A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic black-box identity testing algorithm for univariate polynomials of the form $\sum_{j=0}^t c_j X^{α_j} (a + b X)^{β_j}$. From our algorithm we derive an exponential lower bound for representations of polynomials such as $\prod_{i=1}^{2^n} (X^i-1)$ under this form. It has been conjectured that these polynomials are hard to compute by general arithmetic circuits. Our result shows that the "hardness from derandomization" approach to lower bounds is feasible for a restricted class of arithmetic circuits. The proof is based on techniques from algebraic number theory, and more precisely on properties of the height function of algebraic numbers.
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 para…
We give improved hitting sets for two special cases of Read-once Oblivious Arithmetic Branching Programs (ROABP). First is the case of an ROABP with known order of the variables. The best previously known hitting set for this case had size $(nw)^{O(\log n)}$ where $n$ is the number of variables and $w$ is the width of the ROABP. Even for a constant-width ROABP, nothing better than a quasi-polynomial bound was known. We improve the hitting-set size for the known-order case to $n^{O(\log w)}$. In particular, this gives the first polynomial-size hitting set for constant-width ROABP (known-order).…
In this paper we study polynomials in $\text{VP}_e$ (polynomial-sized formulas) and in $ΣΠΣ$ (polynomial-size depth-$3$ circuits) whose orbits, under the action of the affine group $\text{GL}_n^{\text{aff}}(\mathbb{F})$, are $\mathit{dense}$ in their ambient class. We construct hitting sets and interpolating sets for these orbits as well as give reconstruction algorithms. As $\text{VP}=\text{VNC}^2$, our results for $\text{VP}_e$ translate immediately to $\text{VP}$ with a quasipolynomial blow up in parameters. If any of our hitting or interpolating sets could be made $\mathit{robust}$ then th…
We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich for boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike the boolean setting, there has been no concrete evidence demonstrating that this is a barrier to obtaining super-polynomial lower bounds for general algebraic circuits, as there is little understanding whether algebraic circuits are expressive enough to support "cryptography" secure against algebraic …
We show that if we can design poly($s$)-time hitting-sets for $Σ\wedge^aΣΠ^{O(\log s)}$ circuits of size $s$, where $a=ω(1)$ is arbitrarily small and the number of variables, or arity $n$, is $O(\log s)$, then we can derandomize blackbox PIT for general circuits in quasipolynomial time. This also establishes that either E$\not\subseteq$\#P/poly or that VP$\ne$VNP. In fact, we show that one only needs a poly($s$)-time hitting-set against individual-degree $a'=ω(1)$ polynomials that are computable by a size-$s$ arity-$(\log s)$ $ΣΠΣ$ circuit (note: $Π$ fanin may be $s$). Alternatively, we claim …
We give a $n^{O(\log n)}$-time ($n$ is the input size) blackbox polynomial identity testing algorithm for unknown-order read-once oblivious algebraic branching programs (ROABP). The best result known for this class was $n^{O(\log^2 n)}$ due to Forbes-Saptharishi-Shpilka (STOC 2014), and that too only for multilinear ROABP. We get rid of their exponential dependence on the individual degree. With this, we match the time-complexity for the unknown order ROABP with the known order ROABP (due to Forbes-Shpilka (FOCS 2013)) and also with the depth-$3$ set-multilinear circuits (due to Agrawal-Saha-S…
Given a hypergraph $H = (V,E)$, what is the smallest subset $X \subseteq V$ such that $e \cap X \neq \emptyset$ holds for all $e \in E$? This problem, known as the hitting set problem, is a basic problem in parameterized complexity theory. There are well-known kernelization algorithms for it, which get a hypergraph $H$ and a number $k$ as input and output a hypergraph $H'$ such that (1) $H$ has a hitting set of size $k$ if, and only if, $H'$ has such a hitting set and (2) the size of $H'$ depends only on $k$ and on the maximum cardinality $d$ of edges in $H$. The algorithms run in polynomial t…
In this paper we study the complexity of constructing a hitting set for the closure of VP, the class of polynomials that can be infinitesimally approximated by polynomials that are computed by polynomial sized algebraic circuits, over the real or complex numbers. Specifically, we show that there is a PSPACE algorithm that given n,s,r in unary outputs a set of n-tuples over the rationals of size poly(n,s,r), with poly(n,s,r) bit complexity, that hits all n-variate polynomials of degree-r that are the limit of size-s algebraic circuits. Previously it was known that a random set of this size is a…
The depth-$3$ model has recently gained much importance, as it has become a stepping-stone to understanding general arithmetic circuits. Its restriction to multilinearity has known exponential lower bounds but no nontrivial blackbox identity tests. In this paper we take a step towards designing such hitting-sets. We define a notion of distance for multilinear depth-$3$ circuits (say, in $n$ variables and $k$ product gates) that measures how far are the partitions from a mere refinement. The $1$-distance strictly subsumes the set-multilinear model, while $n$-distance captures general multilinea…
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. 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 efficien…
In their paper on the "chasm at depth four", Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2^o(m) also admit arithmetic circuits of depth four and size 2^o(m). This theorem shows that for problems such as arithmetic circuit lower bounds or black-box derandomization of identity testing, the case of depth four circuits is in a certain sense the general case. In this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits. For instance, we show that if the permanent of…
We consider the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of monomials. We show that these problems are complete for subclasses of the counting hierarchy which had few or no known natural complete problems. We also study these questions for circuits computing multilinear polynomials.
