Computer Algebra in the Service of Mathematical Physics and Number Theory

Chudnovsky, D. V.; Chudnovsky, G. V.

doi:10.1201/9781003072157-5

Cited by 21 publications

(20 citation statements)

References 1 publication

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“…where N is assumed to be a large positive integer? This question was recognized as a very important one in complexity theory, as well as in various applications to algorithmic number theory: Atkin-Swinnerton-Dyer congruences, integer factorization, discrete logarithm and point-counting [10,3].…”

Section: Introductionmentioning

confidence: 99%

Fast coefficient computation for algebraic power series in positive characteristic

et al. 2019

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We revisit Christol's theorem on algebraic power series in positive characteristic and propose yet another proof for it. This new proof combines several ingredients and advantages of existing proofs, which make it very wellsuited for algorithmic purposes. We apply the construction used in the new proof to the design of a new efficient algorithm for computing the N th coefficient of a given algebraic power series over a perfect field of characteristic p. It has several nice features: it is more general, more natural and more efficient than previous algorithms. Not only the arithmetic complexity of the new algorithm is linear in log N and quasi-linear in p, but its dependency with respect to the degree of the input is much smaller than in the previously best algorithm. Moreover, when the ground field is finite, the new approach yields an even faster algorithm, whose bit complexity is linear in log N and quasi-linear in √ p.

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Section: Introductionmentioning

confidence: 99%

Fast coefficient computation for algebraic power series in positive characteristic

et al. 2019

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“…org; Marc Mezzarobba, LIX, CNRS, École polytechnique, Institut polytechnique de Paris, 91200, Palaiseau, France, marc@ mezzarobba.net; Nobuki Takayama, Kobe University, Kobe, Japan, takayama@math.kobe-u.ac.jp; Tristan VacconUniversité de Limoges;, CNRS, XLIM UMR 7252, Limoges, France, tristan.vaccon@unilim.fr. ingredient for reaching a quasi-optimal complexity is an adaptation to the -adic setting of the socalled bit-burst method introduced by Chudnovsky and Chudnovsky [8,9], building on the binary splitting technique [e.g. , 16] (see also [2, §178]) and other ideas dating back to Brent's work on elementary functions [6].…”

Section: Introductionmentioning

confidence: 99%

Fast evaluation of some p-adic transcendental functions

Caruso,

Mezzarobba,

Takayama

et al. 2021

Preprint

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We design algorithms for computing values of many -adic elementary and special functions, including logarithms, exponentials, polylogarithms, and hypergeometric functions. All our algorithms feature a quasi-linear complexity with respect to the target precision and most of them are based on an adaptation to the -adic setting of the binary splitting and bit-burst strategies.CCS Concepts: • Computing methodologies → Algebraic algorithms.

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“…We evaluate y (for two values of the parameter a) at N points z. We can compute each such function value in p 1+o(1) bit operations and p 1+o(1) space using the bit-burst algorithm [CC90], employing arithmetic in the number field Q(s). This bound holds uniformly for the required values of a and z, by the same argument as in [Mez12, Corollary 1], using the facts that |z| ≤ p 1+o(1) and that A as well as the defining polynomial of Q(s) have fixed degree and coefficients of bit size O(log p).…”

Section: Introductionmentioning

confidence: 99%

“…The only interesting point in the proof of Theorem 1 is the use of the bit-burst algorithm instead of naive summation, which allows us to compute the function Γ(a, z) in quasilinear rather than quadratic time. The bit-burst algorithm for holonomic functions has been known since the 1980s [CC90,vdH99,vdH01,Mez11,Mez12], and since the 1970s in special cases [Bre76a], yet we are not aware of a correct complexity bound of this kind in the literature for Dirichlet L-functions or even for the special case of the Riemann zeta function.…”

Section: Introductionmentioning

confidence: 99%

Rapid computation of special values of Dirichlet $L$-functions

Johansson¹

2021

Preprint

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We consider computing the Riemann zeta function ζ(s) and Dirichlet L-functions L(s, χ) to p-bit accuracy for large p. Using the approximate functional equation together with asymptotically fast computation of the incomplete gamma function, we observe that p 3/2+o(1) bit complexity can be achieved if s is an algebraic number of fixed degree and with algebraic height bounded by O(p). This is an improvement over the p 2+o(1) complexity of previously published algorithms and yields, among other things, p 3/2+o(1) complexity algorithms for Stieltjes constants and n 3/2+o(1) complexity algorithms for computing the nth Bernoulli number or the nth Euler number exactly.

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Computer Algebra in the Service of Mathematical Physics and Number Theory

Cited by 21 publications

References 1 publication

Fast coefficient computation for algebraic power series in positive characteristic

Fast coefficient computation for algebraic power series in positive characteristic

Fast evaluation of some p-adic transcendental functions

Rapid computation of special values of Dirichlet $L$-functions

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