We show that the GCD of two univariate polynomials can be computed by (piece-wise) algebraic circuits of constant depth and polynomial size over any sufficiently large field, regardless of the characteristic. This extends a recent result of Andrews & Wigderson who showed such an upper bound over fields of zero or large characteristic. Our proofs are based on a recent work of Bhattacharjee, Kumar, Rai, Ramanathan, Saptharishi \& Saraf that shows closure of constant depth algebraic circuits under factorization. On our way to the proof, we show that any $n$-variate symmetric polynomial $P$ that has a small constant depth algebraic circuit can be written as the composition of a small constant depth algebraic circuit with elementary symmetric polynomials. This statement is a constant depth version of a result of Bläser & Jindal, who showed this for algebraic circuits of unbounded depth. As an application of our techniques, we also strengthen the closure results for factors of constant-depth circuits in the work of Bhattacharjee et al. over fields for small characteristic.

We show that algebraic formulas and constant-depth circuits are closed under taking factors. In other words, we show that if a multivariate polynomial over a field of characteristic zero has a small constant-depth circuit or formula, then all its factors can be computed by small constant-depth circuits or formulas respectively.

While efficient randomized algorithms for factorization of polynomials given by algebraic circuits have been known for decades, obtaining an even slightly non-trivial deterministic algorithm for this problem has remained an open question of great interest. This is true even when the input algebraic circuit has additional structure, for instance, when it is a constant-depth circuit. Indeed, no efficient deterministic algorithms are known even for the seemingly easier problem of factoring sparse polynomials or even the problem of testing the irreducibility of sparse polynomials.In this work, we make progress on these questions: we design a deterministic algorithm that runs in subexponential time, and when given as input a constant-depth algebraic circuit C over the field of rational numbers, it outputs algebraic circuits (of potentially unbounded depth) for all the irreducible factors of C, together with their multiplicities. In particular, we give the first subexponential time deterministic algorithm for factoring sparse polynomials.For our proofs, we rely on a finer understanding of the structure of power series roots of constant-depth circuits and the analysis of the Kabanets-Impagliazzo generator. In particular, we show that the Kabanets-Impagliazzo generator constructed using low-degree hard polynomials (explicitly constructed in the work of Limaye, Srinivasan & Tavenas) preserves not only the non-zeroness of small constant-depth circuits (as shown by Chou, Kumar & Solomon), but also their irreducibility and the irreducibility of their factors.

We prove that the most natural low-degree test for polynomials over finite fields is “robust” in the high-error regime for linear-sized fields. Specifically we consider the “local” agreement of a function$f:\mathbb{F}_{q}^{m}\rightarrow \mathbb{F}_{q}$from the space of degree-d polynomials, i.e., the expected agreement of the function from univariate degree-d polynomials over a randomly chosen line in$\mathbb{F}_{q}^{m}$, and prove that if this local agreement is$\varepsilon\geq\Omega((d/q)^{\tau}))$for some fixed$\tau > 0$, then there is a global degree-d polynomial$Q:\mathbb{F}_{q}^{m}\rightarrow \mathbb{F}_{q}$with agreement nearly$\varepsilon$with$f$. This settles a long-standing open question in the area of low-degree testing, yielding an$O(d)$-query robust test in the “high-error” regime (i.e., when$\varepsilon < 1/2)$. The previous results in this space either required$\varepsilon > 1/2$(Polishchuk & Spielman, STOC 1994), or$q=\Omega(d^{4})$(Arora & Sudan, Combinatorica 2003), orneeded to measure local distance on 2-dimensional “planes” rather than one-dimensional lines leading to$\Omega(d^{2})$-query complexity (Raz & Safra, STOC 1997). Our analysis follows the spirit of most previous analyses in first analyzing the low-variable case$(m=O(1))$and then “boot-strapping” to general multivariate settings. Our main technical novelty is a new analysis in the bivariate setting that exploits a previously known connection between multivariate factorization and finding (or testing) low-degree polynomials, in a non “black-box” manner. This connection was used roughly in a black-box manner in the work of Arora & Sudan — and we show that opening up this black box and making some delicate choices in the analysis leads to our essentially optimal analysis. A second contribution is a bootstrapping analysis which manages to lift analyses for$m=2$directly to analyses for general$m$, where previous works needed to work with$m=3$or$m=4$— arguably this bootstrapping is significantly simpler than those in prior works.

For every constant d, we design a subexponential time deterministic algorithm that takes as input a multivariate polynomial f given as a constant depth algebraic circuit over the field of rational numbers, and outputs all irreducible factors of f of degree at most d together with their respective multiplicities. Moreover, if f is a sparse polynomial, then the algorithm runs in quasipolynomial time.

We design nearly-linear time numerical algorithms for the problem of multivariate multipoint evaluation over the fields of rational, real and complex numbers. We consider both exact and approximate versions of the algorithm. The input to the algorithms are (1) coefficients of an m-variate polynomial f with degree d in each variable, and (2) points $\mathbf{a}_{1}, \ldots, \mathbf{a}_{N}$ each of whose coordinate has absolute value bounded by one. Approximate version: Given additionally an accuracy parameter t, the algorithm computes rational numbers $\beta_{1}, \ldots, \beta_{N}$ such that $\left|f\left(\mathbf{a}_{i}\right)-\beta_{i}\right| \leq 1 / 2^{t}$ for all i, and has a running time of $\left(\left(N m+d^{m}\right) t\right)^{1+o(1)}$ for all m and all sufficiently large d. Exact version (when over rationals): Given additionally a bound s on the bit-complexity of all the rational numbers in the input and output, the algorithm computes the rational numbers $f\left(\mathbf{a}_{1}\right), \ldots, f\left(\mathbf{a}_{N}\right)$, in time $\left(\left(N m+d^{m}\right) s\right)^{1+o(1)}$ for all m and all sufficiently large d. Our results also naturally extend to the case when the input is over the field of real or complex numbers under an appropriate standard model of representation of field elements in such fields.Prior to this work, a nearly-linear time algorithm for multivariate multipoint evaluation (exact or approximate) over any infinite field appears to be known only for the case of univariate polynomials, and was discovered in a recent work of Moroz [Proc. 62nd FOCS, 2021]. In this work, we extend this result from the univariate to the multivariate setting. However, our algorithm is based on ideas that seem to be conceptually different from those of Moroz [Proc. 62nd FOCS, 2021] and crucially relies on a recent algorithm of Bhargava, Ghosh, Guo, Kumar & Umans [Proc. 63rd FOCS, 2022] for multivariate multipoint evaluation over finite fields, and known efficient algorithms for the problems of rational number reconstruction and fast Chinese remaindering in computational number theory.

We study the following natural question on random sets of points in 𝔽₂^m: Given a random set of k points Z = {z₁, z₂, … , z_k} ⊆ 𝔽₂^m, what is the dimension of the space of degree at most r multilinear polynomials that vanish on all points in Z? We show that, for r ≤ γ m (where γ > 0 is a small, absolute constant) and k = (1-ε)⋅binom(m, ≤ r) for any constant ε > 0, the space of degree at most r multilinear polynomials vanishing on a random set Z = {z_1,…, z_k} has dimension exactly binom(m, ≤ r) - k with probability 1 - o(1). This bound shows that random sets have a much smaller space of degree at most r multilinear polynomials vanishing on them, compared to the worst-case bound (due to Wei (IEEE Trans. Inform. Theory, 1991)) of binom(m, ≤ r) - binom(log₂ k, ≤ r) ≫ binom(m, ≤ r) - k. Using this bound, we show that high-degree Reed-Muller codes (RM(m,d) with d > (1-γ) m) "achieve capacity" under the Binary Erasure Channel in the sense that, for any ε > 0, we can recover from (1-ε)⋅binom(m, ≤ m-d-1) random erasures with probability 1 - o(1). This also implies that RM(m,d) is also efficiently decodable from ≈ binom(m, ≤ m-(d/2)) random errors for the same range of parameters.

Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class VNP requires algebraic circuits of super-polynomial size. In a recent work of Chatterjee and the authors (FOCS 2020), it was shown that the subclasses of VP and VNP consisting of polynomials with bounded integer coefficients do have equations with small algebraic circuits. Their work left open the possibility that these results could perhaps be extended to all of VP or VNP. The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations.

A hitting-set generator (HSG) is a polynomial map ${{\mathsf{Gen}}}:{\mathbb{F}}^k \to {\mathbb{F}}^n$ such that for all $n$-variate polynomials $C$ of small enough circuit size and degree, if $C$ is nonzero, then $C\circ {{\mathsf{Gen}}}$ is nonzero. In this paper, we give a new construction of such an HSG assuming that we have an explicit polynomial of sufficient hardness. Formally, we prove the following result over any field ${\mathbb{F}}$ of characteristic zero: Let $k\in {\mathbb{N}}$ and $\delta > 0$ be arbitrary constants. Suppose ${\left\{ {P_d} \right\}}_{d\in {\mathbb{N}}}$ is an explicit family of $k$-variate polynomials such that $\deg P_d = d$ and $P_d$ requires algebraic circuits of size $d^\delta$. Then, there are explicit hitting sets of polynomial size for the class ${\mathsf{VP}}$. This is the first HSG in the algebraic setting that yields a complete derandomization of polynomial identity testing (PIT) for general circuits from a suitable algebraic hardness assumption. Unlike the prior constructions of such maps [N. Nisan and A. Wigderson, J. Comput. System Sci., 49 (1994), pp. 149--167; V. Kabanets and R. Impagliazzo, Comput. Complexity, 13 (2004), pp. 1--46; M. Agrawal, S. Ghosh, and N.Saxena, Proc. Natl. Acad. Sci. USA, 116 (2019), pp. 8107--8118; M. Kumar, R. Saptharishi, and A. Tengse, Proceedings of the \textup30th Annual ACM-SIAM Symposium on Discrete Algorithms, 2019, pp. 639--646], our construction is purely algebraic and does not rely on the notion of combinatorial designs. As a direct consequence, we show that even saving a single point from the “trivial” explicit, exponential sized hitting sets for constant-variate polynomials of low individual degree which are computable by small circuits implies a deterministic polynomial time algorithm for PIT. More precisely, we show the following: Let $k\in {\mathbb{N}}$ and $\delta > 0$ be arbitrary constants. Suppose for every $s$ large enough, there is an explicit hitting set of size at most $((s+1)^k - 1)$ for the class of $k$-variate polynomials of individual degree $s$ that are computable by size $s^\delta$ circuits. Then there is an explicit hitting set of size ${\operatorname{poly}}(s)$ for the class of $s$-variate polynomials, of degree $s$, that are computable by size $s$ circuits. As a consequence, we give a deterministic polynomial time construction of hitting sets for algebraic circuits, if a strengthening of the $\tau$-conjecture of Shub and Smale [M. Shub and S. Smale, Duke Math. J., 81 (1995), pp. 47--54; S. Smale, Math. Intelligencer, 20 (1998), pp. 7--15] is true.

For every constant , we show that there is a family {PN, c} of polynomials whose degree and algebraic circuit complexity are polynomially bounded in the number of variables, that satisfies the following properties: For every family {fn} of polynomials in VP, where fn is an n variate polynomial of degree at most ncwith bounded integer coefficients and for N=nc+nn, PN, c vanishes on the coefficient vector of fn. There exists a family {hn} of polynomials where hn is an n variate polynomial of degree at most ncwith bounded integer coefficients such that for N=nc+nn, PN, c does not vanish on the coefficient vector of hn. In other words, there are efficiently computable equations for polynomials in VP that have small integer coefficients. In fact, we also prove an analogous statement for the seemingly larger class VNP. Thus, in this setting of polynomials with small integer coefficients, this provides evidence against a natural proof like barrier for proving algebraic circuit lower bounds, a framework for which was proposed in the works of Forbes, Shpilka and Volk [1], and Grochow, Kumar, Saks and Saraf [2]. Our proofs are elementary and rely on the existence of (non-explicit) hitting sets for VP (and VNP) to show that there are efficiently constructible, low degree equations for these classes and also extend to finite fields of small size. Our proofs are elementary and rely on the existence of (non-explicit) hitting sets for VP (and VNP) to show that there are efficiently constructible, low degree equations for these classes and also extend to finite fields of small size.

