Polynomial identity testing (PIT) is the problem of checking whether a given arithmetic circuit is the zero circuit. PIT ranks as one of the most important open problems in the intersection of algebra and computational complexity. In the last few years, there has been an impressive progress on this problem but a complete solution might take a while. In this article we give a soft survey exhibiting the ideas that have been useful. 1

Abstract. The parallel complexity class NC 1 has many equivalent models such as polynomial size formulae and bounded width branching programs. Caussinus et al. [CMTV98] considered arithmetizations of two of these classes, #NC 1 and #BWBP. We further this study to include arithmetization of other classes. In particular, we show that counting paths in branching programs over visibly pushdown automata is in FLogDCFL, while counting proof-trees in logarithmic width formulae has the same power as #NC 1. We also consider polynomial-degree restrictions of SC i, denoted sSC i, and show that the Boolean class sSC 1 is sandwiched between NC 1 and L, whereas sSC 0 equals NC 1. On the other hand, the arithmetic class #sSC 0 contains #BWBP and is contained in FL, and #sSC 1 contains #NC 1 and is in SC 2. We also investigate some closure properties of the newly defined arithmetic classes. 1

One of the fundamental problems of computational algebra is to classify polynomials according to the hardness of computing them. Recently, this problem has been related to another important problem: Polynomial identity testing. Informally, the problem is to decide if a certain succinct representation of a polynomial is zero or not. This problem has been extensively studied owing to its connections with various areas in theoretical computer science. Several efficient randomized algorithms have been proposed for the identity testing problem over the last few decades but an efficient deterministic algorithm is yet to be discovered. It is known that such an algorithm will imply hardness of computing an explicit polynomial. In the last few years, progress has been made in designing deterministic algorithms for restricted circuits, and also in understanding why the problem is hard even for small depth. In this article, we survey important results for the polynomial identity testing problem and its connection with classification of polynomials. 1

that the Boolean AND function can not be computed by polynomial size constant depth circuits built from modular counting gates, i.e., by CC 0 circuits. In this work we show that the AND function can be computed by uniform probabilistic CC 0 circuits that use only O(log n) random bits. This may be viewed as evidence contrary to the conjecture. As a consequence of our construction we get that all of ACC 0 can be computed by probabilistic CC 0 circuits that use only O(log n) random bits. Thus, if one were able to derandomize such circuits, we would obtain a collapse of circuit classes giving ACC 0 = CC 0. We present a derandomization of probabilistic CC 0 circuits using AND and OR gates to obtain ACC 0 = AND ◦ OR ◦ CC 0 = OR ◦ AND ◦ CC 0. AND and OR gates of sublinear fan-in suffice. Both these results hold for uniform as well as non-uniform circuit classes. For non-uniform circuits we obtain the stronger conclusion that ACC 0 = rand − ACC 0 = rand − CC 0 = rand(log n)−CC 0, i.e., probabilistic ACC 0 circuits can be simulated by probabilistic CC 0 circuits using only O(log n) random bits. As an application of our results we obtain a characterization of ACC 0 by constant width planar nondeterministic branching programs, improving a previous characterization for the quasipolynomial size setting. I.

Abstract The boolean circuit complexity classes AC0 ` AC0[m] ` T C0 ` N C1 have been studied in-tensely. Other than N C1, they are defined by constant-depth circuits of polynomial size and unboundedfan-in over some set of allowed gates. One reason for interest in these classes is that they contain the boundary marking the limits of current lower bound technology: such technology exists for AC0 andsome of the classes AC0[m], while the other classes AC0[m] as well as T C0 lack such technology.Continuing a line of research originating from Valiant's work on the counting class]P, the arithmeticcircuit complexity classes]AC0 and]NC1 have recently been studied. In this paper, we define andinvestigate the classes]AC0[m] and]T C0. Just as the boolean classes AC0[m] and T C0 give a refinedview of N C1, our new arithmetic classes, which fall into the inclusion chain]AC0 `]AC0[m] `]T C0 `]N C1, refine]NC1. These new classes (along with]AC0) are also defined by constant-depthcircuits, but the allowed gates compute arithmetic functions. We also introduce the classes Diff AC0[m](differences of two]AC0[m] functions), which generalize the class Diff AC0 studied in previous work.We study the structure of three hierarchies: the]AC0[m] hierarchy, the Diff AC0[m] hierarchy,and a hierarchy of language classes. We prove class separations and containments where possible, and demonstrate relationships among the various hierarchies. For instance, we prove that the hierarchy ofclasses]AC0[m] has exactly the same structure as the hierarchy of classes AC0[m]: AC0[m] ` AC0[m0] iff]AC0[m] `]AC0[m0] We also investigate closure properties of our new classes, which generalize those appearing in previ-ous work on

this paper describes the complete problems that motivate interest in these classes, discusses some surprising recent discoveries, and points out open problems where progress can reasonably be expected. 3 Definitions

The boolean circuit complexity classes AC 0 # AC 0 [m] # TC 0 # NC 1 have been studied intensely. Other than NC 1 , they are defined by constant-depth circuits of polynomial size and unbounded fan-in over some set of allowed gates. One reason for interest in these classes is that they contain the boundary marking the limits of current lower bound technology: such technology exists for AC 0 and some of the classes AC 0 [m], while the other classes AC 0 [m] as well as TC 0 lack such technology. Continuing a line of research originating from Valiant's work on the counting class #P , the arithmetic circuit complexity classes #AC 0 and #NC 1 have recently been studied. In this paper, we define and investigate the classes #AC 0 [m] and #TC 0 . Just as the boolean classes AC 0 [m] and TC 0 give a refined view of NC 1 , our new arithmetic classes, which fall into the inclusion chain #AC 0 # #AC 0 [m] # #TC 0 # #NC 1 , refine #NC 1 . These new classes (along with #AC 0 ) are also defined by constant-depth circuits, but the allowed gates compute arithmetic functions. We also introduce the classes Diff AC 0 [m] (differences of two AC 0 [m] functions), which generalize the class Diff AC 0 studied in previous work. We study the structure of three hierarchies: the #AC 0 [m] hierarchy, the Diff AC 0 [m] hierarchy, and a hierarchy of language classes. We prove class separations and containments where possible, and demonstrate relationships among the various hierarchies. For instance, we prove that the hierarchy of classes #AC 0 [m] has exactly the same structure as the hierarchy of classes AC 0 [m]: AC 0 [m] # AC 0 [m # ] iff #AC 0 [m] # #AC 0 [m # ] We also investigate closure properties of our new classe...

this paper describes the complete problems that motivate interest in these classes, discusses some surprising recent discoveries, and points out open problems where progress can reasonably be expected