DISTRIBUTION OF ZEROS OF NONDEGENERATE FUNCTIONS ON SHORT CUTTINGS
https://doi.org/10.22405/2226-8383-2017-18-4-106-114
Abstract
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.
About the Authors
V. I. BernikBelarus
Minsk.
N. V. Budarina
Russian Federation
Moscow.
A. V. Lunevich
Belarus
Minsk.
H. O’Donnel
Russian Federation
York.
References
1. Ibragimov, 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. 486-493
2. Zaporozhets, D. N. Ibragimov, I. A.2010 “On random surface area“ Journal of Mathematical Sciences, vol. 176, pp. 190-202.
3. 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.
4. Beresnevich, V. 1999, “On approximation of real numbers by real algebraic numbers“, Acta Arith Vol. 90, no. 8, pp. 97-112.
5. Beresnevich, V., Bernik V. 1996 “On a metrical theorem of W. Shmidt. Acta Arith“ Acta Arith, vol. 75, pp. 219-233.
6. Beresnevich, V. A. 2002 “Grasher type theorem for convergence on maifolds“, Acta Matth. Hung, vol. 94(1—2), pp. 99-130.
7. Baker, R. 1976 “Metric diophantine approximation on manifolds“ J. Lond. Math. Soc., vol. 14, pp. 43-48.
8. 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.
9. 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.
10. 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.
11. 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.
12. Kleinbok, D., Margulis, G. 1998 “Flow on homogeneous spaces and Diophantine approximation on manifolds“, Ann. of Math., vol. 148 no. 2, pp. 339-360
13. Mahker K. 1932 “¨Uber das Mass der Menge aller S-Zhlen“, Math. Ann., vol. 106, pp. 131-139.
14. Pyartly, A. 1969 “Diophantine approximation on submanifolds of euclidion space“, Funk. Analis and its application, vol. 3, no. 4, pp. 303-306
15. Shmidt, W. 1964 “Metrische Satze u¨ber simultane Approximationen abhangiger Grossen“, Monatsh. Math., vol. 68, pp. 145-166
16. Sprindzuk, V. 1980 “Achievements and problems of the theory of Diophantine approximations“, Uspekhi mat. Baur. vol. 35, no. 4, pp. 3-68.
17. Sprindzuk, V. 1969 “Mahler problem in metric theory numbers“, Eng. trans. Amer. Math. Soc. Providence
18. G¨otze, F. Koleda, D., Zaporozhets, D. 2017 “Distribution of complex algebraic numbers“, Proc. Amer. Math. Soc., vol. 145, no. 1, (), 61-71.
19. Bernik 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. 140–147
20. Bernik 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. 14-47
21. Beresnevich V., Bernik V., Go¨tze F. 2016 “Integral polynomials with small discriminants and resultants“, Adv. Math., vol. 298, pp. 393-412.
22. 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.
Review
For citations:
Bernik V.I., Budarina N.V., Lunevich A.V., O’Donnel H. DISTRIBUTION OF ZEROS OF NONDEGENERATE FUNCTIONS ON SHORT CUTTINGS. Chebyshevskii Sbornik. 2017;18(4):106-114. (In Russ.) https://doi.org/10.22405/2226-8383-2017-18-4-106-114
Views:
651
Email this article 