We study the power of negation in the Boolean and algebraic settings and show the following results.

We design the first efficient polynomial identity testing algorithms over the nonassociative polynomial algebra. In particular, multiplication among the formal variables is commutative but it is not associative. This complements the strong lower bound results obtained over this algebra by Hrubeš, Yehudayoff, and Wigderson [Pavel Hrubes et al., 2010] and Fijalkow, Lagarde, Ohlmann, and Serre [Fijalkow et al., 2021] from the identity testing perspective. Our main results are the following: - We construct nonassociative algebras (both commutative and noncommutative) which have no low degree identities. As a result, we obtain the first Amitsur-Levitzki type theorems [A. S. Amitsur and J. Levitzki, 1950] over nonassociative polynomial algebras. As a direct consequence, we obtain randomized polynomial-time black-box PIT algorithms for nonassociative polynomials which allow evaluation over such algebras. - On the derandomization side, we give a deterministic polynomial-time identity testing algorithm for nonassociative polynomials given by arithmetic circuits in the white-box setting. Previously, such an algorithm was known with the additional restriction of noncommutativity [Vikraman Arvind et al., 2017]. - In the black-box setting, we construct a hitting set of quasipolynomial-size for nonassociative polynomials computed by arithmetic circuits of small depth. Understanding the black-box complexity of identity testing, even in the randomized setting, was open prior to our work.

We give new lower bounds for the fragments of the Ideal Proof System (IPS) introduced by Grochow and Pitassi [Joshua A. Grochow and Toniann Pitassi, 2018]. The Ideal Proof System is a central topic in algebraic proof complexity developed in the context of Nullstellensatz refutation [Paul Beame et al., 1994] and simulates Extended Frege efficiently. Our main results are as follows. - mult-IPS_{Lin'}: We prove nearly quadratic-size formula lower bound for multilinear refutation (over the Boolean hypercube) of a variant of the subset-sum axiom polynomial. Extending this, we obtain a nearly matching qualitative statement for a constant degree target polynomial. - IPS_{Lin'}: Over the fields of characteristic zero, we prove exponential-size sum-of-ROABPs lower bound for the refutation of a variant of the subset-sum axiom polynomial. The result also extends over the fields of positive characteristics when the target polynomial is suitably modified. The modification is inspired by the recent results [Tuomas Hakoniemi et al., 2024; Amik Raj Behera et al., 2025]. The mult-IPS_{Lin'} lower bound result is obtained by combining the quadratic-size formula lower bound technique of Kalorkoti [Kalorkoti, 1985] with some additional ideas. The proof technique of IPS_{Lin'} lower bound result is inspired by the recent lower bound result of Chatterjee, Kush, Saraf and Shpilka [Prerona Chatterjee et al., 2024].

Let$X=X_{1} \sqcup X_{2} \sqcup \ldots \sqcup X_{k}$be a partitioned set of variables such that the variables in each part$X_{i}$are noncommuting but for any$i\neq j$, the variables$x\in X_{i}$commute with the variables$x^{\prime}\in X_{j}$. Given as input a square matrix$T$whose entries are linear forms over$\mathbb{Q}\langle X\rangle$〉, we consider the problem of checking if$T$is invertible or not over the universal skew field of fractions of the partially commutative polynomial ring$\mathbb{Q}\langle X\rangle$[1]. In this paper, we design a deterministic polynomial-time algorithm for this problem for constant$k$. The special case$k=1$is the noncommutative Edmonds' problem (NSINGULAR) which has a deterministic polynomial-time algorithm by recent results [2]–[4]. En-route, we obtain the first deterministic polynomial-time algorithm for the equivalence testing problem of$k$-tape weighted automata (for constant$k$) resolving a longstanding open problem [5], [6]. Algebraically, the equivalence problem reduces to testing whether a partially commutative rational series over the partitioned set$X$is zero or not [6]. Decidability of this problem was established by Harju and Karhumäki [5]. Prior to this work, a randomized polynomial-time algorithm for this problem was given by Worrell [6] and, subsequently, a deterministic quasipolynomial-time algorithm was also developed [7].

