{"title":"Real Lines on Random Cubic Surfaces","authors":"Rida Ait El Manssour, Mara Belotti, Chiara Meroni","doi":"10.1007/s40598-021-00182-y","DOIUrl":null,"url":null,"abstract":"<div><p>We give an explicit formula for the expectation of the number of real lines on a random invariant cubic surface, i.e., a surface <span>\\(Z\\subset {\\mathbb {R}}{\\mathrm {P}}^3\\)</span> defined by a random gaussian polynomial whose probability distribution is invariant under the action of the orthogonal group <i>O</i>(4) by change of variables. Such invariant distributions are completely described by one parameter <span>\\(\\lambda \\in [0,1]\\)</span> and as a function of this parameter the expected number of real lines equals: </p><div><div><span>$$\\begin{aligned} E_\\lambda =\\frac{9(8\\lambda ^2+(1-\\lambda )^2)}{2\\lambda ^2+(1-\\lambda )^2}\\left( \\frac{2\\lambda ^2}{8\\lambda ^2+(1-\\lambda )^2}-\\frac{1}{3}+\\frac{2}{3}\\sqrt{\\frac{8\\lambda ^2+(1-\\lambda )^2}{20\\lambda ^2+(1-\\lambda )^2}}\\right) . \\end{aligned}$$</span></div></div><p>This result generalizes previous results by Basu et al. (Math Ann 374(3–4):1773–1810, 2019) for the case of a Kostlan polynomial, which corresponds to <span>\\(\\lambda =\\frac{1}{3}\\)</span> and for which <span>\\(E_{\\frac{1}{3}}=6\\sqrt{2}-3.\\)</span> Moreover, we show that the expectation of the number of real lines is maximized by random purely harmonic cubic polynomials, which corresponds to the case <span>\\(\\lambda =1\\)</span> and for which <span>\\(E_1=24\\sqrt{\\frac{2}{5}}-3\\)</span>.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-021-00182-y","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Arnold Mathematical Journal","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40598-021-00182-y","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 3
Abstract
We give an explicit formula for the expectation of the number of real lines on a random invariant cubic surface, i.e., a surface \(Z\subset {\mathbb {R}}{\mathrm {P}}^3\) defined by a random gaussian polynomial whose probability distribution is invariant under the action of the orthogonal group O(4) by change of variables. Such invariant distributions are completely described by one parameter \(\lambda \in [0,1]\) and as a function of this parameter the expected number of real lines equals:
This result generalizes previous results by Basu et al. (Math Ann 374(3–4):1773–1810, 2019) for the case of a Kostlan polynomial, which corresponds to \(\lambda =\frac{1}{3}\) and for which \(E_{\frac{1}{3}}=6\sqrt{2}-3.\) Moreover, we show that the expectation of the number of real lines is maximized by random purely harmonic cubic polynomials, which corresponds to the case \(\lambda =1\) and for which \(E_1=24\sqrt{\frac{2}{5}}-3\).
我们给出了一个关于随机不变三次曲面上实线数期望的显式公式,即由随机高斯多项式定义的曲面(Z\subet{\mathbb{R}}{\math rm{P}}}^3\),其概率分布在正交群O(4)的作用下通过变量的变化而不变。这种不变分布完全由一个参数\(\lambda\in[0,1]\)来描述,并且作为该参数的函数,期望的实数等于:$$\beagin{aligned}E_\lambda=\frac{9(8\lambda^2+(1-\lambda)^2)}{2\lambda^2+1-\lambda)^2}{20\lambda ^2+(1-\lambda)^2}\right)。\end{aligned}$$这个结果推广了Basu等人以前的结果。(Math Ann 374(3-4):1773–181019)对于Kostlan多项式的情况,该多项式对应于\(λ=\frac{1}{3}\),并且\(E_{2}-3.\)此外,我们证明了实线数的期望通过随机纯谐波三次多项式最大化,这对应于情况\(λ=1),并且对于情况\(E_1=24\sqrt{\frac{2}{5}}-3)。
期刊介绍:
The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.