chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2015-16-1-153-162cheb-36Research ArticleСтатьиArticleКАК ЗАВИСЯТ ДИСКРИМИНАНТЫ ЦЕЛОЧИСЛЕННЫХ МНОГОЧЛЕНОВ ОТ ВЗАИМНОГО РАСПОЛОЖЕНИЯ КОРНЕЙ?HOW DOES THE DISCRIMINANT OF INTEGER POLYNOMIALS DEPEND ON THE DISTRIBUTION OF ROOTS?БударинаН. В.BudarinaN. V.

 

 

 

(Dublin),  (Minsk), (Dublin)

БерникВ. И.BernikV. I.О’ДоннеллХ.O’DonnellH.Дублинский технологический институтRussian FederationИнститут математики НАН БеларусиBelarus201515062016161153162Copyright © Бударина Н.В., Берник В.И., О’Доннелл Х., 20162016Бударина Н.В., Берник В.И., О’Доннелл Х.Budarina N.V., Bernik V.I., O’Donnell 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/36

Пусть n ∈ N – фиксированное число, Q > 1 – некоторый натуральный параметр, и Pn(Q) обозначает множество целочисленных многочленов степени n и высоты, не превосходящей Q. Для заданного многочлена P(x) = anx n + · · · + a0 ∈ Z[x] степени n, число D(P) = a 2n−2 n ∏ 16i< |D(P)| 6 Q 2n−2−2v . Первые результаты по оценкам количества многочленов с заданными дискриминантами получил Х. Давенпорт в 1961 году, что имело важное значение при решении проблемы Малера. В данной работе впервые найдена точная верхняя и нижняя оценка для #P3(Q, v) при дополнительном условии на взаимное расположение корней полиномов. Интересно, что величина #Pn(Q, v) принимает наибольшее значение, когда все корни многочленов близки друг к другу. Если же близки только k, 2 6 k < n, корней, то величина #Pn(Q, v) будет меньше.

Let n ∈ N be fixed, Q > 1 be some natural parameter, and Pn(Q) denote the set of integer polynomials of degree n and height of at most Q. Given a polynomial P(x) = anx n + · · · + a0 ∈ Z[x] of degree n, the discriminant of P(x) is defined by D(P) = a 2n−2 n ∏ 16i< |D(P)| 6 Q 2n−2−2v . The first results for the estimate of the number of polynomials with given discriminants were received by H. Davenport in 1961, which were crucial to the solving of the problem of Mahler. In this paper for the first time we obtain the exact upper and lower bounds for #P3(Q, v) with the additional condition on the distribution of the roots of the polynomials. It is interesting that the value of #Pn(Q, v) has the largest value when all the roots of polynomials are close to each other. If there are only k, 2 6 k < n, close roots to each other then the value of #Pn(Q, v) will be less.

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