𝔽) and 𝐛 ∈ 𝔽ⁿ}. In this paper, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of: 1) Low-individual-degree polynomials computable by commutative ROABPs. This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials. 2) Multilinear polynomials computable by constant-width ROABPs. This implies a quasi-polynomial time hitting set for the orbits of the family {IMM_{3,d}}_{d ∈ ℕ}, which is complete for arithmetic formulas. 3) Polynomials computable by constant-depth, constant-occur formulas. This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in, and also polynomial-time hitting sets for the orbits of the power symmetric and the sum-product polynomials. 4) Polynomials computable by occur-once formulas.

We show an Ω̃(n^2.5) lower bound for general depth four arithmetic circuits computing an explicit n-variate degree-Θ(n) multilinear polynomial over any field of characteristic zero. To our knowledge, and as stated in the survey [Amir Shpilka and Amir Yehudayoff, 2010], no super-quadratic lower bound was known for depth four circuits over fields of characteristic ≠ 2 before this work. The previous best lower bound is Ω̃(n^1.5) [Abhijat Sharma, 2017], which is a slight quantitative improvement over the roughly Ω(n^1.33) bound obtained by invoking the super-linear lower bound for constant depth circuits in [Ran Raz, 2010; Victor Shoup and Roman Smolensky, 1997]. Our lower bound proof follows the approach of the almost cubic lower bound for depth three circuits in [Neeraj Kayal et al., 2016] by replacing the shifted partials measure with a suitable variant of the projected shifted partials measure, but it differs from [Neeraj Kayal et al., 2016]’s proof at a crucial step - namely, the way "heavy" product gates are handled. Loosely speaking, a heavy product gate has a relatively high fan-in. Product gates of a depth three circuit compute products of affine forms, and so, it is easy to prune Θ(n) many heavy product gates by projecting the circuit to a low-dimensional affine subspace [Neeraj Kayal et al., 2016; Amir Shpilka and Avi Wigderson, 2001]. However, in a depth four circuit, the second (from the top) layer of product gates compute products of polynomials having arbitrary degree, and hence it was not clear how to prune such heavy product gates from the circuit. We show that heavy product gates can also be eliminated from a depth four circuit by projecting the circuit to a low-dimensional affine subspace, unless the heavy gates together account for Ω̃(n^2.5) size. This part of our argument is inspired by a well-known greedy approximation algorithm for the weighted set-cover problem.

We develop algorithms for writing a polynomial as sums of powers of low degree polynomials in the non-degenerate case. This problem generalizes symmetric tensor decomposition which is widely studied, having many applications in machine learning. Our algorithm for this more general problem allows us to solve the moment problem for mixtures of zero-mean Gaussians in the nondegenerate case. Our algorithm is based on a scheme for obtaining a learning algorithm for an arithmetic circuit model from lower bound for the same model, provided certain non-degeneracy conditions hold. The scheme reduces the learning problem to the problem of decomposing two vector spaces under the action of a set of linear operators, where the spaces and the operators are derived from the input circuit and the complexity measure used in a typical lower bound proof. The non-degeneracy conditions are certain restrictions on how the spaces decompose. Such a scheme is present in a rudimentary form in an earlier work of Kayal and Saha. Here, we make it more general and detailed, and potentially applicable to learning other circuit models. An exponential lower bound on the representation above is known using the shifted partials measure. However, the number of linear operators in shifted partials is exponential and also the non-degeneracy condition emerging out of this measure is unlikely to be satisfied by a random such circuit when the number of variables is large with respect to the degree. We bypass this hurdle by proving a lower bound (which is nearly as strong as the previous bound) using a novel variant of the partial derivatives measure, namely affine projections of partials (APP). The non-degeneracy conditions appearing from this new measure are satisfied by a random circuit of the above kind. The APP measure could be of independent interest for proving other lower bounds.

Equivalence testing for a polynomial family {g_m} over a field F is the following problem: Given black-box access to an n-variate polynomial f(x), where n is the number of variables in g_m, check if there exists an A in GL(n,F) such that f(x) = g_m(Ax). If yes, then output such an A. The complexity of equivalence testing has been studied for a number of important polynomial families, including the determinant (Det) and the two popular variants of the iterated matrix multiplication polynomial: IMM_{w,d} (the (1,1) entry of the product of d many w $\times$ w symbolic matrices) and Tr-IMM_{w,d} (the trace of the product of d many w $\times$ w symbolic matrices). The families Det, IMM and Tr-IMM are VBP-complete, and so, in this sense, they have the same complexity. But, do they have the same equivalence testing complexity? We show that the answer is 'yes' for Det and Tr-IMM (modulo the use of randomness). The result is obtained by connecting the two problems via another well-studied problem called the full matrix algebra isomorphism problem (FMAI). In particular, we prove the following: 1. Testing equivalence of polynomials to Tr-IMM_{w,d}, for d$\geq$ 3 and w$\geq$ 2, is randomized polynomial-time Turing reducible to testing equivalence of polynomials to Det_w, the determinant of the w $\times$ w matrix of formal variables. (Here, d need not be a constant.) 2. FMAI is randomized polynomial-time Turing reducible to equivalence testing (in fact, to tensor isomorphism testing) for the family of matrix multiplication tensors {Tr-IMM_{w,3}}. These in conjunction with the randomized poly-time reduction from determinant equivalence testing to FMAI [Garg,Gupta,Kayal,Saha19], imply that FMAI, equivalence testing for Tr-IMM and for Det, and the $3$-tensor isomorphism problem for the family of matrix multiplication tensors are randomized poly-time equivalent under Turing reductions.

The determinant polynomial Det_n(x) of degree n is the determinant of a n x n matrix of formal variables. A polynomial f is equivalent to Det_n(x) over a field F if there exists a A in GL(n^2,F) such that f = Det_n(A * x). Determinant equivalence test over F is the following algorithmic task: Given black-box access to a f in F[x], check if f is equivalent to Det_n(x) over F, and if so then output a transformation matrix A in GL(n^2,F). In (Kayal, STOC 2012), a randomized polynomial time determinant equivalence test was given over F = C. But, to our knowledge, the complexity of the problem over finite fields and over Q was not well understood. In this work, we give a randomized poly(n,log |F|) time determinant equivalence test over finite fields F (under mild restrictions on the characteristic and size of F). Over Q, we give an efficient randomized reduction from factoring square-free integers to determinant equivalence test for quadratic forms (i.e. the n=2 case), assuming GRH. This shows that designing a polynomial-time determinant equivalence test over Q is a challenging task. Nevertheless, we show that determinant equivalence test over Q is decidable: For bounded n, there is a randomized polynomial-time determinant equivalence test over Q with access to an oracle for integer factoring. Moreover, for any n, there is a randomized polynomial-time algorithm that takes input black-box access to a f in Q[x] and if f is equivalent to Det_n over Q then it returns a A in GL(n^2,L) such that f = Det_n(A * x), where L is an extension field of Q and [L : Q] <= n. The above algorithms over finite fields and over Q are obtained by giving a polynomial-time randomized reduction from determinant equivalence test to another problem, namely the full matrix algebra isomorphism problem. We also show a reduction in the converse direction which is efficient if n is bounded. These reductions, which hold over any F (under mild restrictions on the characteristic and size of F), establish a close connection between the complexity of the two problems. This then leads to our results via applications of known results on the full algebra isomorphism problem over finite fields (Rónyai, STOC 1987 and Rónyai, J. Symb. Comput. 1990) and over Q (Ivanyos {et al}., Journal of Algebra 2012 and Babai {et al}., Mathematics of Computation 1990).

