Discriminant and root separation of integral polynomials

Standard

Discriminant and root separation of integral polynomials. / Götze, F.; Zaporozhets, D.

In: Journal of Mathematical Sciences (United States), Vol. 219, No. 5, 01.12.2016, p. 700-706.

Research output: Contribution to journal › Article › peer-review

Author

Götze, F. ; Zaporozhets, D. / Discriminant and root separation of integral polynomials. In: Journal of Mathematical Sciences (United States). 2016 ; Vol. 219, No. 5. pp. 700-706.

BibTeX

@article{47a5289c075d44ecb5e85ce19194c6d7,

title = "Discriminant and root separation of integral polynomials",

abstract = "Consider a random polynomial GQ (x) = ξQ,n xn + ξQ,n−1 xn−1 + · · · + ξQ,0 with independent coefficients that are uniformly distributed on 2Q+1 integer points {−Q, …, Q}. Denote by D(GQ) the discriminant of GQ. We show that there exists a constant Cn depending on n only such that for all Q ≥ 2, the distribution of D(GQ) can be approximated as follows: (Formula presented) where ϕn denotes the probability density function of the discriminant of a random polynomial of degree n with independent coefficients that are uniformly distributed on [−1, 1]. Let Δ(GQ) denote the minimal distance between complex roots of GQ. As an application, we show that for any ε > 0 there exists a constant δn > 0 such that Δ(GQ) is stochastically bounded from below/above for all sufficiently large Q in the following sense: (Formula presented) Bibliography: 14 titles.",

author = "F. G{\"o}tze and D. Zaporozhets",

year = "2016",

month = dec,

day = "1",

doi = "10.1007/s10958-016-3139-9",

language = "English",

volume = "219",

pages = "700--706",

journal = "Journal of Mathematical Sciences",

issn = "1072-3374",

publisher = "Springer Nature",

number = "5",

}

RIS

TY - JOUR

T1 - Discriminant and root separation of integral polynomials

AU - Götze, F.

AU - Zaporozhets, D.

PY - 2016/12/1

Y1 - 2016/12/1

N2 - Consider a random polynomial GQ (x) = ξQ,n xn + ξQ,n−1 xn−1 + · · · + ξQ,0 with independent coefficients that are uniformly distributed on 2Q+1 integer points {−Q, …, Q}. Denote by D(GQ) the discriminant of GQ. We show that there exists a constant Cn depending on n only such that for all Q ≥ 2, the distribution of D(GQ) can be approximated as follows: (Formula presented) where ϕn denotes the probability density function of the discriminant of a random polynomial of degree n with independent coefficients that are uniformly distributed on [−1, 1]. Let Δ(GQ) denote the minimal distance between complex roots of GQ. As an application, we show that for any ε > 0 there exists a constant δn > 0 such that Δ(GQ) is stochastically bounded from below/above for all sufficiently large Q in the following sense: (Formula presented) Bibliography: 14 titles.

AB - Consider a random polynomial GQ (x) = ξQ,n xn + ξQ,n−1 xn−1 + · · · + ξQ,0 with independent coefficients that are uniformly distributed on 2Q+1 integer points {−Q, …, Q}. Denote by D(GQ) the discriminant of GQ. We show that there exists a constant Cn depending on n only such that for all Q ≥ 2, the distribution of D(GQ) can be approximated as follows: (Formula presented) where ϕn denotes the probability density function of the discriminant of a random polynomial of degree n with independent coefficients that are uniformly distributed on [−1, 1]. Let Δ(GQ) denote the minimal distance between complex roots of GQ. As an application, we show that for any ε > 0 there exists a constant δn > 0 such that Δ(GQ) is stochastically bounded from below/above for all sufficiently large Q in the following sense: (Formula presented) Bibliography: 14 titles.

UR - http://www.scopus.com/inward/record.url?scp=85046874639&partnerID=8YFLogxK

U2 - 10.1007/s10958-016-3139-9

DO - 10.1007/s10958-016-3139-9

M3 - Article

AN - SCOPUS:85046874639

VL - 219

SP - 700

EP - 706

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 5

ER -

ID: 126286703