Let C be a depth-3 circuit with n variables, degree d and top fanin k (called sps(k,d,n) circuits) over base field F. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests if C is identically zero. Klivans & Spielman (STOC 2001) observed that the problem is open even when k is a constant. This case has been subjected to a serious study over the past few years, starting from the work of Dvir & Shpilka (STOC 2005). We give the first polynomial time blackbox algorithm for this problem. Our algorithm runs in time poly(nd^k), regardless of the base field. The only field for which polynomial time algorithms were previously known is F=Q (Kayal & Saraf, FOCS 2009, and Saxena & Seshadhri, FOCS 2010). This is the first blackbox algorithm for depth-3 circuits that does not use the rank based approaches of Karnin & Shpilka (CCC 2008). We prove an important tool for the study of depth-3 identities. We design a blackbox polynomial time transformation that reduces the number of variables in a sps(k,d,n) circuit to k variables, but preserves the identity structure.
Polynomial identity testing (PIT) is a major open problem in theoretical computer science. The input is an arithmetic circuit that computes a polynomial p(x 1 , x 2 , . . . , x n ) over a base field F. We wish to check if p is the zero polynomial, or in other words, is identically zero. We may be provided with an explicit circuit, or may only have blackbox access. In the latter case, we can only evaluate the polynomial p at various domain points. The main goal is to devise a deterministic blackbox polynomial time algorithm for PIT. One of the main reasons for interest in this problem is the connection between PIT algorithms and circuit lower bounds (Heintz & Schnorr [HS80], Kabanets & Impagliazzo [KI03] and Agrawal [Agr05,Agr06]). Refer to surveys for a detailed treatment of PIT [Sax09,AS09].
Since the problem of PIT is very hard, restricted versions of it have been studied. One common and natural variant is that of the bounded depth circuits. Results of Agrawal & Vinay [AV08] justify this restriction. They essentially show that an efficient blackbox identity test for depth-4 circuits leads to (almost) the complete resolution of PIT and also provides exponential lower bounds. Raz [Raz10] showed that even lower bounds for depth-3 circuits imply super-polynomial lower bounds for general arithmetic formulas. Not surprisingly, the problem of PIT is still wide open for the special case of depth-3 circuits.
A depth-3 circuit C over a field F is of the form C(x 1 , . . . , x n ) = k i=1 T i , where T i (a multiplication term) is a product of at most d linear polynomials with coefficients in F. The size of the circuit C can be expressed in three parameters: the number of variables n, the degree d, and the top fanin k. Such a circuit is referred to as a ΣΠΣ(k, d, n) circuit. Even when the top fanin k is constant, blackbox polynomial time algorithms were not known1 .
The study of PIT algorithms for depth-3 circuits was initiated by Dvir & Shpilka [DS05], who gave a quasi-polynomial time non-blackbox algorithm. The first non-trivial blackbox algorithm was given by Karnin & Shpilka [KS08]. There have been many recent results in this area by Kayal & Saxena [KS06], Saxena & Seshadhri [SS09,SS10a], and Kayal & Saraf [KS09b]. Our main result is the first polynomial time blackbox tester for bounded top fanin depth-3 circuits over any field.
Theorem 1 There exists a deterministic blackbox poly(nd k ) time algorithm for PIT on ΣΠΣ(k, d, n) circuits, regardless of the base field F.
Table 1 details the time complexities2 of previous algorithms. For convenience, we do not give the list of all algorithms, but only the important milestones for the case of arbitrary fields. We stress that the time complexities bound the number of bit operations.
The only field for which such polynomial time algorithms were previously known was Q. This was a breakthrough result of Kayal & Saraf [KS09b], which was followed by improvements in [SS10a]. These used beautiful incidence geometry theorems for the reals, but analogues of these results are either unknown or false for other fields. Since the best running time of these algorithms is poly(nd k 2 ), we get an improved algorithm for this case as well.
As Table 1 shows, even for the simple case of k = 3 and F = F 2 , no deterministic polynomialtime blackbox PIT algorithm was known. Kayal & Saxena [KS06] gave a non-blackbox algorithm (over all fields), which runs in poly(nd k ) time [KS06]. Theorem 1 closes the gap between blackbox and non-blackbox algorithms.
Throughout the following discussion, we will think of k as a constant. Hence, when we refer to polynomial time, the dependence on k will be ignored.
Dvir & Shpilka [DS05] introduced a powerful idea. They defined the notion of the rank of a ΣΠΣ(k, d, n) circuit. We will not explain this precisely here but merely say that this is the number of “free variables” in a ΣΠΣ circuit. They proved the remarkable fact that the rank of every identity3 is small. This led to the reduction of PIT for general ΣΠΣ(k, d, n) circuits to PIT on ΣΠΣ circuits over few variables. They developed a non-blackbox quasi-polynomial time algorithm through their rank bounds. Karnin & Shpilka [KS08] used the idea of rank to devise blackbox algorithms for ΣΠΣ circuits. Their algorithms had a running time that depended exponentially in the rank. Hence, constant rank bounds would lead to polynomial time algorithms. Unfortunately, Kayal & Saxena [KS06] gave constructions (extended in [SS09]) showing that for ΣΠΣ identities over finite fields, the rank could be unbounded (as large as k log d). This means that the best running time one could hope for over finite fields via this approach was d k log d . Tighter rank bounds from [SS09,SS10a] gave algorithms that almost match this running time. For the special case when the field F is Q, Kayal & Saraf [KS09b] proved a constant rank bound, establishing the first polynomial time blackbox algorithm for this case. Refer to [SS10
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