In a Nisan-Wigderson design polynomial (in short, a design polynomial), every pair of monomials share a few common variables. A useful example of such a polynomial, introduced in [Neeraj Kayal et al., 2014], is the following: NW_{d,k}({x}) = sum_{h in F_d[z], deg(h) <= k}{ prod_{i=0}^{d-1}{x_{i, h(i)}}}, where d is a prime, F_d is the finite field with d elements, and k << d. The degree of the gcd of every pair of monomials in NW_{d,k} is at most k. For concreteness, we fix k = ceil[sqrt{d}]. The family of polynomials NW := {NW_{d,k} : d is a prime} and close variants of it have been used as hard explicit polynomial families in several recent arithmetic circuit lower bound proofs. But, unlike the permanent, very little is known about the various structural and algorithmic/complexity aspects of NW beyond the fact that NW in VNP. Is NW_{d,k} characterized by its symmetries? Is it circuit-testable, i.e., given a circuit C can we check efficiently if C computes NW_{d,k}? What is the complexity of equivalence test for NW, i.e., given black-box access to a f in F[{x}], can we check efficiently if there exists an invertible linear transformation A such that f = NW_{d,k}(A * {x})? Characterization of polynomials by their symmetries plays a central role in the geometric complexity theory program. Here, we answer the first two questions and partially answer the third. We show that NW_{d,k} is characterized by its group of symmetries over C, but not over R. We also show that NW_{d,k} is characterized by circuit identities which implies that NW_{d,k} is circuit-testable in randomized polynomial time. As another application of this characterization, we obtain the "flip theorem" for NW. We give an efficient equivalence test for NW in the case where the transformation A is a block-diagonal permutation-scaling matrix. The design of this algorithm is facilitated by an almost complete understanding of the group of symmetries of NW_{d,k}: We show that if A is in the group of symmetries of NW_{d,k} then A = D * P, where D and P are diagonal and permutation matrices respectively. This is proved by completely characterizing the Lie algebra of NW_{d,k}, and using an interplay between the Hessian of NW_{d,k} and the evaluation dimension.

A homogeneous depth three circuit C computes a polynomial f = T1 + T2 + ... + Ts, where each Ti is a product of d linear forms in n variables over some underlying field F. Given black-box access to f, can we efficiently reconstruct (i.e. proper learn) a homogeneous depth three circuit computing f? Learning various subclasses of circuits is natural and interesting from both theoretical and practical standpoints and in particular, properly learning homogeneous depth three circuits efficiently is stated as an open problem in a work by Klivans and Shpilka (COLT 2003) and is well-studied. Unfortunately, there is substantial amount of evidence to show that this is a hard problem in the worst case. We give a (randomized) poly(n,d,s)-time algorithm to reconstruct non-degenerate homogeneous depth three circuits for n = Ω(d2) (with some additional mild requirements on s and the characteristic of F). We call a circuit C as non-degenerate if the dimension of the partial derivative space of f equals the sum of the dimensions of the partial derivative spaces of the terms T1, T2, …, Ts. In this sense, the terms are “independent” of each other in a non-degenerate circuit. A random homogeneous depth three circuit (where the coefficients of the linear forms are chosen according to the uniform distribution or any other reasonable distribution) is almost surely non-degenerate. In comparison, previous learning algorithms for this circuit class were either improper (with an exponential dependence on d), or they only worked for s < n (with a doubly exponential dependence of the running time on s). The main contribution of this work is to formulate the following paradigm for efficiently handling addition gates and to successfully implement it for the class of homogeneous depth three circuits. The problem of finding the children of an addition gate with large fan-in s is first reduced to the problem of decomposing a suitable vector space U into a (direct) sum of simpler subspaces U1, U2, …, Us. One then constructs a suitable space of operators S consisting of linear maps acting on U such that analyzing the simultaneous global structure of S enables us to efficiently decompose U. In our case, we exploit the structure of the set of low rank matrices in S and of the invariant subspaces of U induced by S. We feel that this paradigm is novel and powerful: it should lead to efficient reconstruction of many other subclasses of circuits for which the efficient reconstruction problem had hitherto looked unapproachable because of the presence of large fan-in addition gates.

An algebraic branching program (ABP) A can be modelled as a product expression X_1 X_2 ... X_d, where X_1 and X_d are 1 x w and w x 1 matrices respectively, and every other X_k is a w x w matrix; the entries of these matrices are linear forms in m variables over a field F (which we assume to be either Q or a field of characteristic poly(m)). The polynomial computed by A is the entry of the 1 x 1 matrix obtained from the product X_1 X_2 ... X_d. We say A is a full rank ABP if the w^2(d-2) + 2w linear forms occurring in the matrices X_1, X_2, ... , X_d are F-linearly independent. Our main result is a randomized reconstruction algorithm for full rank ABPs: Given blackbox access to an m-variate polynomial f of degree at most m, the algorithm outputs a full rank ABP computing f if such an ABP exists, or outputs 'no full rank ABP exists' (with high probability). The running time of the algorithm is polynomial in m and b, where b is the bit length of the coefficients of f. The algorithm works even if X_k is a w_{k-1} x w_k matrix (with w_0 = w_d = 1), and v = (w_1, ..., w_{d-1}) is unknown. The result is obtained by designing a randomized polynomial time equivalence test for the family of iterated matrix multiplication polynomial IMM_{v,d}, the (1,1)-th entry of a product of d rectangular symbolic matrices whose dimensions are according to v in N^{d-1}. At its core, the algorithm exploits a connection between the irreducible invariant subspaces of the Lie algebra of the group of symmetries of a polynomial f that is equivalent to IMM_{v,d} and the 'layer spaces' of a full rank ABP computing f. This connection also helps determine the group of symmetries of IMM_{v,d} and show that IMM_{v,d} is characterized by its group of symmetries.

In a multi- k -ic depth three circuit every variable appears in at most k of the linear polynomials in every product gate of the circuit. This model is a natural generalization of multilinear depth three circuits that allows the formal degree of the circuit to exceed the number of underlying variables (as the formal degree of a multi- k -ic depth three circuit can be kn where n is the number of variables). The problem of proving lower bounds for depth three circuits with high formal degree has gained in importance following a work by Gupta et al. ( 2013 ) on depth reduction to high formal degree depth three circuits. In this work, we show an exponential lower bound for multi- k -ic depth three circuits for any arbitrary constant k .

We show here a $2^{\Omega(\sqrt{d} \cdot \log N)}$ size lower bound for homogeneous depth four arithmetic formulas over fields of characteristic zero. That is, we give an explicit family of polynomials of degree $d$ on $N$ variables (with $N = d^3$ in our case) with 0, 1-coefficients such that for any representation of a polynomial $f$ in this family of the form $ f = \sum_{i} \prod_{j} Q_{ij}, $ where the $Q_{ij}$'s are homogeneous polynomials (recall that a polynomial is said to be homogeneous if all its monomials have the same degree), it must hold that $ \sum_{i, j} (\text{number of monomials of~} Q_{ij}) \geq 2^{\Omega (\sqrt{d} \cdot \log N)}. $ The abovementioned family, which we refer to as the Nisan--Wigderson design-based family of polynomials, is in the complexity class $\mathsf{VNP}$. Our work builds on recent lower bound results and yields an improved quantitative bound as compared to the quasi-polynomial lower bound of [N. Kayal et al., in Symposium on Theory of Computing, ACM, New York, 2014, pp. 119--127] and the $N^{\Omega(\log \log N)}$ lower bound in the independent work of [M. Kumar and S. Saraf, in Automata, Languages, and Programming, Part I, Springer, Berlin, 2014, pp. 751--762].

We show an exponential separation between two well-studied models of algebraic computation, namely read-once oblivious algebraic branching programs (ROABPs) and multilinear depth three circuits. In particular we show the following: 1. There exists an explicit n-variate polynomial computable by linear sized multilinear depth three circuits (with only two product gates) such that every ROABP computing it requires 2^{Omega(n)} size. 2. Any multilinear depth three circuit computing IMM_{n,d} (the iterated matrix multiplication polynomial formed by multiplying d, n * n symbolic matrices) has n^{Omega(d)} size. IMM_{n,d} can be easily computed by a poly(n,d) sized ROABP. 3. Further, the proof of 2 yields an exponential separation between multilinear depth four and multilinear depth three circuits: There is an explicit n-variate, degree d polynomial computable by a poly(n,d) sized multilinear depth four circuit such that any multilinear depth three circuit computing it has size n^{Omega(d)}. This improves upon the quasi-polynomial separation result by Raz and Yehudayoff [2009] between these two models. The hard polynomial in 1 is constructed using a novel application of expander graphs in conjunction with the evaluation dimension measure used previously in Nisan [1991], Raz [2006,2009], Raz and Yehudayoff [2009], and Forbes and Shpilka [2013], while 2 is proved via a new adaptation of the dimension of the partial derivatives measure used by Nisan and Wigderson [1997]. Our lower bounds hold over any field.

