Integral polynomials with small discriminants and resultants

TY - JOUR

T1 - Integral polynomials with small discriminants and resultants

AU - Beresnevich, Victor

AU - Bernik, Vasili

AU - Goetze, Friedrich

N1 - © 2016, Authors

PY - 2016/8/6

Y1 - 2016/8/6

N2 - Let n∈N be fixed, Q>1 be a real parameter and Pn(Q) denote the set of polynomials over Z of degree n and height at most Q. In this paper we investigate the following counting problems regarding polynomials with small discriminant D(P) and pairs of polynomials with small resultant R(P 1, P 2):(i)given 0≤v≤n-1 and a sufficiently large Q, estimate the number of polynomials P∈Pn(Q) such that0<|D(P)|≤Q 2n-2-2v; (ii)given 0≤w≤n and a sufficiently large Q, estimate the number of pairs of polynomials P1,P2∈Pn(Q) such that0<|R(P1,P2)|≤Q 2n-2w. Our main results provide lower bounds within the context of the above problems. We believe that these bounds are best possible as they correspond to the solutions of naturally arising linear optimisation problems. Using a counting result for the number of rational points near planar curves due to R. C. Vaughan and S. Velani we also obtain the complementary optimal upper bound regarding the discriminants of quadratic polynomials.

AB - Let n∈N be fixed, Q>1 be a real parameter and Pn(Q) denote the set of polynomials over Z of degree n and height at most Q. In this paper we investigate the following counting problems regarding polynomials with small discriminant D(P) and pairs of polynomials with small resultant R(P 1, P 2):(i)given 0≤v≤n-1 and a sufficiently large Q, estimate the number of polynomials P∈Pn(Q) such that0<|D(P)|≤Q 2n-2-2v; (ii)given 0≤w≤n and a sufficiently large Q, estimate the number of pairs of polynomials P1,P2∈Pn(Q) such that0<|R(P1,P2)|≤Q 2n-2w. Our main results provide lower bounds within the context of the above problems. We believe that these bounds are best possible as they correspond to the solutions of naturally arising linear optimisation problems. Using a counting result for the number of rational points near planar curves due to R. C. Vaughan and S. Velani we also obtain the complementary optimal upper bound regarding the discriminants of quadratic polynomials.

KW - Algebraic numbers

KW - Counting discriminants and resultants of polynomials

KW - Metric theory of Diophantine approximation

KW - Polynomial root separation

UR - http://www.scopus.com/inward/record.url?scp=84965108162&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2016.04.022

DO - 10.1016/j.aim.2016.04.022

M3 - Article

SN - 0001-8708

VL - 298

SP - 393

EP - 412

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -