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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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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
Shallow circuits with highpowered inputs
 Proceedings of the Second Symposium on Innovations in Computer Science
, 2011
"... Abstract: 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 ..."
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Cited by 8 (5 self)
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Abstract: 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?
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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Cited by 5 (1 self)
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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
A Hitting Set Construction, with Applications to Arithmetic Circuit Lower Bounds
, 2009
"... Abstract. A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic blackbox identity testing algorithm for univariate polynomials of the form Pt j=0 cjXα j β (a + bX) j. From our algorithm we derive Q an exponential l ..."
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Abstract. A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic blackbox identity testing algorithm for univariate polynomials of the form Pt j=0 cjXα j β (a + bX) j. From our algorithm we derive Q an exponential lower bound for representations of polynomials such as n 2 i=1 (Xi − 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. 1
computational complexity INTERPOLATION OF SHIFTEDLACUNARY POLYNOMIALS
"... Abstract. Given a “black box ” function to evaluate an unknown rational polynomial f ∈ Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t ∈ Z>0, the shift α ∈ Q, the exponent ..."
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Abstract. Given a “black box ” function to evaluate an unknown rational polynomial f ∈ Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t ∈ Z>0, the shift α ∈ Q, the exponents 0 ≤ e1 <e2 < ·· · <et, and the coefficients c1,...,ct ∈ Q \{0} such that f(x) =c1(x − α) e1 + c2(x − α) e2 + ···+ ct(x − α) et. The computed sparsity t is absolutely minimal over any shifted power basis. The novelty of our algorithm is that the complexity is polynomial in the (sparse) representation size, which may be logarithmic in the degree of f. Our method combines previous celebrated results on sparse interpolation and computing sparsest shifts, and provides a way to handle polynomials with extremely high degree which are, in some sense, sparse in information.
SPARSE POLYNOMIAL INTERPOLATION AND THE FAST EUCLIDEAN ALGORITHM
, 2012
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