Abstract

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(P1,P2):(i)given0≤v≤n−1and a sufficiently large Q, estimate the number of polynomialsP∈Pn(Q)such that0<|D(P)|≤Q2n−2−2v;(ii)given0≤w≤nand a sufficiently large Q, estimate the number of pairs of polynomialsP1,P2∈Pn(Q)such that0<|R(P1,P2)|≤Q2n−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.

Highlights

  • Throughout this paper n will denote a positive integer

  • In this paper we investigate the following counting problems regarding polynomials with small discriminant D(P ) and pairs of polynomials with small resultant R(P1, P2): (i) given 0 ≤ v ≤ n − 1 and a sufficiently large Q, estimate the number of polynomials

  • Much less is known about Problem 2, some counting estimates are implicit in various papers, e.g. upper bounds for pairs of irreducible polynomials are a vital ingredient in [9]

Read more

Summary

Introduction

Throughout this paper n will denote a positive integer. In what follows, given a polynomial P = anxn + · · · + a0 ∈ Z[x] of degree n, let. Much less is known about Problem 2, some counting estimates are implicit in various papers, e.g. upper bounds for pairs of irreducible polynomials are a vital ingredient in [9]. It was shown in [5] that for any integers n ≥ 2 and m ∈ [0, n − 1] there exists a constant ρ depending on n only such that. In this paper we develop a general approach which enables us to determine the optimal configuration that maximises the number of choices for polynomials while keeping their discriminants/resultants under a given bound.

Rational points near planar curves and the quadratic case
Auxiliary lemmas
Proof of Theorem 1
Proof of Theorem 2
Comparing the estimates for different degrees

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.