A hitting-set generator (HSG) is a polynomial map Gen:Fk→ Fnsuch that for all n-variate polynomials Q of small enough circuit size and degree, if Q is non-zero, then Q o Gen is non-zero. In this paper, we give a new construction of such a HSG assuming that we have an explicit polynomial of sufficient hardness in the sense of approximative or border complexity. Formally, we prove the following result over any characteristic zero field F: Suppose P(z1,..., zk) is an explicit k-variate degree d polynomial that is not in the border of circuits of size s. Then, there is an explicit hitting-set generator Gen(P): F2k→ Fnsuch that every non-zero n-variate degree D polynomial Q(x) in the border of size s' circuits satisfies Q ≠ 0 ⇒ Q o Gen(P) ≠ 0 provided n10kd Ds'0 be a constant and k be a large enough constant. Suppose, for every s ≥ k, there is an explicit hitting set of size sk-δfor all degree s polynomials in the border of k-variate size s algebraic circuits. Then, there is an explicit hitting set of size poly(s) for the border s-variate algebraic circuits of size s and degree s. Unlike the prior constructions of such maps (e.g.[NW94], [KI04], [AGS19], [KST19]), our construction is purely algebraic and does not rely on the notion of combinatorial designs.

We study the question of algebraic rank or transcendence degree preserving homomorphisms over finite fields. This concept was first introduced by Beecken et al. [Malte Beecken et al., 2013] and exploited by them and Agrawal et al. [Manindra Agrawal et al., 2016] to design algebraic independence based identity tests using the Jacobian criterion over characteristic zero fields. An analogue of such constructions over finite characteristic fields were unknown due to the failure of the Jacobian criterion over finite characteristic fields. Building on a recent criterion of Pandey, Saxena and Sinhababu [Anurag Pandey et al., 2018], we construct explicit faithful maps for some natural classes of polynomials in fields of positive characteristic, when a certain parameter called the inseparable degree of the underlying polynomials is bounded (this parameter is always 1 in fields of characteristic zero). This presents the first generalisation of some of the results of Beecken, Mittmann and Saxena [Malte Beecken et al., 2013] and Agrawal, Saha, Saptharishi, Saxena [Manindra Agrawal et al., 2016] in the positive characteristic setting.

We show that any $n$-variate polynomial computable by a syntactically multilinear circuit of size $\operatorname{poly}(n)$ can be computed by a depth-$4$ syntactically multilinear ($ΣΠΣΠ$) circuit of size at most $\exp\left({O\left(\sqrt{n\log n}\right)}\right)$. For degree $d = ω(n/\log n)$, this improves upon the upper bound of $\exp\left({O(\sqrt{d}\log n)}\right)$ obtained by Tavenas~\cite{T15} for general circuits, and is known to be asymptotically optimal in the exponent when $d < n^ε$ for a small enough constant $ε$. Our upper bound matches the lower bound of $\exp\left({Ω\left(\sqrt{n\log n}\right)}\right)$ proved by Raz and Yehudayoff~\cite{RY09}, and thus cannot be improved further in the exponent. Our results hold over all fields and also generalize to circuits of small individual degree. More generally, we show that an $n$-variate polynomial computable by a syntactically multilinear circuit of size $\operatorname{poly}(n)$ can be computed by a syntactically multilinear circuit of product-depth $Δ$ of size at most $\exp\left(O\left(Δ\cdot (n/\log n)^{1/Δ} \cdot \log n\right)\right)$. It follows from the lower bounds of Raz and Yehudayoff (CC 2009) that in general, for constant $Δ$, the exponent in this upper bound is tight and cannot be improved to $o\left(\left(n/\log n\right)^{1/Δ}\cdot \log n\right)$.

The classical lemma of Ore-DeMillo-Lipton-Schwartz-Zippel states that any nonzero polynomial f(xi, …, xn) of degree at most s will evaluate to a nonzero value at some point on a grid with |S| > s. Thus, there is a deterministic polynomial identity test (PIT) for all degrees size-s algebraic circuits in n variables that runs in time poly(s) · (s + 1)n. In a surprising recent result, Agrawal, Ghosh and Saxena (STOC 2018) showed any deterministic blackbox PIT algorithm for degree-s, size-s, n-variate circuits with running time as bad as (sn0.5−δ) Huge(n), where δ > 0 and Huge(n) is an arbitrary function, can be used to construct blackbox PIT algorithms for degree-s size s circuits with running time sexp(exp(O(log* s))). Agrawal et al. asked if a similar conclusion followed if their hypothesis was weakened to having deterministic PIT with running time so(n) · Huge(n). In this paper, we answer their question in the affirmative. We show that, given a deterministic blackbox PIT that runs in time so(n) · Huge(n) for all degree-s size-s algebraic circuits over n variables, we can obtain a deterministic blackbox PIT that runs in time sexp(exp(O(log* s))) for all degree-s size-s algebraic circuits over n variables. In other words, any blackbox PIT with just a slightly nontrivial exponent of s compared to the trivial sO(n) test can be used to give a nearly polynomial time blackbox PIT algorithm.

Surprising and beautiful depth reduction results show that sufficiently strong lower bounds for bounded-depth arithmetic circuits imply superpolynomial lower bounds for general arithmetic circuits. Motivated by this, in the last few years, there has been renewed interest in the question of proving superpolynomial lower bounds for bounded-depth circuits, starting with homogeneous depth-4 circuits. Following a sequence of recent results, we now know an $n^{\Omega(\sqrt{d})}$ lower bound for homogeneous depth-4 circuits, for an explicit polynomial of degree $d$ in $n$ variables. It is also known that any asymptotic improvement in the exponent of this lower bound implies a superpolynomial lower bound for general arithmetic circuits. An intriguing fact is that in spite of all this recent progress which seems to bring us to the edge of the chasm at depth-4, it appears to shine very little light on the question of lower bounds for even slight generalizations of homogeneous depth-4 circuits. Indeed, superquadratic lower bounds are not known for even nonhomogeneous depth-4 circuits, or homogeneous depth-5 circuits, and supercubic lower bounds are not known for nonhomogeneous depth-3 circuits. In this paper, we study homogeneous depth-5 circuits, with the aim of proving superpolynomial lower bounds for them and understanding their computational power and limitations when compared to homogeneous depth-4 circuits. We prove the following results. Depth-$5$ versus depth-$4$ circuits. We show that there is a family of polynomials $\{P_n\}$, where $P_n$ is a polynomial in $n$ variables of degree at most $d=O(\log^2n)$, such that $P_n$ can be computed by linear sized homogeneous depth-5 circuits; $P_n$ can be computed by $\operatorname{poly}(n)$ sized nonhomogeneous depth-3 circuits; and any homogeneous depth-4 circuit computing $P_n$ must have size at least $n^{\Omega(\sqrt{d})}$. This shows that the parameters for the depth reduction results of M. Agrawal and V. Vinay, P. Koiran, and S. Tavenas are tight for extremely restricted classes of arithmetic circuits, for instance homogeneous depth-5 circuits and nonhomogeneous depth-3 circuits, and over an appropriate range of parameters. This also qualitatively improves a result of Kumar and Saraf, which showed that the parameters of depth reductions are optimal for algebraic branching programs. Depth-$5$ circuits over small fields. We show that there is an explicit family $\{P_d\}$ of polynomials, where $P_d$ is of degree $d$ in $n=d^{O(1)}$ variables, such that over all finite fields $\mathbb{F}_q$, any homogeneous depth-5 circuit which computes $P_d$ must have size at least $\exp(\Omega_q(\sqrt{d}))$. To the best of our knowledge, this is the first superpolynomial lower bound for this class for any field $\mathbb{F}_q\neq\mathbb{F}_2$. Our proof builds on the ideas developed on the way to proving lower bounds for homogeneous depth-4 circuits and for nonhomogeneous depth-3 circuits over finite fields. Our key insight is to look at the space of shifted partial derivatives of a polynomial as a space of functions from $\mathbb{F}_q^n\rightarrow\mathbb{F}_q$ as opposed to looking at them as a space of formal polynomials and builds over a tighter analysis of the lower bound of Kumar and Saraf.

