chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2017-18-4-106-114cheb-381Research ArticleСтатьиArticleРАСПРЕДЕЛЕНИЕ НУЛЕЙ НЕВЫРОЖДЕННЫХ ФУНКЦИЙ НА КОРОТКИХ ОТРЕЗКАХDISTRIBUTION OF ZEROS OF NONDEGENERATE FUNCTIONS ON SHORT CUTTINGSБерникВ. И.BernikV. I.

Минск.

Minsk.

БударинаН. В.BudarinaN. V.

Москва.

Moscow.

ЛуневичА. В.LunevichA. V.

Минск.

Minsk.

О’ДоннелХ.O’DonnelH.

Йорк.

York.

201708032018184106114Copyright © Берник В.И., Бударина Н.В., Луневич А.В., О’Доннел Х., 20182018Берник В.И., Бударина Н.В., Луневич А.В., О’Доннел Х.Bernik V.I., Budarina N.V., Lunevich A.V., O’Donnel 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/381

В работе получены оценки сверху и снизу количества нулей функций специального вида, а также оценка меры множества точек в которых такие фукции принимают малые значения. Пусть f1 (x), ..., fn (x) функции определенные на интервале I, n + 1 раз дифференцируемы и вронскиан из производных почти везде на I отличен от 0. Такие функции называются невырожденными. Задача о распределении нулей функции F (x) = anfn (x)+ ... +a1f1 (x)+a0, aj ∈ Z, 1 ≤ j ≤ n имеет важное значение в метрической теории диофантовых приближений.

Пусть Q > 1 достаточно большое целое число, а интервал I имеет длину Q−γ, 0 ≤ γ < 1. В работе получены оценки сверху и снизу для количества нулей функции F (x) на интервале I, при |aj| ≤ Q, 0 ≤ γ < 1. При γ = 0 такие оценки были получены А. С. Пяртли, В. Г. Спринджуком, В. И. Берником, В. В. Бересневичем, Н. В. Будариной.

The paper presents newly obtained upper and lower bounds for the number of zeros for functions of a special type, as well as an estimate for the measure of the set where these functions attain small values. Let f1 (x), ..., fn (x) be functions differentiable on the interval I, n+1 times and Wronskian from derivatives almost everywhere on I is different from 0. Such functions are called nondegenerate. The problem of the distribution of the zeros of the function F (x) = anfn (x) + ... + a1f1 (x) + a0, aj ∈ Z, 1 ≤ j ≤ n is important in the metric theory of Diophantine approximations.

Let Q > 1 be a sufficiently large integer, and the interval I has length Q−γ, 0 ≤ γ < 1. We obtain upper and lower bounds for the number of zeros of the function F (x) on the interval I, with |aj| ≤ Q, 0 ≤ γ < 1. For γ = 0 such estimates were obtained by A. S. Pyartli, V. G. Sprindzhuk, V. I. Bernik, V. V. Beresnevitch, N. V. Budarina.

невырожденные функциинули невырожденных функцийnondegenerate functionsonszeros of nondegenerate functionsonsReferencesИбрагимов, И. А., Маслова, Н. Б. О среднем числе вещественных нулей случайных полиномов. II. Коэффициенты с ненулевым средним // Теория вероятн. и ее примен., 1971 vol. 16, P. 595-503Ibragimov, I. A., Maslova, N. B., 1971 “ On the Expected Number of Real Zeros of Random Polynomials. II. Coefficients With Non-Zero Means Theory Probab. Appl., vol. 16, pp. 486-493Запорожец, Д. Н. Ибрагимов, И. А. О площади случайной поверхности. Вероятность и статистика // Зап. научн. сем. ПОМИ, 2010, vol. 384, P. 154-1750.Zaporozhets, D. N. Ibragimov, I. A.2010 “On random surface area“ Journal of Mathematical Sciences, vol. 176, pp. 190-202.Берник, В. И., Гётце, Ф. Распределение действительных алгебраических чисел произвольной степени в коротких интервалах // Изв. РАН. Сер. матем., 2015, vol. 79, no.1, P. 21-42.Берник, В. И., Гётце, Ф. 2015 “Distribution of real algebraic numbers of arbitrary degree in short intervals“ Izvestiya: Mathematics, vol. 79, no.1, P. 21-42.Beresnevich, V. On approximation of real numbers by real algebraic numbers // Acta Arith, 1999, Vol. 90, no. 8, P. 97-112.Beresnevich, V. 1999, “On approximation of real numbers by real algebraic numbers“, Acta Arith Vol. 90, no. 8, pp. 97-112.Beresnevich, V., Bernik V. On a metrical theorem of W. Shmidt. Acta Arith // Acta Arith, 1996, vol. 75, P. 219-233.Beresnevich, V., Bernik V. 1996 “On a metrical theorem of W. Shmidt. Acta Arith“ Acta Arith, vol. 75, pp. 219-233.Beresnevich, V. A. Grasher type theorem for convergence on maifolds // Acta Matth. Hung, 2002, vol. 94(1—2), P. 99-130.Beresnevich, V. A. 2002 “Grasher type theorem for convergence on maifolds“, Acta Matth. Hung, vol. 94(1—2), pp. 99-130.Baker, R. Metric diophantine aProximation on manifolds // J. Lond. Math. Soc., 1976, vol. 14, P. 43-48.Baker, R. 1976 “Metric diophantine approximation on manifolds“ J. Lond. Math. Soc., vol. 14, pp. 43-48.Berink, V. On the exact order of approximation of zero by the values of integer-valued polynomials // Acta. Arith., 1989, vol. 53, no. 1, P. 17-28.Berink, V. 1989 “On the exact order of approximation of zero by the values of integer-valued polynomials“, Acta. Arith., vol. 53, no. 1, pp. 17-28.Berink, V. Kleinbok, D., Marguli Y. 2001 Khinchine-type theorems on manifolds: the convergence case for standart and multiplicative versions // Jntern. Math. Res., vol. 9, P. 453-486.Berink, V. Kleinbok, D., Marguli Y. 2001 “Khinchine-type theorems on manifolds: the convergence case for standart and multiplicative versions“, Jntern. Math. Res., vol. 9, pp. 453486.Berink, V., Go¨tze, F. Distribution of real algebraic numbers of arbitary degree in short intervals // Jzv. Math. RAN., 2015, vol. 79, no. 1, P. 18-39.Berink, V., G¨otze, F. 2015 “Distribution of real algebraic numbers of arbitary degree in short intervals“, Jzv. Math. RAN., vol. 79, no. 1, pp. 18-39.Berink, V., Gusakova, A., G¨otze F. On ponts with algebraically conjugate coordinates close to smooth curves // Moscow Journal of Combinations and Number Theory, 2016, vol. 6, iss. 2-3, P. 56-101.Berink, V., Gusakova, A., G¨otze F. 2016 “On ponts with algebraically conjugate coordinates close to smooth curves“, Moscow Journal of Combinations and Number Theory, vol. 6, iss. 2-3, pp. 56-101.Kleinbok, D., Margulis, G. Flow on homogeneous spaces and Diophantine approximation on manifolds // Ann. of Math., 1998, vol. 148 no. 2, P. 339-360Kleinbok, D., Margulis, G. 1998 “Flow on homogeneous spaces and Diophantine approximation on manifolds“, Ann. of Math., vol. 148 no. 2, pp. 339-360Mahker K. ¨Uber das Mass der Menge aller S-Zhlen // Math. Ann.,1932, vol. 106, P. 131-139.Mahker K. 1932 “¨Uber das Mass der Menge aller S-Zhlen“, Math. Ann., vol. 106, pp. 131-139.Pyartly, A. Diophantine approximation on submanifolds of euclidion space // Funk. Analis and its application, 1969, vol. 3, no. 4, P. 303-306Pyartly, A. 1969 “Diophantine approximation on submanifolds of euclidion space“, Funk. Analis and its application, vol. 3, no. 4, pp. 303-306Shmidt, W. Metrische Satze u¨ber simultane Approximationen abhangiger Grossen // Monatsh. Math.,1964, vol. 68, P. 145-166Shmidt, W. 1964 “Metrische Satze u¨ber simultane Approximationen abhangiger Grossen“, Monatsh. Math., vol. 68, pp. 145-166Sprindzuk, V. Achievements and problems of the theory of Diophantine approximations // Uspekhi mat. Baur.,1980, vol. 35, no. 4, P. 3-68.Sprindzuk, V. 1980 “Achievements and problems of the theory of Diophantine approximations“, Uspekhi mat. Baur. vol. 35, no. 4, pp. 3-68.Sprindzuk, V. Mahler problem in metric theory numbers, Eng. trans. // Amer. Math. Soc. Providence, 1969.Sprindzuk, V. 1969 “Mahler problem in metric theory numbers“, Eng. trans. Amer. Math. Soc. ProvidenceG¨otze, F. Koleda, D., Zaporozhets, D. Distribution of complex algebraic numbers // Proc. Amer. Math. Soc., 2017, vol. 145, no. 1, (), 61-71.G¨otze, F. Koleda, D., Zaporozhets, D. 2017 “Distribution of complex algebraic numbers“, Proc. Amer. Math. Soc., vol. 145, no. 1, (), 61-71.Bernik A., G¨otzeb F., Kukso O. Bad-approximable points and distribution of discriminants of the product of linear integer polynomials // Чебышевский сб., 2007m vol. 8, no. 2, P 140–147Bernik A., G¨otzeb F., Kukso O. 2007 “Bad-approximable points and distribution of discriminants of the product of linear integer polynomials“, Chebyshevskii Sbornik, vol. 8, no. 2, pp. 140–147Bernik A., G¨otzeb F., Gusakova A. On the distribution of points with algebraically conjugate coordinates in a neighborhood of smooth curves // Записки ПОМИ, 2016, P. 14-47Bernik A., G¨otzeb F., 2016 “Gusakova A. On the distribution of points with algebraically conjugate coordinates in a neighborhood of smooth curves“, Zapiski POMI, pp. 14-47Beresnevich V., Bernik V., G¨otze F. Integral polynomials with small discriminants and resultants // Adv. Math., 2016, vol. 298, P. 393-412.Beresnevich V., Bernik V., Go¨tze F. 2016 “Integral polynomials with small discriminants and resultants“, Adv. Math., vol. 298, pp. 393-412.Koleda, D. V. On the density function of the distribution of real algebraic numbers // Journal de Theorie des Nombres de Bordeaux, 2017, vol. 29, P. 179-200.Koleda, D. V. 2017 “On the density function of the distribution of real algebraic numbers“ Journal de Theorie des Nombres de Bordeaux, vol. 29, pp. 179-200.

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