Let r be an integer. Let us call a polynomial f as a multi-r-ic polynomial if the degree of f with respect to any variable is at most r (this generalizes the notion of multilinear polynomials). We investigate arithmetic circuits in which the output is syntactically forced to be a multi-r-ic polynomial and refer to these as multi-r-ic circuits. Specifically, first define the formal degree of a node a with respect to a variable x inductively as follows. For a leaf it is 1 if a is labelled with x and zero otherwise; for an internal node labelled with * (respectively +) it is the sum of (respectively the maximum of) the formal degrees of the children with respect to x. We call an arithmetic circuit as a multi-r-ic circuit if the formal degree of the output node with respect to any variable is at most r. We prove lower bounds for various subclasses of multi-r-ic circuits.

Shpilka & Wigderson (IEEE conference on computational complexity, vol 87, 1999) had posed the problem of proving exponential lower bounds for (nonhomogeneous) depth-three arithmetic circuits with bounded bottom fanin over a field $${{\mathbb{F}}}$$ of characteristic zero. We resolve this problem by proving a $${N^{\Omega(\frac{d}{\tau})}}$$ lower bound for (nonhomogeneous) depth-three arithmetic circuits with bottom fanin at most $${\tau}$$ computing an explicit $${N}$$ -variate polynomial of degree $${d}$$ over $${{\mathbb{F}}}$$ . Meanwhile, Nisan & Wigderson (Comp Complex 6(3):217–234, 1997) had posed the problem of proving super-polynomial lower bounds for homogeneous depth-five arithmetic circuits. Over fields of characteristic zero, we show a lower bound of $${N^{\Omega(\sqrt{d})}}$$ for homogeneous depth-five circuits (resp. also for depth-three circuits) with bottom fanin at most $${N^{\mu}}$$ , for any fixed $${\mu < 1}$$ . This resolves the problem posed by Nisan and Wigderson only partially because of the added restriction on the bottom fanin (a general homogeneous depth-five circuit has bottom fanin at most $${N}$$ ).

We present a single common tool to strictly subsume all known cases of polynomial time black box polynomial identity testing (PIT), that have been hitherto solved using diverse tools and techniques, over fields of zero or large characteristic. In particular, we show that polynomial (in the size of the circuit) time hitting-set generators for identity testing of the two seemingly different and well studied models---depth-3 circuits with bounded top fanin, and constant-depth constant-read multilinear formulas---can be constructed using one common algebraic-geometry theme: Jacobian captures algebraic independence. By exploiting the Jacobian, we design the first efficient hitting-set generators for broad generalizations of the above-mentioned models, namely, (a) depth-3 ($\Sigma \Pi \Sigma$) circuits with constant transcendence degree of the polynomials computed by the product gates (no bounded top fanin restriction), and (b) constant-depth constant-occur formulas (no multilinear restriction). Constant occur of a variable, as we define it, is a more general concept than constant read. Also, earlier work on the latter model assumed that the formula is multilinear. Thus, our work goes further beyond the related results obtained by Saxena and Seshadhri [STOC, ACM, New York, 2011, pp. 431--440], Saraf and Volkovich [STOC, ACM, New York, 2011, pp. 421--430], Anderson, van Melkebeek, and Volkovich, [IEEE Conference on Computational Complexity, IEEE, Piscataway, NJ, 2011, pp. 273--282], Beecken, Mittmann, and Saxena [ICALP, Springer, New York, 2011, pp. 134--148] and Grenet et al. [Proceedings of the 30th Foundations of Software Technology and Theoretical Computer Science (FSTTCS), Schloss Dagstuhl--Liebniz--Zentrum für Informatik, Wadern, Germany, 2011, pp. 127--139] and brings them under one unifying technique. In addition, using the same Jacobian-based approach, we prove exponential lower bounds for the immanant (which includes permanent and determinant) on the same depth-3 and depth-4 models for which we give efficient PIT algorithms. Our results reinforce the intimate connection between identity testing and lower bounds by exhibiting a concrete mathematical tool---the Jacobian. The Jacobian is equally effective in solving both the problems on certain interesting and previously well-investigated (but not well understood) models of computation.