We study the class of non-commutative Unambiguous circuits or Unique-Parse-Tree (UPT) circuits, and a related model of Few-Parse-Trees (FewPT) circuits (which were recently introduced by Lagarde, Malod and Perifel [LMP16] and Lagarde, Limaye and Srinivasan [LLS17]) and give the following constructions: (1) An explicit hitting set of quasipolynomial size for UPT circuits, (2) An explicit hitting set of quasipolynomial size for FewPT circuits (circuits with constantly many parse tree shapes), (3) An explicit hitting set of polynomial size for UPT circuits (of known parse tree shape), when a parameter of preimage-width is bounded by a constant. The above three results are extensions of the results of [AGKS15], [GKST15] and [GKS16] to the setting of UPT circuits, and hence also generalize their results in the commutative world from read-once oblivious algebraic branching programs (ROABPs) to UPT-set-multilinear circuits. The main idea is to study shufflings of non-commutative polynomials, which can then be used to prove suitable depth reduction results for UPT circuits and thereby allow a careful translation of the ideas in [AGKS15], [GKST15] and [GKS16].

In this paper, we show exponential lower bounds for the class of homogeneous depth-5 circuits over all small finite fields. More formally, we show that there is an explicit family {P_d} of polynomials in VNP, where P_d is of degree d in n = d^{O(1)} variables, such that over all finite fields GF(q), any homogeneous depth-5 circuit which computes P_d must have size at least exp(Omega_q(sqrt{d})). To the best of our knowledge, this is the first super-polynomial lower bound for this class for any non-binary field. Our proof builds up on the ideas developed on the way to proving lower bounds for homogeneous depth-4 circuits [Gupta et al., Fournier et al., Kayal et al., Kumar-Saraf] and for non-homogeneous depth-3 circuits over finite fields [Grigoriev-Karpinski, Grigoriev-Razborov]. Our key insight is to look at the space of shifted partial derivatives of a polynomial as a space of functions from GF(q)^n to GF(q) as opposed to looking at them as a space of formal polynomials and builds over a tighter analysis of the lower bound of Kumar and Saraf [Kumar-Saraf].

Reed-Muller (RM) codes encode an m-variate polynomial of degree at most r by evaluating it on all points in {0,1}m. We denote this code by RM(r,m). The minimum distance of RM(r,m) is 2m-rand so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result. In this paper we give an efficient algorithm (in the block length n=2m) for decoding random errors in RM codes far beyond the minimum distance. Specifically, for low-rate codes (of degree r=o(√m)), we can correct a random set of (1/2-o(1))n errors with high probability. For high rate codes (of degree m-r for r=o(√m/log m)), we can correct roughly mr/2errors. More generally, for any integer r, our algorithm can correct any error pattern in RM(m-(2r+2),m), for which the same erasure pattern can be corrected in RM(m-(r+1),m). The results above are obtained by applying recent results of Abbe, Shpilka, and Wigderson (STOC, 2015) and Kudekar et al. (STOC, 2016) regarding the ability of RM codes to correct random erasures. The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding RM codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the error-locating pair method of Pellikaan, Duursma, and Kötter that generalizes the Berlekamp-Welch algorithm for decoding Reed-Solomon codes.

