HOW DOES THE DISCRIMINANT OF INTEGER POLYNOMIALS DEPEND ON THE DISTRIBUTION OF ROOTS?
https://doi.org/10.22405/2226-8383-2015-16-1-153-162
Abstract
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.
About the Authors
N. V. Budarina
Дублинский технологический институт
Russian Federation
(Dublin), (Minsk), (Dublin)
Russian Federation
(Dublin), (Minsk), (Dublin)
V. I. Bernik
Институт математики НАН Беларуси
Belarus
Belarus
H. O’Donnell
Дублинский технологический институт
Russian Federation
Russian Federation
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Review
For citations:
Budarina N.V., Bernik V.I., O’Donnell H. HOW DOES THE DISCRIMINANT OF INTEGER POLYNOMIALS DEPEND ON THE DISTRIBUTION OF ROOTS? Chebyshevskii Sbornik. 2015;16(1):153-162. (In Russ.) https://doi.org/10.22405/2226-8383-2015-16-1-153-162
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