We show here a 2Ω(√d ⋅ log N) size lower bound for homogeneous depth four arithmetic formulas. That is, we give an explicit family of polynomials of degree d on N variables (with N = d3 in our case) with 0, 1-coefficients such that for any representation of a polynomial f in this family of the form f = Σi ∏j Qij, where the Qij's are homogeneous polynomials (recall that a polynomial is said to be homogeneous if all its monomials have the same degree), it must hold that ∑i, j (Number of monomials of Qij)) ≥2Ω(√d ⋅log N). The above mentioned family, which we refer to as the Nisan-Wigderson design-based family of polynomials, is in the complexity class VNP. Our work builds on recent lower bound results [1], [2], [3], [4], [5] and yields an improved quantitative bound as compared to the quasi-polynomial lower bound from an earlier work of the same authors and the NΩ(log log N) lower bound in the independent work of [7].

We show that any depth-4 homogeneous arithmetic formula computing the Iterated Matrix Multiplication polynomial IMMn,d -- the (1, 1)-th entry of the product of d generic n × n matrices -- has size nΩ(log n), if d = Ω (log2 n). More-over, any depth-4 homogeneous formula computing the determinant polynomial Detn -- the determinant of a generic n × n matrix -- has size nΩ(log n).

We consider arithmetic formulas consisting of alternating layers of addition (+) and multiplication (×) gates such that the fanin of all the gates in any fixed layer is the same. Such a formula Φ which additionally has the property that its formal/syntactic degree is at most twice the (total) degree of its output polynomial, we refer to as a regular formula. As usual, we allow arbitrary constants from the underlying field F on the incoming edges to a + gate so that a + gate can in fact compute an arbitrary F-linear combination of its inputs. We show that there is an (n2 + 1)-variate polynomial of degree 2n in VNP such that any regular formula computing it must be of size at least nΩ(log n).

We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition X1⊔⋅⋅⋅⊔ Xd of the variable indices [n] that the top product layer respects, i.e. C(term{x})=∑i=1k ∏j=1d fi,j(term{x}Xj), where fi,j is a sparse polynomial in F[term{x}Xj]. Extending this definition to any depth - we call a depth-D formula C (consisting of alternating layers of Σ and Π gates, with a Σ-gate on top) a set-depth-D formula if every Π-layer in C respects a (unknown) partition on the variables; if D is even then the product gates of the bottom-most Π-layer are allowed to compute arbitrary monomials. In this work, we give a hitting-set generator for set-depth-D formulas (over any field) with running time polynomial in exp((D2log s) Δ - 1), where s is the size bound on the input set-depth-D formula. In other words, we give a quasi-polynomial time blackbox polynomial identity test for such constant-depth formulas. Previously, the very special case of D=3 (also known as set-multilinear depth-3 circuits) had no known sub-exponential time hitting-set generator. This was declared as an open problem by Shpilka & Yehudayoff (FnT-TCS 2010); the model being first studied by Nisan & Wigderson (FOCS 1995) and recently by Forbes & Shpilka (STOC 2012 & ECCC TR12-115). Our work settles this question, not only for depth-3 but, up to depth εlog s / log log s, for a fixed constant ε < 1. The technique is to investigate depth-D formulas via depth-(D-1) formulas over a Hadamard algebra, after applying a 'shift' on the variables. We propose a new algebraic conjecture about the low-support rank-concentration in the latter formulas, and manage to prove it in the case of set-depth-D formulas.

We give an $N\cdot \log N\cdot 2^{O(\log^*N)}$ time algorithm to multiply two $N$-bit integers that uses modular arithmetic for intermediate computations instead of arithmetic over complex numbers as in Fürer's algorithm, which also has the same and so far the best known complexity. The previous best algorithm using modular arithmetic (by Schönhage and Strassen) has complexity $O(N \cdot \log N \cdot \log\log N)$. The advantage of using modular arithmetic as opposed to complex number arithmetic is that we can completely evade the task of bounding the truncation error due to finite approximations of complex numbers, which makes the analysis relatively simple. Our algorithm is based upon Fürer's algorithm, but uses fast Fourier transform over multivariate polynomials along with an estimate of the least prime in an arithmetic progression to achieve this improvement in the modular setting. It can also be viewed as a $p$-adic version of Fürer's algorithm.

