sign in sign up
<<
Home , Configuration (geometry), Incidence (geometry), Incidence geometry

From Sylvester-Gallai Configurations to Rank Bounds: Improved Black-box Identity Test for Depth-3 Circuits

Nitin Saxena C. Seshadhri Hausdorff Center for IBM Almaden Bonn - 53115, Germany San Jose - 95126, USA [email protected] [email protected]

Abstract—We study the problem of identity testing for depth- A depth-3 circuitC over a field F is of the form 3 k d k circuits of top fanin and degree .Wegiveanew C(x1, ··· ,xn)= i=1 Ti, where Ti (a multiplication structure theorem for such identities. A direct application of O(k) term) is a product of at most d linear polynomials with dk our theorem improves the known deterministic -time coefficients in F. We are especially interested in the case black-box identity test over rationals (Kayal & Saraf, FOCS O k2 F = Q. In this section, we will just assume this unless 2009) to one that takes d ( )-time. Our structure theorem essentially says that the number of independent variables in a explicitly mentioned otherwise. The size of the circuit real depth-3 identity is very small. This theorem affirmatively C can be expressed in three parameters: the number settles the strong rank conjecture posed by Dvir & Shpilka of variables n, the degree d, and the top fanin (or the (STOC 2005). number of terms) k. Such a circuit is referred to as a We devise a powerful algebraic framework and develop tools ΣΠΣ(k, d, n) circuit. PIT algorithms for depth-3 circuits to study depth-3 identities. We use these tools to show that any depth-3 identity contains a much smaller nucleus identity were first studied by Dvir & Shpilka [DS06]. There that contains most of the “complexity” of the main identity. have been many recent results in this area by Kayal & The special properties of this nucleus allow us to get almost Saxena [KS07] (in the non-black-box setting), Karnin & optimal rank bounds for depth-3 identities. Shpilka [KS08], Saxena & Seshadhri [SS09], and Kayal & Keywords-depth-3 circuit; identities; Sylvester-Gallai; inci- Saraf [KS09b]. Our main result is a better black-box tester k2 dence configuration; Chinese remaindering; ideal theory for ΣΠΣ circuits over Q. We get a running time of nd , an exponential improvement (in k) over the previous best kk I. INTRODUCTION of nd [KS09b]. Table I details the time complexities of previous algorithms. These time complexities are actually Polynomial identity testing (PIT) ranks as one of the most bounds on the total number of bit operations. Also, the important open problems in the intersection of algebra and running times are technically polynomial in the stated times. computer science. We are provided an arithmetic circuit that computes a polynomial p(x1,x2, ··· ,xn) over a field F, and Theorem 1: Consider circuits over Q. There exists a we wish to test if p is identically zero (in other words, if p deterministic black-box algorithm for PIT on ΣΠΣ(k, d, n) 2 is the zero polynomial). In the black-box setting, we do not circuits, whose time complexity is poly(ndk ). have access to the circuit. We are only allowed to evaluate the polynomial p at various domain points. The main goal is Table I: Depth-3 Black-box PIT algorithms over Q to devise a deterministic (preferably black-box) polynomial Paper Time complexity time algorithm for PIT. Heintz & Schnorr [HS80], Kabanets k2 k−2 [KS08] nd(2 log d) & Impagliazzo [KI04] and Agrawal [Agr05], [Agr06] have 3 [SS09] ndk log d shown connections between deterministic algorithms for k (k ) identity testing and circuit lower bounds, emphasizing the [KS09b] nd k2 importance of this problem. For a detailed exposition, see This paper nd surveys [Sax09], [AS09]. Even for the special case of depth-3 circuits, this ques- This is the first result that gives a time complexity both tion is still open. This may seem quite depressing. It is. polynomial in d and singly-exponential in k for Q.Thisis Nonetheless, there exist concrete results that justify both not too far from the best non-black-box algorithm for ΣΠΣ our ignorance and the acceptance of results on depth-3 PIT circuits, which runs in poly(ndk) time [KS07]. This result in major publishing venues. Agrawal and Vinay [AV08] closes the gap (almost) between black-box and non-black- showed that an efficient black-box identity test for depth- box algorithms. 4 essentially leads to subexponential lower bounds. All these results go via rank bounds for depth-3 identities, introduced by Dvir & Shpilka [DS06]. This is a very inter- about depth-3 identities over any field. Every such identity esting quantity associated with these circuits, and roughly contains a nucleus identity expressed on few variables. This speaking, bounds the maximum number of “free variables” nucleus, in some sense, captures all the complexity of the that can be present in a depth-3 identity. If a ΣΠΣ(k, d, n) original identity, and has some very special properties. A circuit has rank r, then there exists a linear transformation better understanding of this nucleus may lead us to the goal that converts this to an equivalent ΣΠΣ(k, d, r) circuit. of a truly polynomial time algorithm. (This linear transformation is very easy to determine.) The remarkable insight of [DS06] was that the rank of every A. Definitions and results ΣΠΣ(k, d, n) identity is very low. Any ΣΠΣ(k, d, r)-circuit We recall that a depth-3 circuit C over a field F is: r k can be completely expanded out in poly(kd ) time. Hence, C(x1,...,xn)= i=1 Ti, where Ti is a product of di low rank bounds for identities imply efficient