Rational Identity Testing (RIT) is the decision problem of determining whether or not a noncommutative rational formula computes zero in the free skew field. It admits a deterministic polynomial-time white-box algorithm [Garg, Gurvits, Oliveira, and Wigderson (2016); Ivanyos, Qiao, Subrahmanyam (2018); Hamada and Hirai (2021)], and a randomized polynomial-time algorithm [Derksen and Makam (2017)] in the black-box setting, via singularity testing of linear matrices over the free skew field. Indeed, a randomized NC algorithm for RIT in the white-box setting follows from the result of Derksen and Makam (2017). Designing an efficient deterministic black-box algorithm for RIT and understanding the parallel complexity of RIT are major open problems in this area. Despite being open since the work of Garg, Gurvits, Oliveira, and Wigderson (2016), these questions have seen limited progress. In fact, the only known result in this direction is the construction of a quasipolynomial-size hitting set for rational formulas of only inversion height two [Arvind, Chatterjee, and Mukhopadhyay (2022)]. In this paper, we significantly improve the black-box complexity of this problem and obtain the first quasipolynomial-size hitting set for all rational formulas of polynomial size. Our construction also yields the first deterministic quasi-NC upper bound for RIT in the white-box setting.

Hrubeš and Wigderson [Hrubeš and Wigderson, 2015] initiated the complexity-theoretic study of noncommutative formulas with inverse gates. They introduced the Rational Identity Testing (RIT) problem which is to decide whether a noncommutative rational formula computes zero in the free skew field. In the white-box setting, there are deterministic polynomial-time algorithms due to Garg, Gurvits, Oliveira, and Wigderson [Ankit Garg et al., 2016] and Ivanyos, Qiao, and Subrahmanyam [Ivanyos et al., 2018]. A central open problem in this area is to design an efficient deterministic black-box identity testing algorithm for rational formulas. In this paper, we solve this for the first nested inverse case. More precisely, we obtain a deterministic quasipolynomial-time black-box RIT algorithm for noncommutative rational formulas of inversion height two via a hitting set construction. Several new technical ideas are involved in the hitting set construction, including concepts from matrix coefficient realization theory [Volčič, 2018] and properties of cyclic division algebras [T.Y. Lam, 2001]. En route to the proof, an important step is to embed the hitting set of Forbes and Shpilka for noncommutative formulas [Michael A. Forbes and Amir Shpilka, 2013] inside a cyclic division algebra of small index.

We prove two results that shed new light on the monotone complexity of the spanning tree polynomial, a classic polynomial in algebraic complexity and beyond. First, we show that the spanning tree polynomials having $n$ variables and defined over constant-degree expander graphs, have monotone arithmetic complexity $2^{Ω(n)}$. This yields the first strongly exponential lower bound on the monotone arithmetic circuit complexity for a polynomial in VP. Before this result, strongly exponential size monotone lower bounds were known only for explicit polynomials in VNP (Gashkov-Sergeev'12, Raz-Yehudayoff'11, Srinivasan'20, Cavalar-Kumar-Rossman'20, Hrubes-Yehudayoff'21). Recently, Hrubes'20 initiated a program to prove lower bounds against general arithmetic circuits by proving $ε$-sensitive lower bounds for monotone arithmetic circuits for a specific range of values for $ε\in (0,1)$. We consider the spanning tree polynomial $ST_{n}$ defined over the complete graph on $n$ vertices and show that the polynomials $F_{n-1,n} - ε\cdot ST_{n}$ and $F_{n-1,n} + ε\cdot ST_{n}$ defined over $n^2$ variables, have monotone circuit complexity $2^{Ω(n)}$ if $ε\geq 2^{-Ω(n)}$ and $F_{n-1,n} = \prod_{i=2}^n (x_{i,1} +\cdots + x_{i,n})$ is the complete set-multilinear polynomial. This provides the first $ε$-sensitive exponential lower bound for a family of polynomials inside VP. En-route, we consider a problem in 2-party, best partition communication complexity of deciding whether two sets of oriented edges distributed among Alice and Bob form a spanning tree or not. We prove that there exists a fixed distribution, under which the problem has low discrepancy with respect to every nearly-balanced partition. This result could be of interest beyond algebraic complexity.

Let \({\mathbb {F}}[X]\) be the polynomial ring in the variables X = { x 1 , x 2 ,…, x n } over a field \({\mathbb {F}}\) . An ideal I = 〈 p 1 ( x 1 ),…, p n ( x n )〉 generated by univariate polynomials \(\{p_{i}(x_{i})\}_{i=1}^{n}\) is a univariate ideal . Motivated by Alon’s Combinatorial Nullstellensatz we study the complexity of univariate ideal membership : Given \(f\in {\mathbb {F}}[X]\) by a circuit and polynomials p i the problem is test if f ∈ I . We obtain the following results. Suppose f is a degree- d , rank- r polynomial given by an arithmetic circuit where ℓ i : 1 ≤ i ≤ r are linear forms in X . We give a deterministic time d O ( r ) ⋅poly( n ) division algorithm for evaluating the (unique) remainder polynomial f ( X )mod I at any point \(\vec {a}\in {\mathbb {F}}^{n}\) . This yields a randomized n O ( r ) algorithm for minimum vertex cover in graphs with rank- r adjacency matrices. It also yields a new n O ( r ) algorithm for evaluating the permanent of a n × n matrix of rank r , over any field \(\mathbb {F}\) . Let f be over rationals with \(\deg (f)=k\) treated as fixed parameter. When the ideal \(I=\left \langle {x_{1}^{e_{1}}, \ldots , x_{n}^{e_{n}}}\right \rangle \) , we can test ideal membership in randomized O ∗ ((2 e ) k ). On the other hand, if each p i has all distinct rational roots we can check if f ∈ I in randomized O ∗ ( n k /2 ) time, improving on the brute-force \(\left (\begin {array}{cc}{n+k}\\ k \end {array}\right )\) -time search. If \(I=\left \langle {p_{1}(x_{1}), \ldots , p_{k}(x_{k})}\right \rangle \) , with k as fixed parameter, then ideal membership testing is W[2]-hard. The problem is MINI[1]-hard in the special case when \(I=\left \langle {x_{1}^{e_{1}}, \ldots , x_{k}^{e_{k}}}\right \rangle \) .

Motivated by equivalence testing of k-tape automata, we study the equivalence testing of weighted automata in the more general setting, over partially commutative monoids (in short, pc monoids), and show efficient algorithms in some special cases, exploiting the structure of the underlying non-commutation graph of the monoid. Specifically, if the edge clique cover number of the non-commutation graph of the pc monoid is a constant, we obtain a deterministic quasi-polynomial time algorithm for equivalence testing. As a corollary, we obtain the first deterministic quasi-polynomial time algorithms for equivalence testing of k-tape weighted automata and for equivalence testing of deterministic k-tape automata for constant k. Prior to this, the best complexity upper bound for these k-tape automata problems were randomized polynomial-time, shown by Worrell [James Worrell, 2013]. Finding a polynomial-time deterministic algorithm for equivalence testing of deterministic k-tape automata for constant k has been open for several years [Emily P. Friedman and Sheila A. Greibach, 1982] and our results make progress. We also consider pc monoids for which the non-commutation graphs have an edge cover consisting of at most k cliques and star graphs for any constant k. We obtain a randomized polynomial-time algorithm for equivalence testing of weighted automata over such monoids. Our results are obtained by designing efficient zero-testing algorithms for weighted automata over such pc monoids.

Valiant (1980) showed that general arithmetic circuits with negation can be exponentially more powerful than monotone ones. We give the first improvement to this classical result: we construct a family of polynomials Pn in n variables, each of its monomials has non-negative coefficient, such that Pn can be computed by a polynomial-size depth-three formula but every monotone circuit computing it has size 2Ω(n1/4/log(n)).

We explore a special case of rational identity testing and algorithmic versions of two theorems on noncommutative polynomials, namely, Amitsur's theorem [S.A Amitsur, 1966] and the Brešar-Klep theorem [Brešar and Klep, 2008] when the input polynomial is given by an algebraic branching program (ABP). Let f be a degree-d n-variate noncommutative polynomial in the free ring Q over rationals. 1) We consider the following special case of rational identity testing: Given a noncommutative ABP as white-box, whose edge labels are linear forms or inverses of linear forms, we show a deterministic polynomial-time algorithm to decide if the rational function computed by it is equivalent to zero in the free skew field Q<(X)>. Given black-box access to the ABP, we give a deterministic quasi-polynomial time algorithm for this problem. 2) Amitsur's theorem implies that if a noncommutative polynomial f is nonzero on k x k matrices then, in fact, f(M_1,M_2,...,M_n) is invertible for some matrix tuple (M_1,M_2,...,M_n) in (M_k(ℚ))^n. While a randomized polynomial time algorithm to find such (M_1,M_2,...,M_n) given black-box access to f is simple, we obtain a deterministic s^{O(log d)} time algorithm for the problem with black-box access to f, where s is the minimum ABP size for f and d is the degree of f. 3) The Brešar-Klep Theorem states that the span of the range of any noncommutative polynomial f on k x k matrices over Q is one of the following: zero, scalar multiples of I_k, trace-zero matrices in M_k(Q), or all of M_k(Q). We obtain a deterministic polynomial-time algorithm to decide which case occurs, given white-box access to an ABP for f. We also give a deterministic s^{O(log d)} time algorithm given black-box access to an ABP of size s for f. Our algorithms work when k >= d. Our techniques are based on some automata theory combined with known techniques for noncommutative ABP identity testing [Ran Raz and Amir Shpilka, 2005; Michael A. Forbes and Amir Shpilka, 2013].

