Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer
Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer
复制标题
机器学习遇上数论：Birch-Swinnerton-Dyer 的数据科学
DOI：
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
Yang
中科院分区：
文献类型：
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作者：
Laura Alessandretti;Andrea Baronchelli;Yang
实证分析往往是猜想诞生的第一步。这就是Birch-Swinnerton-Dyer（BSD）猜想的情况，它描述了椭圆曲线上的有理点，这是数学中最著名的未解决问题之一。在这里，我们扩展了原来的经验方法，分析克雷莫纳数据库的数量相关的BSD，检查超过250万椭圆曲线的数据科学，机器学习和拓扑数据分析的最新技术手段。关键数量，如排名，维尔斯特拉斯系数，周期，导体，玉川数，调节器和秩序的泰特-Shafarevich组产生一个高维点云的统计特性，我们调查。我们揭示的模式和分布的秩与维尔斯特拉斯系数，以及Beta分布的BSD比的数量。通过梯度提升树，机器学习被应用于寻找各种量之间的相互关系。我们预计，我们的方法将引发进一步的研究大型数据集的统计特性在数论和更一般的纯数学。
Empirical analysis is often the first step towards the birth of a conjecture. This is the case of the Birch-Swinnerton-Dyer (BSD) Conjecture describing the rational points on an elliptic curve, one of the most celebrated unsolved problems in mathematics. Here we extend the original empirical approach, to the analysis of the Cremona database of quantities relevant to BSD, inspecting more than 2.5 million elliptic curves by means of the latest techniques in data science, machine-learning and topological data analysis. Key quantities such as rank, Weierstrass coefficients, period, conductor, Tamagawa number, regulator and order of the Tate-Shafarevich group give rise to a high-dimensional point-cloud whose statistical properties we investigate. We reveal patterns and distributions in the rank versus Weierstrass coefficients, as well as the Beta distribution of the BSD ratio of the quantities. Via gradient boosted trees, machine learning is applied in finding inter-correlation amongst the various quantities. We anticipate that our approach will spark further research on the statistical properties of large datasets in Number Theory and more in general in pure Mathematics.
L 函数和模块化形式
DOI：
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发表时间：
2006
期刊：
影响因子：
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作者：
Juergen Herzog;Yukihide Takayama;Naoki Terai;H.Katsurada;Mutsumi Amasaki;H.Katsurada;H. Katsurada;Shoetsu Ogata;S. Boecherer;Atsushi Noma;M.Amasaki;H. Katsurada;S.Ogata;H. Katsurada and H. Kawamura;A.Noma;S. Boecherer;C.Miyazaki;桂田英典;N.Terai 他;宮崎 誓;H. Katsurada;宮崎誓;Chikashi Miyazaki;H. Katsurada;C.Miyazaki;H. Katsurada;Chikashi Miyazaki;H. Katsurada;宮崎誓;Chikashi Miyazaki;千吉良直紀;宮崎誓;千吉良直紀;H. Katsurada;桂田英典;H. Katsurada;千吉良直紀;H. Katsurada;H. Katsurada;H. Katsurada and H. Kawamura;H. Katsurada and H. Kawamura;H. Katsuradan;H. Katsurada and H. Kawamura;H. Katsurada and H. Kawamura;H. Katsurada
通讯作者：
H. Katsurada