non-black-box linear polynomials i,j over F. For the purposes of studying PIT algorithms. identities we can assume, by homogenization, that i,j ’s are Karnin & Shpilka [KS08] showed how small rank bounds linear forms (i.e. linear polynomials with a zero constant for identities imply efficient black-box PIT algorithms. This coefficient) and ∀i, di = d. It will be convenient to state our opened the door for black-box algorithms for depth-3 PIT. results in terms of arbitrary fields. Indeed, all known algorithms for this problem come as a Definition 2: [DS06] consequence of their result. Rank bounds have also found • Simple Circuit: C is a simple circuit if there is no applications in learning ΣΠΣ circuits [Shp09], [KS09a]. nonzero linear form dividing all the Ti’s. Hence, the rank and file of researchers studying this problem • Minimal Circuit: C is a minimal circuit if for every are interested in proving small rank bounds. As mentioned  proper subset S ⊂ [k], Ti is nonzero. earlier, we focus on the field Q. Dvir & Shpilka [DS06] i∈S • initiated this of work by showing that the rank of a Rank of a circuit: The coefficients of i,j form an 2 F simple, minimal ΣΠΣ(k, d) identity is 2O(k )(log d)k−2. n-dimensional vector over .Therank of the circuit, There are basic constructions of rank Ω(k) identities over rk(C), is defined as the rank of the set of all linear Q [DS06]. Dvir & Shpilka [DS06] conjectured that the forms i,j viewed as vectors. rank should be bounded by poly(k). This rank bound was The rank of a circuit can be interpreted as the minimum improved to O(k3 log d) by Saxena & Seshadhri [SS09]. number of independent variables required to express C.The Kayal & Saraf [KS09b] achieved a breakthrough by proving definition of simple and minimal circuits are used to remove a rank bound independent of d. Their bound was kO(k).We certain pathological cases. The rank question is: for a simple finally settle the Dvir-Shpilka conjecture and show a rank and minimal ΣΠΣ(k, d, n) identity over field F, what is the bound of O(k2). maximal possible rank? A trivial upper bound on the rank The advances of Kayal & Saraf were obtained through (for any ΣΠΣ-circuit) is kd, since that is the total number the use of theorems, like the famous of linear forms involved in C. A substantially smaller rank Sylvester-Gallai theorem. This theorem states that for any bound than kd shows that identities do not have as many set S of points in the Euclidean , not all collinear, “degrees of freedom” as general circuits. there exists a line passing through exactly two points in Before we state our results, it will be helpful to explain S. Generalizations of this to higher are called Sylvester-Gallai configurations. A set of points S with the Sylvester-Gallai theorems (see survey [BM90]). This theo- property that every line through two points of S passes rem and its generalizations have connections to rank bounds through a third in S is called a Sylvester-Gallai for depth-3 circuits. The result of [KS09b] gave an intricate configuration. The famous Sylvester-Gallai theorem states combinatorial construction that converts depth-3 identities to that the only Sylvester-Gallai configuration in R2 is a set sets of colored points in Euclidean space. This allowed the of collinear points. This basic theorem about point-line use of Sylvester-Gallai theorems to bound the rank. incidences was extended to higher dimensions [Han65], Our contribution comes through a new algebraic frame- [BE67]. We define the notion of Sylvester-Gallai rank work for studying depth-3 identities. This has many ben- bounds. This is a clean and convenient way of expressing efits. Firstly, it allows for a much more “efficient” use of these theorems. Sylvester-Gallai theorems to bound the rank. This leads to nearly optimal rank bounds. Secondly, the connection Definition 3: Let S be a finite subset of the projective between Sylvester-Gallai theorems and rank bounds is far space FPn. Alternately, S is a set of non-zero vectors in more transparent, at the loss of some color from the the- Fn+1 without multiples: no two vectors in S are scalar orems. Theorem 4 gives a simple formula that relates the multiples of each other. Suppose, for every set V ⊂ S of k depth-3 rank to Sylvester-Gallai bounds. A nice byproduct linearly independent vectors, the linear span of V contains of this connection is the improvement of rank bounds over at least k +1 vectors of S. Then, the set S is said to be arbitrary fields. Thirdly, we get a deep structural theorem SGk-closed. The largest possible rank of an SGk-closed set of at most this, nothing was previously known. One of our auxiliary n m vectors in F (for any n) is denoted by SGk(F,m). theorems, of independent interest, gives a high-dimensional The Sylvester-Gallai theorem states Higher dimensional Sylvester-Gallai bound for all fields. Applying the stronger analogues [Han65], [BE67] can be interpreted to say version of Theorem 4, we get our rank bound. SGk(R,m) ≤ 2(k − 1). Our main theorem is a simple, clean expression of how Sylvester-Gallai influences Theorem 7 (SGk for all fields): For any field F and >1 identities. We state this for general fields. k, m ∈ N ,SGk(F,m) ≤ 9k lg m. (There is a construction that shows that SGk(Fp,m)=Ω(k · logp m).) Theorem 4 (From SGk to Rank): Let |F| >d. The rank Let C be a ΣΠΣ(k, d) circuit, over an arbitrary field F, 2 of a simple and minimal ΣΠΣ(k, d) identity over F is at that is simple, minimal and zero. Then, rk(C) < 3k (lg 2d). 