chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2023-24-1-219-227cheb-1485Research ArticleКраткие сообщенияОкрестность Вороного главной совершенной формы от пяти переменныхThe neighborhood of the Voronoi main perfect form from five variablesГуломовОтабек ХудайбердиевичGulomovOtabek Hudaiberdievich

кандидат физико-математических наук, доцент

candidate of physical and mathematical sciences, associate professor

otabek10@mail.ruИнститут математики им. В. И. Романовского Академии наук Республики УзбекистанУзбекистанV. I. Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of UzbekistanUzbekistan202325052023241219227Copyright © Гуломов О.Х., 20232023Гуломов О.Х.Gulomov O.H.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.chebsbornik.ru/jour/article/view/1485

Вороной получил для совершенных форм три результата. Во-первых, он доказал, что форма, отвечающая плотнейшей упаковке, является совершенной. Во-вторых, он установил, что совершенных форм от данного числа переменных конечное число. И самое главное, в-третьих, Вороной предложил метод нахождения всех совершенных форм. Этот метод опирается на так называемый совершенный полиэдр, весьма сложный многомерныймногогранник, введенный Вороным. В принципе, найдя методом Вороного все совершенные формы, можно вычислить плотности для конечного числа соответствующих упаковок и выделить те, которые отвечают максимальному значению. Классической задачи Вороного отыскания совершенных форм, тесно связанной с известной проблемой Эрмита арифметические минимумы положительных квадратичных форм. Они появились и в работахС.Л.Соболева и Х.М. Шадиметова в связи с построением решетчатых оптимальных кубатурных формул. В настоящей работы предлагается усовершенствованные алгоритма Воро-ного для вычислении окрестности Вороного совершенной формы от много переменных и с помощью этого алгоритма вычислена окрестность Вороного главной совершенной формы от пяти переменных.

Voronoi obtained three results for perfect forms. First, he proved that the form corresponding to the closest packing is perfect. Secondly, he established that there are a finite number of perfect forms from a given number of variables. And most importantly, thirdly, Voronoi proposed a method for finding all perfect forms. This method relies on the so-called perfect polyhedron, a highly complex multidimensional polyhedron introduced by Voronoi. In principle, having found all perfect forms by the Voronoi method, one can calculate the densities for a finite number of corresponding packings and single out those that correspond to the maximum value.The classical Voronoi problem of finding perfect forms, closely related to Hermite’s well-known problem of arithmetic minima of positive quadratic forms. They also appeared in the works of S.L. Sobolev and Kh.M. Shadimetov in connection with the construction of lattice optimal cubature formulas. In this paper, we propose an improved Voronoi algorithm for calculating the Voronoi neighborhood of a perfect form in many variables, and using this algorithm, theVoronoi neighborhood of the main perfect form in five variables is calculated.

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The authors declare that there are no conflicts of interest present.