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On The Complexity Of Computing Determinants
 COMPUTATIONAL COMPLEXITY
, 2001
"... We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n n matrix A with integer entries in (n and (n bi ..."
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Cited by 61 (19 self)
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We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n n matrix A with integer entries in (n and (n bit operations; here denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors for positive real constants C 1 , C 2 , C 3 . The bit complexity (n results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n and O(n ) ring additions, subtractions and multiplications.
It Is Easy to Determine Whether a Given Integer Is Prime
, 2004
"... The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be super ..."
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Cited by 17 (2 self)
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The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers... It frequently happens that the trained calculator will be sufficiently rewarded by reducing large numbers to their factors so that it will compensate for the time spent. Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated... It is in the nature of the problem
A CASE OF DEPTH3 IDENTITY TESTING, SPARSE FACTORIZATION AND DUALITY
"... Polynomial identity testing (PIT) problem is known to be challenging even for constant depth arithmetic circuits. In this work, we study the complexity of two special but natural cases of identity testing first is a case of depth3 PIT, the other of depth4 PIT. Our first problem is a vast generali ..."
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Cited by 10 (5 self)
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Polynomial identity testing (PIT) problem is known to be challenging even for constant depth arithmetic circuits. In this work, we study the complexity of two special but natural cases of identity testing first is a case of depth3 PIT, the other of depth4 PIT. Our first problem is a vast generalization of: Verify whether a bounded top fanin depth3 circuit equals a sparse polynomial (given as a sum of monomial terms). Formally, given a depth3 circuit C, having constant many general product gates and arbitrarily many semidiagonal product gates, test if the output of C is identically zero. A semidiagonal product gate in C computes a product of the form m · ∏b, where m is a i=1 ℓei i monomial, ℓi is an affine linear polynomial and b is a constant. We give a deterministic polynomial time test, along with the computation of leading monomials of semidiagonal circuits over local rings. The second problem is on verifying a given sparse polynomial factorization, which is a classical question (von zur Gathen, FOCS 1983): Given multivariate sparse polynomials f, g1,..., gt explicitly, check if f = ∏ t i=1 gi. For the special case when every gi is a sum of univariate polynomials, we give a deterministic polynomial time test. We characterize the factors of such gi’s and even show how to test the divisibility of f by the powers of such polynomials. The common tools used are Chinese remaindering and dual representation. The dual representation of polynomials (Saxena, ICALP 2008) is a technique to express a productofsums of univariates as a sumofproducts of univariates. We generalize this technique by combining it with a generalized Chinese remaindering to solve these two problems (over any field).
EFFICIENTLY CERTIFYING Noninteger Powers
"... We describe a randomized algorithm that, given an integer a, produces a certificate that the integer is not a pure power of an integer in expected (log a) 1+o(1) bit operations under the assumption of the Generalized Riemann Hypothesis. The certificate can then be verified in deterministic (log a) ..."
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We describe a randomized algorithm that, given an integer a, produces a certificate that the integer is not a pure power of an integer in expected (log a) 1+o(1) bit operations under the assumption of the Generalized Riemann Hypothesis. The certificate can then be verified in deterministic (log a) 1+o(1) time. The certificate constitutes for each possible prime exponent p a prime number qp, such that a mod qp is a pth nonresidue. We use an effective version of the Chebotarev density theorem to estimate the density of such prime numbers qp.
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"... A good new millenium for the primes Andrew Granville* is the Canadian Research Chair in number theory at the Université de Montréal. A good new millenium for the primes Prime numbers, the building blocks from which integers are made, are central to much of mathematics. Understanding their distributi ..."
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A good new millenium for the primes Andrew Granville* is the Canadian Research Chair in number theory at the Université de Montréal. A good new millenium for the primes Prime numbers, the building blocks from which integers are made, are central to much of mathematics. Understanding their distribution is one of the most natural, and hence oldest, problems in mathematics. Once the ancient Greeks had determined that there are infinitely many then it was natural to ask how many there are up to any given point, perhaps a very large point, how many there are in certain special subsequences (for example, primes of the form “a square plus one”), and how to identify primes quickly. If one examines tables of primes then they appear to be “randomly distributed” though, as Bob Vaughan once put it, “we do not yet know what random means”. Answering these questions has thus proved to be difficult, each success requiring new, far reaching ideas and methods. During the last thirty years there had been few new results of this type but then, in the last decade, several surprises, some of which we will discuss here: