English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $n\in\mathbb{N}$ be fixed, $Q>1$ be a real parameter and$\mathcal{P}_n(Q)$ denote the set of polynomials over $\mathbb{Z}$ of degree$n$ and height at most $Q$. In this paper we investigate the following countingproblems regarding polynomials with small discriminant $D(P)$ and pairs ofpolynomials with small resultant $R(P_1,P_2)$:  (i) given $0\le v\le n-1$ and a sufficiently large $Q$, estimate the numberof polynomials $P\in\mathcal{P}_n(Q)$ such that $$0<|D(P)|\le Q^{2n-2-2v};$$(ii) given $0\le w\le n$ and a sufficiently large $Q$, estimate the number ofpairs of polynomials $P_1,P_2\in\mathcal{P}_n(Q)$ such that $$0<|R(P_1,P_2)|\leQ^{2n-2w}.$$ Our main results provide lower bounds within the context of theabove problems. We believe that these bounds are best possible as theycorrespond to the solutions of naturally arising linear optimisation problems.Using a counting result for the number of rational points near planar curvesdue to R.C.Vaughan and S.Velani we also obtain the complementary optimal upperbound regarding the discriminants of quadratic polynomials.

Integral polynomials with small discriminants and resultants