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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(P _{1}, P _{2}):(i)given 0≤v≤n1 and a sufficiently large Q, estimate the number of polynomials P∈Pn(Q) such that0<D(P)≤Q ^{2n22v}; (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 ^{2n2w}. 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.
Original language  English 

Pages (fromto)  393412 
Number of pages  20 
Journal  Advances in Mathematics 
Volume  298 
Early online date  9 May 2016 
DOIs  
Publication status  Published  6 Aug 2016 
Bibliographical note
© 2016, AuthorsKeywords
 Algebraic numbers
 Counting discriminants and resultants of polynomials
 Metric theory of Diophantine approximation
 Polynomial root separation
Profiles
Projects
 1 Finished

Programme GrantNew Frameworks in metric Number Theory
1/06/12 → 30/11/18
Project: Research project (funded) › Research