Hrubeš and Wigderson [Pavel Hrubeš and Avi Wigderson, 2014] initiated the study of noncommutative arithmetic circuits with division computing a noncommutative rational function in the free skew field, and raised the question of rational identity testing. For noncommutative formulas with inverses the problem can be solved in deterministic polynomial time in the white-box model [Ankit Garg et al., 2016; Ivanyos et al., 2018]. It can be solved in randomized polynomial time in the black-box model [Harm Derksen and Visu Makam, 2017], where the running time is polynomial in the size of the formula. The complexity of identity testing of noncommutative rational functions, in general, remains open for noncommutative circuits with inverses. We solve the problem for a natural special case. We consider expressions in the free group algebra F(X,X^{-1}) where X={x_1, x_2, ..., x_n}. Our main results are the following. 1) Given a degree d expression f in F(X,X^{-1}) as a black-box, we obtain a randomized poly(n,d) algorithm to check whether f is an identically zero expression or not. The technical contribution is an Amitsur-Levitzki type theorem [A. S. Amitsur and J. Levitzki, 1950] for F(X, X^{-1}). This also yields a deterministic identity testing algorithm (and even an expression reconstruction algorithm) that is polynomial time in the sparsity of the input expression. 2) Given an expression f in F(X,X^{-1}) of degree D and sparsity s, as black-box, we can check whether f is identically zero or not in randomized poly(n,log s, log D) time. This yields a randomized polynomial-time algorithm when D and s are exponential in n.

Let C be an arithmetic circuit of poly(n) size given as input that computes a polynomial f in F[X], where X={x_1,x_2,...,x_n} and F is any field where the field arithmetic can be performed efficiently. We obtain new algorithms for the following two problems first studied by Koutis and Williams [Ioannis Koutis, 2008; Ryan Williams, 2009; Ioannis Koutis and Ryan Williams, 2016]. - (k,n)-MLC: Compute the sum of the coefficients of all degree-k multilinear monomials in the polynomial f. - k-MMD: Test if there is a nonzero degree-k multilinear monomial in the polynomial f. Our algorithms are based on the fact that the Hadamard product f o S_{n,k}, is the degree-k multilinear part of f, where S_{n,k} is the k^{th} elementary symmetric polynomial. - For (k,n)-MLC problem, we give a deterministic algorithm of run time O^*(n^(k/2+c log k)) (where c is a constant), answering an open question of Koutis and Williams [Ioannis Koutis and Ryan Williams, 2016]. As corollaries, we show O^*(binom{n}{downarrow k/2})-time exact counting algorithms for several combinatorial problems: k-Tree, t-Dominating Set, m-Dimensional k-Matching. - For k-MMD problem, we give a randomized algorithm of run time 4.32^k * poly(n,k). Our algorithm uses only poly(n,k) space. This matches the run time of a recent algorithm [Cornelius Brand et al., 2018] for k-MMD which requires exponential (in k) space. Other results include fast deterministic algorithms for (k,n)-MLC and k-MMD problems for depth three circuits.

We study the arithmetic circuit complexity of some well-known family of polynomials through the lens of parameterized complexity. Our main focus is on the construction of explicit algebraic branching programs (ABP) for determinant and permanent polynomials of the rectangular symbolic matrix in both commutative and noncommutative settings. The main results are: - We show an explicit O^*(binom{n}{downarrow k/2})-size ABP construction for noncommutative permanent polynomial of k x n symbolic matrix. We obtain this via an explicit ABP construction of size O^*(binom{n}{downarrow k/2}) for S_{n,k}^*, noncommutative symmetrized version of the elementary symmetric polynomial S_{n,k}. - We obtain an explicit O^*(2^k)-size ABP construction for the commutative rectangular determinant polynomial of the k x n symbolic matrix. - In contrast, we show that evaluating the rectangular noncommutative determinant over rational matrices is #W[1]-hard.

Tavenas (Proceedings of mathematical foundations of computer science (MFCS), 2013) has recently proved that any $$n^{O(1)}$$ -variate and degree n polynomial in $$\mathsf {VP}$$ can be computed by a depth-4 $$\Sigma \Pi \Sigma \Pi $$ circuit of size $$2^{O(\sqrt{n}\log n)}$$ . So, to prove $$\mathsf {VP}\ne \mathsf {VNP}$$ it is sufficient to show that an explicit polynomial in $$\mathsf {VNP}$$ of degree n requires $$2^{\omega (\sqrt{n}\log n)}$$ size depth-4 circuits. Soon after Tavenas’ result, for two different explicit polynomials, depth-4 circuit-size lower bounds of $$2^{\Omega (\sqrt{n}\log n)}$$ have been proved (see Kayal et al. in Proceedings of symposium on theory of computing, ACM, 2014b. http://doi.acm.org/10.1145/2591796.2591847 ; Fournier et al. in Proceedings of symposium on theory of computing, ACM, 2014). In particular, using a combinatorial design Kayal et al. (2014b) construct an explicit polynomial in $$\mathsf {VNP}$$ that requires depth-4 circuits of size $$2^{\Omega (\sqrt{n}\log n)}$$ and Fournier et al. (Proceedings of symposium on theory of computing, ACM, 2014) show that the iterated matrix multiplication polynomial (which is in $$\mathsf {VP}$$ ) also requires $$2^{\Omega (\sqrt{n}\log n)}$$ size depth-4 circuits. In this paper, we identify a simple combinatorial property such that any polynomial f that satisfies this property would achieve a similar depth-4 circuit-size lower bound. In particular, it does not matter whether f is in $$\mathsf {VP}$$ or in $$\mathsf {VNP}$$ . As a result, we get a simple unified lower-bound analysis for the above-mentioned polynomials. Another goal of this paper is to compare our current knowledge of the depth-4 circuit-size lower bounds and the determinantal complexity lower bounds. Currently, the best known determinantal complexity lower bound is $$\Omega (n^2)$$ for permanent of a $$n\times n$$ matrix (which is a $$n^2$$ -variate and degree n polynomial) due to Cai et al. (Proceedings of symposium on theory of computing, ACM, 2008). We prove that the determinantal complexity of the iterated matrix multiplication polynomial is $$\Omega (dn)$$ where d is the number of matrices and n is the dimension of the matrices. In particular, our result settles the determinantal complexity of the iterated matrix multiplication polynomial to $$\Theta (dn)$$ . To the best of our knowledge, a $$\Theta (n)$$ bound for the determinantal complexity for the iterated matrix multiplication polynomial was known only for any constant $$d>1$$ , due to Jansen (Theory Comput Syst 49(2):343–354, 2011).

Let $G =\langle S\rangle$ be a solvable permutation group given as input by the generating set $S$, that is, $G$ is a solvable subgroup of the symmetric group $S_n$. We give a deterministic polynomial-time algorithm that computes an expanding generating set $T$ of size $\tilde{O}(n^2(1/\lambda)^{c})$ for $G$ such that the undirected Cayley graph ${\rm Cay}_u(G,T)$ is a $\lambda$-spectral expander and the constant $c$ is at most $8$ (the $\tilde{O}$ notation suppresses $\log ^{O(1)}n$ and $\log ^{O(1)}(1/\lambda)$ factors). As a byproduct of our proof, we get a new explicit construction of $\varepsilon$-bias spaces of size $\tilde{O}(n (\log d)^{O(1)}(1/\varepsilon)^{c})$ for the groups $\mathbb{Z}_d^n$ and $c\leq 8$. The earlier known size bound was $O((d + n/\varepsilon^2)^{11/2})$ given by [ Y. Azar, R. Motwani, and J. Naor , Combinatorica, 18 (1998), pp. 151--171]. We also note that for any permutation group $G\le S_n$ given by a generating set, in deterministic polynomial time we can compute an expanding generating set $T$ of size $\left({n}/{\lambda}\right)^{O(1)}$ such that ${\rm Cay}_u(G,T)$ is a $\lambda$-spectral expander where the $O(1)$ notation involves a large constant.

In this paper we show that black-box polynomial identity testing for noncommutative polynomials f∈𝔽⟨z1,z2,…,zn⟩ of degree D and sparsity t, can be done in randomized (n,logt,logD) time. As a consequence, given a circuit C of size s computing a polynomial f∈𝔽⟨ z1,z2,…,zn⟩ with at most t non-zero monomials, then testing if f is identically zero can be done by a randomized algorithm with running time polynomial in s and n and logt. This makes significant progress on a question that has been open for over ten years. Our algorithm is based on automata-theoretic ideas that can efficiently isolate a monomial in the given polynomial. In particular, we carry out the monomial isolation using nondeterministic automata.

We give the first subexponential time deterministic polynomial identity testing algorithm for depth-4 multilinear circuits with a small top fan-in. More accurately, our algorithm works for depth-4 multilinear circuits with a plus gate at the top (also known as $\Sigma\Pi\Sigma\Pi$ circuits) and has a running time of $\exp(\mathrm{poly}(\log(n),\log(s),k))$ where $n$ is the number of variables, $s$ is the size of the circuit, and $k$ is the fan-in of the top gate. In particular, when the circuit is of polynomial (or quasi-polynomial) size, our algorithm runs in quasi-polynomial time. Prior to this work, sub-exponential time deterministic algorithms were known for depth-$3$ circuits with small top fan-in and for very restricted versions of depth-$4$ circuits. The main ingredient in our proof is a new structural theorem for multilinear $\Sigma\Pi\Sigma\Pi(k)$ circuits. Roughly, this theorem shows that any nonzero multilinear $\Sigma\Pi\Sigma\Pi(k)$ circuit contains an “embedded” nonzero multilinear $\Sigma\Pi\Sigma(k)$ circuit. Using ideas from previous works on identity testing of sums of read-once formulas and of depth-3 multilinear circuits, we are able to exploit this structure and obtain an identity testing algorithm for multilinear $\Sigma\Pi\Sigma\Pi(k)$ circuits.

In this paper we give the first construction of a pseudorandom generator, with seed length O(log n), for CC0[p], the class of constant-depth circuits with unbounded fan-in MODpgates, for some prime p. More accurately, the seed length of our generator is O(log n) for any constant error ϵ > 0. In fact, we obtain our generator by fooling distributions generated by low degree polynomials, over Fp, when evaluated on the Boolean cube. This result significantly extends previous constructions that either required a long seed or that could only fool the distribution generated by linear functions over Fp, when evaluated on the Boolean cube. Enroute of constructing our PRG, we prove two structural results for low degree polynomials over finite fields that can be of independent interest. 1) Let f be an n-variate degree d polynomial over Fp. Then, for every ϵ > 0 there exists a subset S ⊂ [n], whose size depends only on d and ϵ, such that Σα∈Fpn:α≠0,αS=0|f̂(α)|2≤ ϵ. Namely, there is a constant size subset S such that the total weight of the nonzero Fourier coefficients that do not involve any variable from S is small. 2) Let f be an n-variate degree d polynomial over Fp. If the distribution of f when applied to uniform zero-one bits is ϵ-far (in statistical distance) from its distribution when applied to biased bits, then for every δ > 0, f can be approximated over zero-one bits, up to error δ, by a function of a small number (depending only on ϵ, δ and d) of lower degree polynomials.

We give the first sub-exponential time deterministic polynomial identity testing algorithm for depth-4 multilinear circuits with a small top fan-in. More accurately, our algorithm works for depth-4 circuits with a plus gate at the top (also known as ΣΠΣΠ circuits) and has a running time of exp(poly(log(n),log(s),k)) where n is the number of variables, s is the size of the circuit and k is the fan-in of the top gate. In particular, when the circuit is of polynomial (or quasi-polynomial) size, our algorithm runs in quasi-polynomial time. In [AV08], it was shown that derandomizing polynomial identity testing for general ΣΠΣΠ circuits implies a derandomization of polynomial identity testing in general arithmetic circuits. Prior to this work sub-exponential time deterministic algorithms were known for depth-$3$ circuits with small top fan-in and for very restricted versions of depth-4 circuits.

Motivated by the quantum algorithm for testing commutativity of black-box groups (Magniez and Nayak, 2007), we study the following problem: Given a black-box finite ring by an additive generating set and a multilinear polynomial over that ring, also accessed as a black-box function (we allow the indeterminates of the polynomial to be commuting or noncommuting), we study the problem of testing if the polynomial is an \emph{identity} for the given ring. We give a quantum algorithm with query complexity sub-linear in the number of generators for the ring, when the number of indeterminates of the input polynomial is small (ideally a constant). Towards a lower bound, we also show a reduction from a version of the collision problem (which is well studied in quantum computation) to a variant of this problem.

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, ldrldrldr , xn} computing a polynomial in F{x1, ldrldrldr , xn} of degree d with at most t monomials, where the variables xiare noncommuting, we give a deterministic polynomial identity test that checks if C equiv 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 isin F{x1, ldrldrldr , 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.