Read-k oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ROABPs). In this work, we give an exponential lower bound of exp(n/k^{O(k)}) on the width of any read-k oblivious ABP computing some explicit multilinear polynomial f that is computed by a polynomial size depth-3 circuit. We also study the polynomial identity testing (PIT) problem for this model and obtain a white-box subexponential-time PIT algorithm. The algorithm runs in time 2^{~O(n^{1-1/2^{k-1}})} and needs white box access only to know the order in which the variables appear in the ABP.

We say that a circuit $C$ over a field $F$ functionally computes an $n$-variate polynomial $P$ if for every $x \in \{0,1\}^n$ we have that $C(x) = P(x)$. This is in contrast to syntactically computing $P$, when $C \equiv P$ as formal polynomials. In this paper, we study the question of proving lower bounds for homogeneous depth-$3$ and depth-$4$ arithmetic circuits for functional computation. We prove the following results : 1. Exponential lower bounds homogeneous depth-$3$ arithmetic circuits for a polynomial in $VNP$. 2. Exponential lower bounds for homogeneous depth-$4$ arithmetic circuits with bounded individual degree for a polynomial in $VNP$. Our main motivation for this line of research comes from our observation that strong enough functional lower bounds for even very special depth-$4$ arithmetic circuits for the Permanent imply a separation between ${\#}P$ and $ACC$. Thus, improving the second result to get rid of the bounded individual degree condition could lead to substantial progress in boolean circuit complexity. Besides, it is known from a recent result of Kumar and Saptharishi [KS15] that over constant sized finite fields, strong enough average case functional lower bounds for homogeneous depth-$4$ circuits imply superpolynomial lower bounds for homogeneous depth-$5$ circuits. Our proofs are based on a family of new complexity measures called shifted evaluation dimension, and might be of independent interest.

In this paper, we show that there is a family of polynomials P_n, where P_n is a polynomial in n variables of degree at most d = O(log^2(n)), such that * P_n can be computed by linear sized homogeneous depth-5 arithmetic circuits, * P_n can be computed by poly(n) sized non-homogeneous depth-3 arithmetic circuits. * Any homogeneous depth-4 arithmetic circuit computing P_n must have size at least n^{Omega(sqrt(d))}. This shows that the parameters for the depth reduction results of [Agrawal-Vinay 08, Koiran 12, Tavenas 13] are tight for extremely restricted classes of arithmetic circuits, for instance homogeneous depth-5 circuits and non-homogeneous depth-3 circuits, and over an appropriate range of parameters, qualitatively improve a result of [Kumar-Saraf 14], which showed that the parameters of depth reductions are optimal for algebraic branching programs. As an added advantage, our proofs are much shorter and simpler than the two known proofs of n^{Omega(sqrt(d))} lower bound for homogeneous depth-4 circuits [Kayal-Limaye-Saha-Srinivasan 14, Kumar-Saraf 14], albeit our proofs only work when d = O(log^2(n)).

Reed-Muller codes encode an m-variate polynomial of degree r by evaluating it on all points in {0,1}m. We denote this code by RM(m,r). The minimal distance of RM(m,r) is 2m−r and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result.

We show that, over ${\mathbb Q}$, if an $n$-variate polynomial of degree $d = n^{O(1)}$ is computable by an arithmetic circuit of size $s$ (respectively, by an arithmetic branching program of size $s$), then it can also be computed by a depth-3 circuit (i.e., a $\Sigma \Pi \Sigma$ circuit) of size $\exp(O(\sqrt{d \log n \log d\log s}))$ (respectively, of size $\exp(O(\sqrt{d \log n \log s}))$. In particular this yields a $\Sigma \Pi \Sigma$ circuit of size $\exp(O(\sqrt{d} \cdot \log d))$ computing the $d \times d$ determinant $\mathsf{Det}_d$. It also means that if we can prove a lower bound of $\exp(\omega(\sqrt{d} \cdot \log d))$ on the size of any $\Sigma \Pi \Sigma$ circuit computing the $d \times d$ permanent $\mathsf{Perm}_d$, then we get superpolynomial lower bounds for the size of any arithmetic branching program computing $\mathsf{Perm}_d$. We then give some further results pertaining to derandomizing polynomial identity testing and circuit lower bounds. The $\Sigma \Pi \Sigma $ circuits that we construct have the property that (some of) the intermediate polynomials have degree much higher than $d$. Indeed such a counterintuitive construction is unavoidable---it is known that in any $\Sigma \Pi \Sigma$ circuit $C$ computing either $\mathsf{Det}_d$ or $\mathsf{Perm}_d$, if every multiplication gate has fanin at most $d$ (or any constant multiple thereof), then $C$ must have size at least $\exp(\Omega(d))$.

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.

A recent result of Moshkovitz [Moshkovitz14] presented an ingenious method to provide a completely elementary proof of the Parallel Repetition Theorem for certain projection games via a construction called fortification. However, the construction used in [Moshkovitz14] to fortify arbitrary label cover instances using an arbitrary extractor is insufficient to prove parallel repetition. In this paper, we provide a fix by using a stronger graph that we call fortifiers. Fortifiers are graphs that have both l_1 and l_2 guarantees on induced distributions from large subsets. We then show that an expander with sufficient spectral gap, or a bi-regular extractor with stronger parameters (the latter is also the construction used in an independent update [Moshkovitz15] of [Moshkovitz14] with an alternate argument), is a good fortifier. We also show that using a fortifier (in particular l_2 guarantees) is necessary for obtaining the robustness required for fortification.

We give deterministic black-box polynomial identity testing algorithms for multilinear read-once oblivious algebraic branching programs (ROABPs), in nO(log2 n) time. Further, our algorithm is oblivious to the order of the variables. This is the first sub-exponential time algorithm for this model. Furthermore, our result has no known analogue in the model of read-once oblivious boolean branching programs with unknown order.

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).

Agrawal and Vinay [2008], Koiran [2012], and Tavenas [2013] have recently shown that an exp (ω(√ n log n )) lower bound for depth four homogeneous circuits computing the permanent with bottom layer of × gates having fanin bounded by √n translates to a superpolynomial lower bound for general arithmetic circuits computing the permanent. Motivated by this, we examine the complexity of computing the permanent and determinant via such homogeneous depth four circuits with bounded bottom fanin. We show here that any homogeneous depth four arithmetic circuit with bottom fanin bounded by √n computing the permanent (or the determinant) must be of size exp,(Ω(√ n )).

Agrawal-Vinay [AV08] and Koiran [Koi12] have recently shown that an exp(ω(√n log2n)) lower bound for depth four homogeneous circuits computing the permanent with bottom layer of × gates having fanin bounded by √n translates to super-polynomial lower bound for general arithmetic circuits computing the permanent. Motivated by this, we examine the complexity of computing the permanent and determinant via such homogeneous depth four circuits with bounded bottom fanin. We show here that any homogeneous depth four arithmetic circuit with bottom fanin bounded by √n computing the permanent (or the determinant) must be of size exp(Ω(√n)).

We show that, over Q, if an n-variate polynomial of degree d = nO(1)is computable by an arithmetic circuit of size s (respectively by an arithmetic branching program of size s) then it can also be computed by a depth three circuit (i.e. a ΣΠΣ-circuit) of size exp(O(√(d log n log d log s))) (respectively of size exp(O(√(d log n log s))). In particular this yields a ΣΠΣ circuit of size exp(O(√(d log d))) computing the d × d determinant Detd. It also means that if we can prove a lower bound of exp(omega(√(d log d))) on the size of any ΣΠΣ-circuit computing the d × d permanent Permdthen we get super polynomial lower bounds for the size of any arithmetic branching program computing Permd. We then give some further results pertaining to derandomizing polynomial identity testing and circuit lower bounds. The ΣΠΣ circuits that we construct have the property that (some of) the intermediate polynomials have degree much higher than d. Indeed such a counterintuitive construction is unavoidable - it is known that in any ΣΠΣ circuit C computing either Detdor Perm_d, if every multiplication gate has fanin at most d (or any constant multiple thereof) then C must have size at least exp(Ω(d)).

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.

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 $\mathbb{F}$.