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Exact \(p\)-adic computation in Magma. (English) Zbl 07312490
Summary: We describe a new arithmetic system for the Magma computer algebra system for working with \(p\)-adic numbers exactly, in the sense that numbers are represented lazily to infinite \(p\)-adic precision. This is the first highly featured such implementation. This has the benefits of increasing user-friendliness and speeding up some computations, as well as forcibly producing provable results. We give theoretical and practical justification for its design and describe some use cases. The intention is that this article will be of benefit to anyone wanting to implement similar functionality in other languages.
11-04 Software, source code, etc. for problems pertaining to number theory
11S82 Non-Archimedean dynamical systems
12-08 Computational methods for problems pertaining to field theory
12J25 Non-Archimedean valued fields
68W30 Symbolic computation and algebraic computation
Full Text: DOI
[1] Berthomieu, J.; Lebreton, R., Relaxed p-adic Hensel lifting for algebraic systems, (ISSAC 2012—Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation (2012), ACM: ACM New York), 59-66 · Zbl 1308.68158
[2] Berthomieu, J.; van der Hoeven, J.; Lecerf, G., Relaxed algorithms for p-adic numbers, J. Théor. Nr. Bordx., 23, 3, 541-577 (2011) · Zbl 1247.11152
[3] Bosma, W.; Cannon, J.; Playoust, C., The Magma algebra system. I. The user language, Computational Algebra and Number Theory (London, 1993). Computational Algebra and Number Theory (London, 1993), J. Symb. Comput., 24, 3-4, 235-265 (1997) · Zbl 0898.68039
[4] Caruso, X., Computations with p-adic numbers (2017)
[5] Cassels, J. W.S., Local Fields (1986), Cambridge University Press · Zbl 0595.12006
[6] Dokchitser, T.; Doris, C., 3-torsion and conductor of genus 2 curves, Math. Comput., 88, 318, 1913-1927 (2019) · Zbl 1460.11091
[7] Doris, C., ExactpAdics: a package for exact p-adic computation (2018)
[8] Doris, C., ExactpAdics: an exact representation of p-adic numbers (2018)
[9] Doris, C., ExactpAdics2: another package for exact p-adic computation (2018)
[10] Doris, C., Genus2Conductor: a package for computing the conductor of curves of genus 2 (2018)
[11] Doris, C., pAdicGaloisGroup: a package for computing Galois groups of p-adic polynomials (2018)
[12] Doris, C., Aspects of p-adic computation (2019), University of Bristol, Ph.D. thesis
[13] Doris, C., Computing the Galois group of a polynomial over a p-adic field, Int. J. Number Theory (2020), In press · Zbl 1453.11154
[14] Hart, W.; Johansson, F.; Pancratz, S., FLINT: Fast Library for Number Theory (2018)
[15] Sinclair, B., Algorithms for enumerating invariants and extensions of local fields (2015), University of North Carolina at Greensboro, Ph.D. thesis
[16] (2020), PARI/GP
[17] (2018), SageMath, the Sage Mathematics Software System
[18] van der Hoeven, J.; Lecerf, G.; Mourrain, B., The Mathemagix computer algebra and analysis system (2018)
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