2 most 2k + k · SGk(F,d). B. History A direct application of the SGk(R,m) bound yields an almost optimal rank bound for real depth-3 identities. For And now, for a brief history of PIT algorithms. The completeness, we state the exact rank bound obtained. We first randomized polynomial time PIT algorithm, which have a slightly stronger version (Theorem 18) of the above was a black-box algorithm, was given (independently) by theorem that gives better constants. Schwartz [Sch80] and Zippel [Zip79]. Randomized algo- rithms that use less randomness were given by Chen & Theorem 5 (Depth-3 Rank Bounds): Let C be a Kao [CK00], Lewin & Vadhan [LV98], and Agrawal & ΣΠΣ(k, d) circuit, over field R, that is simple, minimal Biswas [AB03]. Klivans & Spielman [KS01] observed that and zero. Then, rk(C) < 3k2. even for depth-3 circuits for bounded top fanin, deterministic identity testing was open. Progress towards this was first As discussed before, an application of this result to made by the quasi-polynomial time algorithm of Dvir & Lemma 4.10 of [KS08] gives a deterministic black-box Shpilka [DS06]. The problem was resolved by a polynomial identity test for ΣΠΣ(k, d, n) circuits over Q. Formally, time algorithm given by Kayal and Saxena [KS07], with a we get the following hitting set generator for ΣΠΣ circuits running time exponential in the top fanin. Both these al- with real coefficients. gorithms were non-black-box. As for black-box algorithms, the authors are quite sure that the reader has heard enough Corollary 6 (Black-box PIT over Q): There is a deter- history. Identity tests are known only for very special depth-4 ministic algorithm that takes as input a triple (k, d, n) of circuits [AM07], [Sax08], [SV09], [KMSV09]. Agrawal and 2 natural numbers and in time poly(ndk ), outputs a hitting Vinay [AV08] showed that an efficient black-box identity set H⊂Zn with the following properties: test for depth-4 circuits will actually give a quasi-polynomial black-box test, and subexponential lower bounds, for circuits R 1) Any ΣΠΣ(k, d, n) circuit C over computes the zero of all depths that compute low degree polynomials. Thus, polynomial iff ∀a ∈H, C(a)=0. 2 understanding depth-3 identities seems to be a natural first H k 2) has at most poly(nd ) points. step towards the goal of PIT. 3) The total bit-length of each point in H is poly(kn log d). II. PROOF OUTLINE,IDEAS, AND ORGANIZATION Our proof of the rank bound comprises of several new 1) Other fields: What about other fields? The rank ideas, both at the conceptual and the technical levels. In- bounds of [DS06] and [SS09] hold for arbitrary fields, stead of giving proofs in this extended abstract, we will whereas the rank bound of [KS09b] holds only for R. only provide the intuition and the overall argument. We It has been observed that for finite fields, the rank of recommend the interested reader to see the full version an ΣΠΣ identity can be as large as Ω(k log d) [KS07], of this paper [SS10]. The full proof of Theorem 4 is [SS09]. Hence, the O(k3 log d) bound proved by [SS09] extremely technical, requires many definitions, and involve is almost optimal. As a small bonus, we give a slight many algebraic arguments. Our attempt is to convey with improvement upon this bound using our approach. This main ideas without getting into too much formalism or requires Sylvester-Gallai theorems over arbitrary fields, mathematical details. We describe all the major milestones, an interesting question in itself. It was shown that many of which are interesting in their own right. Indeed, SG2(C,m) ≤ 3 [EPS06], and certain lower bounds for it is the authors’ opinion that the reader has little to gain locally decodable codes implied SG2(F,m)=O(log m). from simply reading the detailed proofs without getting the (Concretely, Corollary 2.9 of [DS06] can be used to prove essence of the ideas. that SG2(F,m)=O(log m). This is an extension of The intuition portion is divided into three subsections, theorems in [GKST02] that prove this for F2. ) Other than each dealing with a separate component of the final proof. Each portion proves an interesting structural theorem. The (Conventionally, sp(∅)={0}.) three notions that are crucially used and developed are: ideal [Matchings] Let U, V be lists of linear forms and I be Chinese remaindering, matchings and Sylvester-Gallai rank a subspace of L(R).AnI-matching π between U, V is a bounds. Related notions have appeared (in some form) in bijection π between lists U, V such that: for all ∈ U, the works of Kayal & Saxena [KS07], Saxena & Seshadhri π() ∈ F∗ + I. [SS09] and Kayal & Saraf [KS09b] respectively, to prove When f,g are multiplication terms, an I-matching different kinds of results. The first two steps set up the between f,g would mean an I-matching between algebraic framework and prove theorems that hold for all L(f),L(g). fields. The third step is where the Sylvester-Gallai theorems are brought in. B. Step 1: Matching the Gates in an Identity A. Notation and definitions We will show that all the multiplication terms of a We will denote the set {1,...,n} by [n].Wefixthe minimal ΣΠΣ identity can be matched by a low rank space base field to be F, so the circuits compute multivariate K, spanned by “few” linear forms in L(R). polynomials in the polynomial ring R := F[x1,...,xn].We ∗ use F to denote F \{0}. Theorem 9 (Matching-Nucleus): Let C = T1 + ···+ Tk A linear form is a linear polynomial in R with zero be a ΣΠΣ(k, d) circuit that is minimal and zero. Then constant term. We will denote the set of all linear forms there exists a K of L(R) such that: n 2 by L(R):={ i=1 aixi | a1,...,an ∈ F}. Clearly, L(R) 1) rk(K)

v2 v3 2 involves increasing the space K (but not by too much) that gives us a C with the right behavior. Specifically, if k v1 T1,...,Tk are linearly independent (i.e. β ∈ F \{0} s.t. i∈[k] βiTi =0), then so are K1,...,Kk .Thefollowing (b) Paths can be seen as an important structural theorem of depth-3 Figure 1 identities.  Theorem 11 (Nucleus): Let C = i∈[k] Ti be a minimal ΣΠΣ(k, d) identity and let {Ti|i ∈I}be a maximal set of linearly independent terms (1 ≤ k := |I|

D. Step 3: Invoking Sylvester-Gallai Theorems We give an example to explain the non-K rank. Let R = F[z1, ··· ,zn,y1, ··· ,ym]. Suppose K = sp(z1, ··· ,zn) As explained in Section I-A, we rephrase the standard and S ⊂ L(R). We can take any element in S and simply Sylvester-Gallai theorems in terms of Sylvester-Gallai drop all the zi terms, i.e. ‘truncate’ the z-part of .This closure and rank bounds (Definition 3). Using some linear gives a set of linear forms over the y variables. The rank of algebra and combinatorial tricks, we prove the first ever these is the non-K rank of S. general Sylvester-Gallai bound for all fields. We are now ready to state the theorem that is proved in Step 3. It basically shows a neat relationship between the Theorem 13 (General Sylvester-Gallai): For any field F >1 non-nucleus part and Sylvester-Gallai. and k, m ∈ N ,SGk(F,m) ≤ 9k lg m. Theorem 16 (Bound for simple, strongly minimal identities): The following definition is very helpful in applying Let |F| >d. The non-nucleus rank of a simple and strongly Sylvester-Gallai rank bounds to our scenario. minimal ΣΠΣ(k, d) identity over F is at most SGk−1(F,d). More specifically, (for nucleus K) the vectors in L(C) \ K · ∈ N>1 Definition 14 (SG operator): [SGk( )] Let k, m . form an SGk−1-closed set. n Suppose a set S ⊆ F has rank greater than SGk(F,m) (where #S ≤ m). Then, by definition, S is not SGk-closed. Observe that this theorem together with Theorem 11 gives In this situation we say the k-dimensional Sylvester-Gallai a complete structure theorem for strongly minimal depth-3 operator SGk(S) (applied on S) returns a set of k linearly identities. One can make suitable claims for identities that independent vectors V in S whose span has no point in S\V . are not strictly minimal. Essentially, we just take a subset of linearly independent terms, say T1,...,Tk , that form Let C be a simple and strongly minimal ΣΠΣ(k, d) iden- a basis for {Ti|i ∈ [k]}. We can now construct strongly 2 tity. Theorem 11 gives us a nucleus K, of rank < 2k , that minimal identities using these terms and apply the above matches T1 to each term Ti. As seen in Step 2, if we look at theorem. Specifically, we get the following. the corresponding multiplication terms Ki := M(LK (Ti)),  ∈  i [k], then they again form a ΣΠΣ(k, d ) “nucleus Definition 17 (Independent-fanin): Let C = i∈[k] Ti  ∗ identity” C = i∈[k] αiKi,forsomeαi’s in F , which is be a ΣΠΣ(k, d) circuit. The independent-fanin of C, simple and strongly minimal. Define the non-nucleus part of ind-fanin(C), is defined to be the size of the maximal c \ ∈ Ti as LK (Ti):=L(Ti) K, for all i [k] (c in the exponent I⊆[k] such that {Ti|i ∈I}are linearly independent c annotates “complement”, since L(Ti)=LK (Ti) LK (Ti)). polynomials. c What can we say about the rank of LK (Ti) ? c  Define the non-nucleus part of C as LK (C):= We now state the following stronger version of the main c i∈[k] LK (Ti). Our goal in Step 3 is to bound theorem. c rk(LK (C) mod K) by 2k when the field is R. This will give c 2 us a rank bound of rk(K)+ rk(LK (C)mod K) < (2k +2k) Theorem 18 (Final bound): Let |F| >d. The rank of a for simple and strongly minimal ΣΠΣ(k, d) identities over simple, minimal ΣΠΣ(k, d), independent-fanin k, identity 2  R. The proof is mainly combinatorial, based on higher is at most 2k +(k − k ) · SGk (F,d). dimensional Sylvester-Gallai theorems and a property of set partitions, with a sprinkling of algebra. Remark: In particular, the rank of a simple, We apply the SGk operator not directly on the forms in minimal ΣΠΣ(k, d) identity over reals is at most 2  2   L(C) but on a suitable truncation of those forms. So we 2k +(k − k ) · SGk (R,d) ≤ 2k +(k − k )2(k − 1) need another definition. < 3k2, proving the main theorem over reals. Likewise, for 2  any F, we get the rank bound of 2k +(k − k ) · SGk (F,d) 2   2 9k2 2 ≤ 2k +(k − k )9k lg d ≤ 2k + 4 lg d<3k lg 2d, operator output (y0 +y1,y0 +y2,y0 +y3) and the fact that C proving the main theorem. is a simple, strongly minimal ΣΠΣ(4,d) identity. Consider the setting given in Figure 2. Suppose the linear forms in Proof Idea for Theorem 16: Basically, we apply the C that are similar to a form in {y0 + yi + K|i ∈ [3]} are SGk(·) operator on the non-nucleus part of the term T1, i.e. exactly those depicted in the figure. All forms within a row a2 an    we treat a linear form i aixi as the point (1, a ,..., a ) ∈ are K-matched. We would like to find forms 1,2,3 with n 1 1  F for the purposes of Sylvester-Gallai and then we con- the following properties: (1) i ≡ cii(mod K) (for some c  sider SGk(LK (T1)) assuming that the non-nucleus rank of constant ci). (2) There exists some j such that no i divides  T1 is more than SGk(F,d). This application of Sylvester- Tj but for each Tl (l = j), some i divides Tl.Inthis   Gallai is much more direct compared to the methods used situation, we can choose 1 = y0 +y1 +z1, 2 = y0 +y2 +z2,  in [KS09b]. There, they effectively needed to prove ver- and 3 = −y0 − y3 + z2. None of these divides T4. Observe sions of Sylvester-Gallai that dealt with colored points and that the triple (y0 + y1 + z1,y0 + y2 + z2,y0 + y3 + z1) does needed a hyperplane decomposition property after applying not satisfy these conditions, since no appropriate Tj can be aSGkO(k) (·) operator on L(C). Since, modulo the nucleus, found. all multiplication terms look essentially the same, it suffices Take C modulo the ideal I := to focus attention on just one of them. Hence, we apply the y0 + y1 + z1,y0 + y2 + z2, −y0 − y3 + z2 . It is easy SGk-operator on a single multiplication term. to see that C ≡ T4(mod I),soI “kills” the first three Assume C is a simple, strongly minimal ΣΠΣ(k, d) terms. Since C is an identity, T4 ∈ I. Thus, there is a    identity with terms {Ti|i ∈ [k]} and let K be its nucleus form ∈ L(T4) such that ∈ sp(1,2,3). Since no form  given by Step 2. It will be convenient for us to fix a linear from i divides T4,so must be a non-trivial combination ∗ form y0 ∈ L(R) and a subspace U of L(R) such that we of these forms. By the matching property, there exists have the following orthogonal decomposition some form ˆ ∈ L(T1) such that trun()=trun(ˆ). In other c L(R)=Fy0 ⊕ U ⊕ K. This means for any form ∈ L(R), words, trun() ∈ trun(LK (T1)). But that contradicts the    there is a unique way to express = αy0 + u + v, where fact that (1,2,3) form an SG3-tuple. This implies that α ∈ F, u ∈ U and v ∈ K. Furthermore, we will assume the non-nucleus rank of C is at most SG3(F,d). c wlog that for every form ∈ LK (T1) the corresponding The approach above worked because we were lucky c    α is nonzero, i.e. each form in LK (T1) is monic wrt y0. enough to find 1,2,3 with the right properties. Can Technically, we do not need the extra variable y0 and we always do this? No, because of repeating forms. Sup- can work in a . Nonetheless, it makes the pose, after going modulo form , the circuit looks like presentation easier. x3y +2x2y2 + xy3 =0. This is not simple, but it does not have to be. We are only guaranteed that the original Definition 19 (trun(·)): Fix a decomposition L(R)= circuit is simple. Once we go modulo , that property is c Fy0 ⊕ U ⊕ K. For any form ∈ LK (T1), there is a unique lost. Now, the choice of any form kills all terms. We will ∗ way to express = αy0 + u + v, where α ∈ F , u ∈ U and use our more powerful Chinese remaindering tools and the v ∈ K. nucleus properties to deal with this. The minimality of the The truncated form trun() is the linear form obtained nucleus identity plays a crucial role here and helps us deal by dropping the K part and normalizing, i.e. trun():= with such situations. We have to prove a special theorem −1 y0 + α u. about partitions of [k] and use strong minimality (which we Given a list of forms S we define trun(S) to be the did not use in the above sketch). corresponding set (thus no repetitions) of truncated forms. III. CONCLUSION

To be precise, we fix a basis {y1,...,yrk(U)} of U In this work we developed the strongest methods, to date, c so that each form in trun(LK (T1)) has representation to study depth-3 identities. The ideal methods hinge on a y0 + i≥1 aiyi (ai’s ∈ F). We view each such form classification of zerodivisors of the ideals generated by gates as the point (1,a1,...,ark(U)) while applying Sylvester- of a ΣΠΣ circuit. That is useful in proving an ideal ver- c Gallai on trun(LK (T1)). Assume, for the sake of contradic- sion of Chinese remaindering tailor-made for ΣΠΣ circuits, c tion, that the non-nucleus rank of T1,rk(trun(LK (T1))) > which is in turn useful to show a connection between all c SGk−1(F,d). Therefore, SGk−1(trun(LK (T1))) gives (k − the gates involved in an identity. As a byproduct, it shows  1) linearly independent forms 1,...,k−1 ∈ (y0 + U) the existence of a low rank nucleus identity C inside any c  whose span contains no other linear form of trun(LK (T1)). given ΣΠΣ(k, d) identity C (when C is not minimal, C For simplicity of exposition, let us fix k =4, K spanned can still be defined but it might not be homogeneous). The by z’s, U spanned by y’s and i = y0 + yi (i ∈ [3]).Note properties of the nucleus identity are an important part of that (by definition) trun(αy0 + i αizi + i βiyi)=y0 + an identity and it might be useful for PIT to understand βi i α yi. We want to derive a contradiction using the SG3- (or classify) it further. Can the rank bound for the nucleus T1 T2 T3 T4

y0 + y1 + z1 2y0 +2y1 y0 + y1 + z1 y0 + y1 + z2

c y0 + y2 + z2 y0 + y2 + z2 y0 + y2 + z1 3y0 +3y2 LK (Ti) y0 + y3 +2z2 y0 + y3 − z1 + z3 −y0 − y3 + z2 y0 + y3 + z1 y0 + y4 − z1 + z2 y0 + y4 + z2 y0 + y4 + z2 y0 + y4 + z3

z2 z1 + z3 z3 − 2z2 z1 LK (Ti)

Figure 2 identity be improved to O(k)? More importantly, can the REFERENCES rank bound for simple minimal real ΣΠΣ(k, d) identities [AB03] M. Agrawal and S. Biswas. Primality and identity be improved to O(k)? The best constructions known, since − testing via Chinese remaindering. Journal of the ACM, [DS06], have rank 4(k 2). Over other fields, our upper 50(4):429–443, 2003. (Conference version in FOCS 2 bound of O(k log d) still leaves some gap in understanding 1999). the exact dependence on k. Of course, the most important question is whether our techniques can help construct a [Agr05] M. Agrawal. Proving lower bounds via pseudo- truly polynomial time deterministic (even non-black-box) random generators. In Proceedings of the 25th Annual Foundations of Software Technology and Theoretical algorithm for PIT. Computer Science (FSTTCS), pages 92–105, 2005. We generalize the notion of Sylvester-Gallai configura- tions to any field and define a parameter SGk(F,m) associ- [Agr06] M. Agrawal. versus permanent. In ated with field F. This number seems to be a fundamental Proceedings of the 25th International Congress of Mathematicians (ICM), volume 3, pages 985–997, property of a field, and as we show, is very closely related to 2006. ΣΠΣ identities. It would be interesting to obtain bounds for SGk(F,m) for different F. For example, as also asked by [AM07] V. Arvind and P. Mukhopadhyay. The monomial ideal [KS09b], can we nontrivially bound the number SGk(F,m) membership problem and polynomial identity testing. for interesting fields: C, finite fields with large characteristic, In Proceedings of the 18th International Symposium on Algorithms and Computation (ISAAC), pages 800– or even p-adic fields? The only known SGk rank bounds are 811, 2007. those for R,SG2(C,m) ≤ 3, and SG2(F,m) ≤ O(log m). We shed (a little) light on SG rank bounds by showing [AS09] M. Agrawal and R. Saptharishi. Classifying polynomi- SGk(F,m)=O(k log m). We conjecture: SGk(F,m) is als and identity testing. Technical report, IIT Kanpur, O(k) for zero characteristic fields, while O(k + k · logp m) http://www.cse.iitk.ac.in/manindra/survey/Identity.pdf, for fields of characteristic p>1. The latter would mean 2009. that when the characteristic is large (p  m), SGk(F,m)= [AV08] M. Agrawal and V. Vinay. Arithmetic circuits: A O(k), matching the bounds for zero characteristic fields. chasm at depth four. In Proceedings of the 49th An- nual Symposium on Foundations of Computer Science (FOCS), pages 67–75, 2008. ACKNOWLEDGEMENTS [BE67] W. Bonnice and M. Edelstein. Flats associated with finite sets in P d. Niew. Arch. Wisk., 15:11–14, 1967. We are grateful to Hausdorff Center for Mathematics, Bonn for the kind support, especially, hosting the second [BM90] P. Borwein and W. O. J. Moser. A survey of Sylvesters problem and its generalizations. Aequationes Mathe- author when part of the work was done. The first author maticae, 40(1):111–135, 1990. thanks Nils Frohberg for several detailed discussions that clarified the topic of and Sylvester- [CK00] Z. Chen and M. Kao. Reducing randomness via irra- Gallai theorems. The second author is extremely grateful tional numbers. SIAM J. on Computing, 29(4):1247– to Ken Clarkson and especially to T. S. Jayram for their 1256, 2000. (Conference version in STOC 1997). suggestions in improving the presentation. We also thank [DS06] Z. Dvir and A. Shpilka. Locally decodable codes with Malte Beecken, Johannes Mittmann (for the high rank SGk 2 queries and polynomial identity testing for depth construction over Fp) and Thomas Thierauf for several 3 circuits. SIAM J. on Computing, 36(5):1404–1434, interesting discussions. 2006. (Conference version in STOC 2005). [EPS06] N. D. Elkies, L. M. Pretorius, and C. J. Swanepoel. [Sax08] N. Saxena. Diagonal circuit identity testing and lower Sylvester-Gallai theorems for complex numbers and bounds. In Proceedings of the 35th Annual Inter- quaternions. Discrete & Computational Geometry, national Colloquium on Automata, Languages and 35(3):361–373, 2006. Programming (ICALP), pages 60–71, 2008.

[GKST02] O. Goldreich, H. Karloff, L. Schulman, and L. Tre- [Sax09] N. Saxena. Progress on polynomial identity testing. visan. Lower bounds for linear locally decodable codes Bulletin of the European Association for Theoretical and private information retrieval. In Proceedings of the Computer Science (EATCS)- Computational Complex- 17th Annual Computational Complexity Conference ity Column, (99):49–79, 2009. (CCC), pages 175–183, 2002. [Sch80] J. T. Schwartz. Fast probabilistic algorithms for [Han65] S. Hansen. A generalization of a theorem of Sylvester verification of polynomial identities. Journal of the on the lines determined by a finite point set. Mathe- ACM, 27(4):701–717, 1980. matica Scandinavia, 16:175–180, 1965. [Shp09] A. Shpilka. Interpolation of depth-3 arithmetic circuits [HS80] J. Heintz and C. P. Schnorr. Testing polynomials with two multiplication gates. SIAM J. Comput., which are easy to compute (extended abstract). In 38(6):2130–2161, 2009. (Conference version in STOC Proceedings of the twelfth annual ACM Symposium 2007). on Theory of Computing, pages 262–272, New York, NY, USA, 1980. [SS09] N. Saxena and C. Seshadhri. An almost optimal rank bound for depth-3 identities. In Proceedings of the 24th Annual Conference on Computational Complexity [KI04] V. Kabanets and R. Impagliazzo. Derandomizing (CCC), pages 137–148, 2009. polynomial identity tests means proving circuit lower bounds. Computational Complexity, 13(1):1–46, 2004. [SS10] N. Saxena and C. Seshadhri. From Sylvester- (Conference version in STOC 2003). Gallai configurations to rank bounds: Improved black-box identity test for depth-3 circuits. [KMSV09] Z. Karnin, P. Mukhopadhyay, A. Shpilka, and Technical Report TR10-013, ECCC, http://eccc.hpi- I. Volkovich. Deterministic identity testing of web.de/report/2010/013/, 2010. depth 4 multilinear circuits with bounded top fan-in. Technical Report TR09-116, ECCC, http://eccc.hpi- [SV09] A. Shpilka and I. Volkovich. Improved polynomial web.de/report/2009/116/, 2009. identity testing for read-once formulas. In Proceedings of the 13th International Workshop on Randomization [KS01] A. Klivans and D. A. Spielman. Randomness effi- and Computation (RANDOM), pages 700–713, 2009. cient identity testing of multivariate polynomials. In Proceedings of the 33rd Symposium on Theory of [Zip79] R. Zippel. Probabilistic algorithms for sparse poly- Computing (STOC), pages 216–223, 2001. nomials. In Proceedings of the International Sympo- sium on Symbolic and Algebraic Manipulation (EU- [KS07] N. Kayal and N. Saxena. Polynomial identity test- ROSAM), pages 216–226, 1979. ing for depth 3 circuits. Computational Complexity, 16(2):115–138, 2007. (Conference version in CCC 2006).

[KS08] Z. Karnin and A. Shpilka. Deterministic black box polynomial identity testing of depth-3 arithmetic cir- cuits with bounded top fan-in. In Proceedings of the 23rd Annual Conference on Computational Complex- ity (CCC), pages 280–291, 2008.

[KS09a] Z. S. Karnin and A. Shpilka. Reconstruction of gen- eralized depth-3 arithmetic circuits with bounded top fan-in. In Proceedings of the 24th Annual Conference on Computational Complexity (CCC), pages 274–285, 2009.

[KS09b] N. Kayal and S. Saraf. Blackbox polynomial identity testing for depth 3 circuits. In Proceedings of the 50th Annual Symposium on Foundations of Computer Science (FOCS), 2009.

[LV98] D. Lewin and S. Vadhan. Checking polynomial identities over any field: Towards a derandomization? In Proceedings of the 30th Annual Symposium on the Theory of Computing (STOC), pages 428–437, 1998.