The motivation for this paper is to study the complexity of constant-width arithmetic circuits. Our main results are the following. 1. For every k > 1, we provide an explicit polynomial that can be computed by a linear-sized monotone circuit of width 2k but has no subexponential-sized monotone circuit of width k. It follows, from the definition of the polynomial, that the constant-width and the constant-depth hierarchies of monotone arithmetic circuits are infinite, both in the commutative and the noncommutative settings. 2. We prove hardness-randomness tradeoffs for identity testing constant-…
We show that the two main reduction notions in arithmetic circuit complexity, p-projections and c-reductions, differ in power. We do so by showing unconditionally that there are polynomials that are VNP-complete under c-reductions but not under p-projections. We also show that the question of which polynomials are VNP-complete under which type of reductions depends on the underlying field.
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 …
In this paper we give a characterization of both Boolean and arithmetic circuit classes of logarithmic depth in the vein of descriptive complexity theory, i.e., the Boolean classes $\textrm{NC}^1$, $\textrm{SAC}^1$ and $\textrm{AC}^1$ as well as their arithmetic counterparts $\#\textrm{NC}^1$, $\#\textrm{SAC}^1$ and $\#\textrm{AC}^1$. We build on Immerman's characterization of constant-depth polynomial-size circuits by formulas of first-order logic, i.e., $\textrm{AC}^0 = \textrm{FO}$, and augment the logical language with an operator for defining relations in an inductive way. Considering sli…
We prove a lower bound of $Ω(n^2/\log^2 n)$ on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial $f(x_1, \ldots, x_n)$. Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff ([RSY08]), who proved a lower bound of $Ω(n^{4/3}/\log^2 n)$ for the same polynomial. Our improvement follows from an asymptotically optimal lower bound for a generalized version of Galvin's problem in extremal set theory.
We prove exponential lower bounds on the size of homogeneous depth 4 arithmetic circuits computing an explicit polynomial in $VP$. Our results hold for the {\it Iterated Matrix Multiplication} polynomial - in particular we show that any homogeneous depth 4 circuit computing the $(1,1)$ entry in the product of $n$ generic matrices of dimension $n^{O(1)}$ must have size $n^{Ω(\sqrt{n})}$. Our results strengthen previous works in two significant ways. Our lower bounds hold for a polynomial in $VP$. Prior to our work, Kayal et al [KLSS14] proved an exponential lower bound for homogeneous depth 4 c…
A hitting-set generator (HSG) is a polynomial map $G:\mathbb{F}^k \to \mathbb{F}^n$ such that for all $n$-variate polynomials $C$ of small enough circuit size and degree, if $C$ is nonzero, then $C\circ G$ 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 over any field of characteristic zero: Let $k\in \mathbb{N}$ and $δ> 0$ be arbitrary constants. Suppose $\{P_d\}_{d\in \mathbb{N}}$ is an explicit family of $k$-variate polynomials such that $\operatorname{deg} P_d = d$ and…
We show that over the field of complex numbers, \emph{every} homogeneous polynomial of degree $d$ can be approximated (in the border complexity sense) by a depth-$3$ arithmetic circuit of top fan-in at most $d+1$. This is quite surprising since there exist homogeneous polynomials $P$ on $n$ variables of degree $2$, such that any depth-$3$ arithmetic circuit computing $P$ must have top fan-in at least $Ω(n)$. As an application, we get a new tradeoff between the top fan-in and formal degree in an approximate analog of the celebrated depth reduction result of Gupta, Kamath, Kayal and Saptharishi …
In this paper, we prove superpolynomial lower bounds for the class of homogeneous depth 4 arithmetic circuits. We give an explicit polynomial in VNP of degree $n$ in $n^2$ variables such that any homogeneous depth 4 arithmetic circuit computing it must have size $n^{Ω(\log \log n)}$. Our results extend the works of Nisan-Wigderson [NW95] (which showed superpolynomial lower bounds for homogeneous depth 3 circuits), Gupta-Kamath-Kayal-Saptharishi and Kayal-Saha-Saptharishi [GKKS13, KSS13] (which showed superpolynomial lower bounds for homogeneous depth 4 circuits with bounded bottom fan-in), Kum…
Motivated by the Hadamard product of matrices we define the Hadamard product of multivariate polynomials and study its arithmetic circuit and branching program complexity. We also give applications and connections to polynomial identity testing. Our main results are the following. 1. We show that noncommutative polynomial identity testing for algebraic branching programs over rationals is complete for the logspace counting class $\ceql$, and over fields of characteristic $p$ the problem is in $\ModpL/\Poly$. 2.We show an exponential lower bound for expressing the Raz-Yehudayoff polynomial as t…
In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds we obtain lower bounds for these models. For depth-3 multilinear formulas, of size $\exp(n^δ)$, we give a hitting set of size $\exp(\tilde{O}(n^{2/3 + 2δ/3}))$. This implies a lower bound of $\exp(\tildeΩ(n^{1/2}))$ for depth-3 multilinear formulas, for some explicit polynomial. For depth-4 multilinear formulas, of size $\exp(n^δ)$, we give a hitting set of size $\exp(\tilde{O}(n^{2/3 + 4δ/3}))$. This implies a lower bound of …
We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits. In particular, we obtain the following results : As our main result, we prove that any homogeneous depth-3 circuit for computing the product of $d$ matrices of dimension $n \times n$ requires $Ω(n^{d-1}/2^d)$ size. This improves the lower bounds by Nisan and Wigderson(1995) when $d=ω(1)$. There is an exp…
In recent years, a very exciting and promising method for proving lower bounds for arithmetic circuits has been proposed. This method combines the method of {\it depth reduction} developed in the works of Agrawal-Vinay [AV08], Koiran [Koi12] and Tavenas [Tav13], and the use of the shifted partial derivative complexity measure developed in the works of Kayal [Kay12] and Gupta et al [GKKS13a]. These results inspired a flurry of other beautiful results and strong lower bounds for various classes of arithmetic circuits, in particular a recent work of Kayal et al [KSS13] showing superpolynomial low…
Powerful skew arithmetic circuits are introduced. These are skew arithmetic circuits with variables, where input gates can be labelled with powers $x^n$ for binary encoded numbers $n$. It is shown that polynomial identity testing for powerful skew arithmetic circuits belongs to $\mathsf{coRNC}^2$, which generalizes a corresponding result for (standard) skew circuits. Two applications of this result are presented: (i) Equivalence of higher-dimensional straight-line programs can be tested in $\mathsf{coRNC}^2$; this result is even new in the one-dimensional case, where the straight-line programs…
A rational number can be naturally presented by an arithmetic computation (AC): a sequence of elementary arithmetic operations starting from a fixed constant, say 1. The asymptotic complexity issues of such a representation are studied e.g. in the framework of the algebraic complexity theory over arbitrary field. Here we study a related problem of the complexity of performing arithmetic operations and computing elementary predicates, e.g. ``='' or ``>'', on rational numbers given by AC. In the first place, we prove that AC can be efficiently simulated by the exact semidefinite programming (SDP…
In this paper, we study the structure of set-multilinear arithmetic circuits and set-multilinear branching programs with the aim of showing lower bound results. We define some natural restrictions of these models for which we are able to show lower bound results. Some of our results extend existing lower bounds, while others are new and raise open questions. More specifically, our main results are the following: (1) We observe that set-multilinear arithmetic circuits can be transformed into shallow set-multilinear circuits efficiently, similar to depth reduction results of [VSBR83,RY08] for mo…
Algorithmic tools for graphs of small treewidth are used to address questions in complexity theory. For both arithmetic and Boolean circuits, it is shown that any circuit of size $n^{O(1)}$ and treewidth $O(\log^i n)$ can be simulated by a circuit of width $O(\log^{i+1} n)$ and size $n^c$, where $c = O(1)$, if $i=0$, and $c=O(\log \log n)$ otherwise. For our main construction, we prove that multiplicatively disjoint arithmetic circuits of size $n^{O(1)}$ and treewidth $k$ can be simulated by bounded fan-in arithmetic formulas of depth $O(k^2\log n)$. From this we derive the analogous statement…
An efficient randomized polynomial identity test for noncommutative polynomials given by noncommutative arithmetic circuits remains an open problem. The main bottleneck to applying known techniques is that a noncommutative circuit of size $s$ can compute a polynomial of degree exponential in $s$ with a double-exponential number of nonzero monomials. In this paper, we report some progress by dealing with two natural subcases (both allow for polynomials of exponential degree and a double exponential number of monomials): (1) We consider \emph{$+$-regular} noncommutative circuits: these are homog…
Arithmetic circuits (AC) are circuits over the real numbers with 0/1-valued input variables whose gates compute the sum or the product of their inputs. Positive AC -- that is, AC representing non-negative functions -- subsume many interesting probabilistic models such as probabilistic sentential decision diagram (PSDD) or sum-product network (SPN) on indicator variables. Efficient algorithms for many operations useful in probabilistic reasoning on these models critically depend on imposing structural restrictions to the underlying AC. Generally, adding structural restrictions yields new tracta…
Assuming the Generalised Riemann Hypothesis (GRH), we show that for all k, there exist polynomials with coefficients in $\MA$ having no arithmetic circuits of size O(n^k) over the complex field (allowing any complex constant). We also build a family of polynomials that can be evaluated in AM having no arithmetic circuits of size O(n^k). Then we investigate the link between fixed-polynomial size circuit bounds in the Boolean and arithmetic settings. In characteristic zero, it is proved that $\NP \not\subset \size(n^k)$, or $\MA \subset \size(n^k)$, or NP=MA imply lower bounds on the circuit siz…
A $ΣΠΣΠ(k)$ circuit $C=\sum_{i=1}^kF_i=\sum_{i=1}^k\prod_{j=1}^{d_i}f_{ij}$ is unmixed if for each $i\in[k]$, $F_i=f_{i1}(x_1)... f_{in}(x_n)$, where each $f_{ij}$ is a univariate polynomial given in the sparse representation. In this paper, we give a polynomial time black-box algorithm of identity testing for the low degree unmixed $ΣΠΣΠ(k)$ circuits. In order to obtain the black-box algorithm, we first show that a special class of low degree unmixed $ΣΠΣΠ(k)$ circuits of size $s$ is $s^{O(k^2)}$-sparse. Then we construct a hitting set $\mathcal{H}$ in polynomial time for the low degree unmix…
We study the class $\textrm{AC}^0$ of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fan-in addition and multiplication gates. No model-theoretic characterization for arithmetic circuit classes is known so far. Inspired by Immerman's characterization of the Boolean class $\textrm{AC}^0$, we remedy this situation and develop such a characterization of $\textrm{AC}^0$. Our characterization can be interpreted as follows: Functions in $\textrm{AC}^0$ are exactly those functions counting winning strategies in first-order model checking games. A consequence of …
In a sequence of seminal results in the 80's, Kaltofen showed that the complexity class VP is closed under taking factors. A natural question in this context is to understand if other natural classes of multivariate polynomials, for instance, arithmetic formulas, algebraic branching programs, bounded depth arithmetic circuits or the class VNP, are closed under taking factors. In this paper, we show that all factors of degree at most $\log^a n$ of polynomials with poly(n) size depth $k$ circuits have poly(n) size circuits of depth at most $O(k + a)$. This partially answers a question of Shpilka…
We prove a downward separation for $\mathsfΣ_2$-time classes. Specifically, we prove that if $Σ_2$E does not have polynomial size non-deterministic circuits, then $Σ_2$SubEXP does not have \textit{fixed} polynomial size non-deterministic circuits. To achieve this result, we use Santhanam's technique on augmented Arthur-Merlin protocols defined by Aydinlioğlu and van Melkebeek. We show that augmented Arthur-Merlin protocols with one bit of advice do not have fixed polynomial size non-deterministic circuits. We also prove a weak unconditional derandomization of a certain type of promise Arthur-M…
This survey presents a necessarily incomplete (and biased) overview of results at the intersection of arithmetic circuit complexity, structured matrices and deep learning. Recently there has been some research activity in replacing unstructured weight matrices in neural networks by structured ones (with the aim of reducing the size of the corresponding deep learning models). Most of this work has been experimental and in this survey, we formalize the research question and show how a recent work that combines arithmetic circuit complexity, structured matrices and deep learning essentially answe…
In recent years, there has been a flurry of activity towards proving lower bounds for homogeneous depth-4 arithmetic circuits, which has brought us very close to statements that are known to imply $\textsf{VP} \neq \textsf{VNP}$. It is open if these techniques can go beyond homogeneity, and in this paper we make some progress in this direction by considering depth-4 circuits of low algebraic rank, which are a natural extension of homogeneous depth-4 circuits. A depth-4 circuit is a representation of an $N$-variate, degree-$n$ polynomial $P$ as \[ P = \sum_{i = 1}^T Q_{i1}\cdot Q_{i2}\cdot \cdo…
We study the problem of obtaining deterministic black-box polynomial identity testing algorithms (PIT) for algebraic branching programs (ABPs) that are read-once and oblivious. This class has an deterministic white-box polynomial identity testing algorithm (due to Raz and Shpilka), but prior to this work there was no known such black-box algorithm. The main result of this work gives the first quasi-polynomial sized hitting sets for size S circuits from this class, when the order of the variables is known. As our hitting set is of size exp(lg^2 S), this is analogous (in the terminology of boole…
Recurrent Graph Neural Networks and Arithmetic Circuits
Institut für Theoretische Informatik, Leibniz Universität Hannover, Germany（汉诺威莱布尼茨大学理论计算机研究所）；School of Computing Science, University of Glasgow, UK（格拉斯哥大学计算机科学学院）
Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit $d^{O(1)}$-variate and degree $d$ polynomial $P_{d}\in VNP$ such that if any depth four circuit $C$ of bounded formal degree $d$ which computes a polynomial of bounded individual degree $O(1)$, that is functionally equivalent to $P_d$, then $C$ must have size $2^{Ω(\sqrt{d}\log{d})}$. The motivation for their work comes from Boolean Circuit Complexity. Based on a characterization for $ACC^0$ circuits by Yao [FOCS, 1985] and Beigel and Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed tha…
We prove super-polynomial lower bounds for low-depth arithmetic circuits using the shifted partials measure [Gupta-Kamath-Kayal-Saptharishi, CCC 2013], [Kayal, ECCC 2012] and the affine projections of partials measure [Garg-Kayal-Saha, FOCS 2020], [Kayal-Nair-Saha, STACS 2016]. The recent breakthrough work of Limaye, Srinivasan and Tavenas [FOCS 2021] proved these lower bounds by proving lower bounds for low-depth set-multilinear circuits. An interesting aspect of our proof is that it does not require conversion of a circuit to a set-multilinear circuit, nor does it involve a random restrictio…
Institute for Theoretical Computer Science（理论计算机科学研究所）；Leibniz University Hanover（汉诺威莱布尼茨大学）；School of Computer Science（计算机科学学院）；University of Sheffield（谢菲尔德大学）
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 U_2(F), the algebra of upper-triangular 2 x 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 U_2(F) are strictly `weaker' than Sigma-Pi-Sigma circuits over F. The equivalence further shows that PIT of dept…
Agrawal and Vinay [AV08] showed how any polynomial size arithmetic circuit can be thought of as a depth four arithmetic circuit of subexponential size. The resulting circuit size in this simulation was more carefully analyzed by Korian [Koiran] and subsequently by Tavenas [Tav13]. We provide a simple proof of this chain of results. We then abstract the main ingredient to apply it to formulas and constant depth circuits, and show more structured depth reductions for them. In an apriori surprising result, Raz [Raz10] showed that for any $n$ and $d$, such that $ ω(1) \leq d \leq O\left(\frac{\log…
A read-once oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial time complexity $n^{O(\log n)}$. In both the cases, our time complexity is double exponential in the number of ROABPs. ROABPs are a generalization of set-multilinear depth-$3$ circuits. The prior results for the sum of constantly many set-multilinear depth-$3$ c…
In this note, we give a short, simple and almost completely self contained proof of a classical result of Kaltofen [Kal86, Kal87, Kal89] which shows that if an $n$ variate degree $d$ polynomial $f$ can be computed by an arithmetic circuit of size $s$, then each of its factors can be computed by an arithmetic circuit of size at most $\textsf{poly}\left(s, n, d\right)$. However, unlike Kaltofen's argument, our proof does not directly give an efficient algorithm for computing the circuits for the factors of $f$.
