Explicit bounds for generators of the class group
Mathematics of Computation, ISSN: 0025-5718, Vol: 87, Issue: 313, Page: 2483-2511
2018
- 4Citations
- 1Captures
Metric Options: Counts1 Year3 YearSelecting the 1-year or 3-year option will change the metrics count to percentiles, illustrating how an article or review compares to other articles or reviews within the selected time period in the same journal. Selecting the 1-year option compares the metrics against other articles/reviews that were also published in the same calendar year. Selecting the 3-year option compares the metrics against other articles/reviews that were also published in the same calendar year plus the two years prior.
Example: if you select the 1-year option for an article published in 2019 and a metric category shows 90%, that means that the article or review is performing better than 90% of the other articles/reviews published in that journal in 2019. If you select the 3-year option for the same article published in 2019 and the metric category shows 90%, that means that the article or review is performing better than 90% of the other articles/reviews published in that journal in 2019, 2018 and 2017.
Citation Benchmarking is provided by Scopus and SciVal and is different from the metrics context provided by PlumX Metrics.
Example: if you select the 1-year option for an article published in 2019 and a metric category shows 90%, that means that the article or review is performing better than 90% of the other articles/reviews published in that journal in 2019. If you select the 3-year option for the same article published in 2019 and the metric category shows 90%, that means that the article or review is performing better than 90% of the other articles/reviews published in that journal in 2019, 2018 and 2017.
Citation Benchmarking is provided by Scopus and SciVal and is different from the metrics context provided by PlumX Metrics.
Article Description
Assuming Generalized Riemann's Hypothesis, Bach proved that the class group CℓK of a number field K may be generated using prime ideals whose norm is bounded by 12 log ΔK, and by (4 + o(1)) log ΔK asymptotically, where ΔK is the absolute value of the discriminant of K. Under the same assumption, Belabas, Diaz y Diaz and Friedman showed a way to determine a set of prime ideals that generates CℓK and which performs better than Bach's bound in computations, but which is asymptotically worse. In this paper we show that CℓK is generated by prime ideals whose norm is bounded by the minimum of 4.01 log ΔK, 4(1 + (2πe)K) log ΔK and 4( logΔK +log logΔK -(γ +log 2π)NK +1+(NK +1) log(7 logΔK)/logΔK ). Moreover, we prove explicit upper bounds for the size of the set determined by Belabas, Diaz y Diaz and Friedman's algorithms, confirming that it has size (logΔK log logΔK). In addition, we propose a different algorithm which produces a set of generators which satisfies the above mentioned bounds and in explicit computations turns out to be smaller than log Δ except for 7 out of the 31292 fields we tested.
Bibliographic Details
http://www.scopus.com/inward/record.url?partnerID=HzOxMe3b&scp=85046969294&origin=inward; http://dx.doi.org/10.1090/mcom/3281; https://www.ams.org/mcom/2018-87-313/S0025-5718-2017-03281-6/; http://www.ams.org/mcom/2018-87-313/S0025-5718-2017-03281-6/S0025-5718-2017-03281-6.pdf; https://www.ams.org/mcom/2018-87-313/S0025-5718-2017-03281-6/S0025-5718-2017-03281-6.pdf; https://dx.doi.org/10.1090/mcom/3281; https://www.ams.org/journals/mcom/2018-87-313/S0025-5718-2017-03281-6/
American Mathematical Society (AMS)
Provide Feedback
Have ideas for a new metric? Would you like to see something else here?Let us know