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Arithmetic Circuits: a survey of recent results and open questions
"... A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last fi ..."
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Cited by 65 (5 self)
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A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions. The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we
Progress on Polynomial Identity Testing
"... 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 ..."
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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
Quasipolynomial hittingset for setdepth formulas
 In STOC
, 2013
"... Abstract. We call a depth4 formula C setdepth4 if there exists a (unknown) partition X1 unionsq · · · unionsq Xd of the variable indices [n] that the top product layer respects, i.e. C(x) = ∑k i=1 ..."
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Abstract. We call a depth4 formula C setdepth4 if there exists a (unknown) partition X1 unionsq · · · unionsq Xd of the variable indices [n] that the top product layer respects, i.e. C(x) = ∑k i=1
Derandomizing polynomial identity testing for multilinear constantread formulae
 Electronic Colloquium on Computational Complexity, Tech. Rep
"... Abstract—We present a polynomialtime deterministic algorithm for testing whether constantread multilinear arithmetic formulae are identically zero. In such a formula each variable occurs only a constant number of times and each subformula computes a multilinear polynomial. Before our work no subex ..."
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Abstract—We present a polynomialtime deterministic algorithm for testing whether constantread multilinear arithmetic formulae are identically zero. In such a formula each variable occurs only a constant number of times and each subformula computes a multilinear polynomial. Before our work no subexponentialtime deterministic algorithm was known for this class of formulae. We also present a deterministic algorithm that works in a blackbox fashion and runs in quasipolynomial time in general, and polynomial time for constant depth. Finally, we extend our results and allow the inputs to be replaced with sparse polynomials. Our results encompass recent deterministic identity tests for sums of a constant number of readonce formulae, and for multilinear depthfour circuits. Keywordsarithmetic circuit; boundeddepth circuit; derandomization; polynomial identity testing; I.
Shallow circuits with highpowered inputs
 PROCEEDINGS OF THE SECOND SYMPOSIUM ON INNOVATIONS IN COMPUTER SCIENCE
, 2011
"... A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic blackbox identity testing algorithm for (highdegree) univariate polynomials would imply a lower b ..."
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A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic blackbox identity testing algorithm for (highdegree) univariate polynomials would imply a lower bound on the arithmetic complexity of the permanent. The lower bounds that are known to follow from derandomization of (lowdegree) multivariate identity testing are weaker. To obtain a lower bound for the permanent it would be sufficient to derandomize identity testing for polynomials of a very specific norm: sums of products of sparse polynomials with sparse coefficients. This observation leads to new versions of the ShubSmale τconjecture on integer roots of univariate polynomials. In particular, we show that a lower bound for the permanent would follow if one could give a polynomial upper bound on the number of real roots of sums of products of sparse polynomials (Descartes ’ rule of signs gives such a bound for sparse polynomials and products thereof). In fact the same lower bound would follow even if one could only prove a slightly superpolynomial upper bound on the number of real roots. This is a consequence of a new result on reduction to depth 4 for arithmetic circuits which we establish in a companion paper. We also show that an even weaker bound on the number of real roots would suffice to obtain a lower bound on the size of depth 4 circuits computing the permanent. These results suggest the intriguing possibility that tools from real analysis might be brought to bear on a longstanding open problem: what is the arithmetic complexity of the permanent polynomial?
Diversification improves interpolation
 In Leykin [21
"... We consider the problem of interpolating an unknown multivariate polynomial with coefficients taken from a finite field or as numerical approximations of complex numbers. Building on the recent work of Garg and Schost, we improve on the bestknown algorithm for interpolation over large finite field ..."
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We consider the problem of interpolating an unknown multivariate polynomial with coefficients taken from a finite field or as numerical approximations of complex numbers. Building on the recent work of Garg and Schost, we improve on the bestknown algorithm for interpolation over large finite fields by presenting a Las Vegas randomized algorithm that uses fewer black box evaluations. Using related techniques, we also address numerical interpolation of sparse polynomials with complex coefficients, and provide the first provably stable algorithm (in the sense of relative error) for this problem, at the cost of modestly more evaluations. A key new technique is a randomization which makes all coefficients of the unknown polynomial distinguishable, producing what we call a diverse polynomial. Another departure from most previous approaches is that our algorithms do not rely on root finding as a subroutine. We show how these improvements affect the practical performance with trial implementations. 1
Algebraic Independence in Positive Characteristic – A padic Calculus
 Trans. Amer. Math. Soc
, 2014
"... Abstract. A set of multivariate polynomials, over a field of zero or large characteristic, can be tested for algebraic independence by the wellknown Jacobian criterion. For fields of other characteristic p> 0, no analogous characterization is known. In this paper we give the first such criterio ..."
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Abstract. A set of multivariate polynomials, over a field of zero or large characteristic, can be tested for algebraic independence by the wellknown Jacobian criterion. For fields of other characteristic p> 0, no analogous characterization is known. In this paper we give the first such criterion. Essentially, it boils down to a nondegeneracy condition on a lift of the Jacobian polynomial over (an unramified extension of) the ring of padic integers. Our proof builds on the functorial de RhamWitt complex, which was invented by Illusie (1979) for crystalline cohomology computations, and we deduce a natural explicit generalization of the Jacobian. This new avatar we call the WittJacobian. In essence, we show how to faithfully differentiate polynomials over Fp (i.e., somehow avoid ∂xp/∂x = 0) and thus capture algebraic independence.
Progress on Polynomial Identity Testing II
"... Abstract. We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years. ..."
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Abstract. We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years.
Faster sparse polynomial interpolation of straightline programs over finite fields
, 2014
"... We present a faster Monte Carlo algorithm for the interpolation of a straightline program to find a sparse polynomial f over an arbitrary finite field of size q. We assume a priori bounds D and T are given on the degree and number of terms of f. The approach presented in this paper is a hybrid of t ..."
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We present a faster Monte Carlo algorithm for the interpolation of a straightline program to find a sparse polynomial f over an arbitrary finite field of size q. We assume a priori bounds D and T are given on the degree and number of terms of f. The approach presented in this paper is a hybrid of the diversified and recursive interpolation algorithms, the two previous fastest known probabilistic methods for this problem. By making effective use of the information contained in the coefficients themselves, this new algorithm improves on the bit complexity of previous methods by a “softOh ” factor of T, logD, or log q. 1
Randomness Efficient Testing of Sparse Black Box Identities of Unbounded Degree over the Reals
"... We construct a hitting set generator for sparse multivariate polynomials over the reals. The seed length of our generator is O(log2(mn/)) where m is the number of monomials, n is number of variables, and 1 − is the hitting probability. The generator can be evaluated in time polynomial in logm, n, ..."
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We construct a hitting set generator for sparse multivariate polynomials over the reals. The seed length of our generator is O(log2(mn/)) where m is the number of monomials, n is number of variables, and 1 − is the hitting probability. The generator can be evaluated in time polynomial in logm, n, and log 1/. This is the first hitting set generator whose seed length is independent of the degree of the polynomial. The seed length of the best generator so far by Klivans and Spielman [16] depends logarithmically on the degree. From this, we get a randomized algorithm for testing sparse black box polynomial identities over the reals using O(log2(mn/)) random bits with running time polynomial in logm, n, and log 1 . We also design a deterministic test with running time Õ(m3n3). Here, the Õnotation suppresses polylogarithmic factors. The previously best deterministic test by Lipton and Vishnoi [18] has a running time that depends polynomially on log δ, where δ is the degree of the black box polynomial.