We present a single common tool to strictly subsume all known cases of polynomial time blackbox polynomial identity testing (PIT), that have been hitherto solved using diverse tools and techniques, over fields of zero or large characteristic. In particular, we show that polynomial time hitting-set generators for identity testing of the two seemingly different and well studied models - depth-3 circuits with bounded top fanin, and constant-depth constant-read multilinear formulas - can be constructed using one common algebraic-geometry theme: Jacobian captures algebraic independence. By exploiting the Jacobian, we design the first efficient hitting-set generators for broad generalizations of the above-mentioned models, namely: - depth-3 (Ω Π Ω) circuits with constant transcendence degree of the polynomials computed by the product gates (no bounded top fanin restriction), and - constant-depth constant-occur formulas (no multilinear restriction). Constant-occur of a variable, as we define it, is a much more general concept than constant-read. Also, earlier work on the latter model assumed that the formula is multilinear. Thus, our work goes further beyond the related results obtained by Saxena & Seshadhri (STOC 2011), Saraf & Volkovich (STOC 2011), Anderson et al. (CCC 2011), Beecken et al. (ICALP 2011) and Grenet et al. (FSTTCS 2011), and brings them under one unifying technique.

The sum of square roots problem over integers is the task of deciding the sign of a non-zero sum, S = Σi=1nδi· √(ai), where δiϵ { +1, -1} and ai's are positive integers that are upper bounded by N (say). A fundamental open question in numerical analysis and computational geometry is whether |S| ≥ 1/2(n·logN)O(1)when S ≠ 0. We study a formulation of this problem over polynomials: Given an expression S = Σi=1nci· √(fi(x)), where ci's belong to a field of characteristic 0 and fi's are univariate polynomials with degree bounded by d and fi(0) ≠ 0 for all i, is it true that the minimum exponent of x which has a nonzero coefficient in the power series S is upper bounded by (n · d)O(1), unless S = 0? We answer this question affirmatively. Further, we show that this result over polynomials can be used to settle (positively) the sum of square roots problem for a special class of integers: Suppose each integer at is of the form, ai= Xdi+ bi1Xdi-1+ ⋯ +bidi, di>; 0, where X is a positive real number and bij's are integers. Let B = maxi,j{|bij|} and d = maxi{di}. If X >; (B + 1)(n·d)O(1)then a non-zero S = Σi=1nδi· √(ai) is lower bounded as |S| ≥ 1/X(n·d)O(1). The constant in the O(1) notation, as fixed by our analysis, is roughly 2. We then consider the following more general problem: given an arithmetic circuit computing a multivariate polynomial f(X) and integer d, is the degree of f(X) less than or equal to d? We give a coRPPP-algorithm for this problem, improving previous results of and.

We study the problem of polynomial identity testing (PIT) for depth $2$ arithmetic circuits over matrix algebra. We show that identity testing of depth $3$ ($\Sigma \Pi \Sigma$) arithmetic circuits over a field $\F$ is polynomial time equivalent to identity testing of depth $2$ ($\Pi \Sigma$) arithmetic circuits over $\mathsf{U}_2(\mathbb{F})$, the algebra of upper-triangular $2\times 2$ matrices with entries from $\F$. Such a connection is a bit surprising since we also show that, as computational models, $\Pi \Sigma$ circuits over $\mathsf{U}_2(\mathbb{F})$ are strictly `weaker' than $\Sigma \Pi \Sigma$ circuits over $\mathbb{F}$. The equivalence further implies that PIT of $\Sigma \Pi \Sigma$ circuits reduces to PIT of width-$2$ commutative \emph{Algebraic Branching Programs}(ABP). Further, we give a deterministic polynomial time identity testing algorithm for a $\Pi \Sigma$ circuit of size $s$ over commutative algebras of dimension $O(\log s/\log\log s)$ over $\F$. Over commutative algebras of dimension $\poly(s)$, we show that identity testing of $\Pi \Sigma$ circuits is at least as hard as that of $\Sigma \Pi \Sigma$ circuits over $\mathbb{F}$.

We study the problem of factoring univariate polynomials over finite fields. Under the assumption of the Extended Riemann Hypothesis (ERH), (Gao, 2001) designed a polynomial time algorithm that fails to factor only if the input polynomial satisfies a strong symmetry property, namely square balance. In this paper, we propose an extension of Gao's algorithm that fails only under an even stronger symmetry property. We also show that our property can be used to improve the time complexity of best deterministic algorithms on most input polynomials. The property also yields a new randomized polynomial time algorithm.