On monic abelian trace-one cubic polynomials

2023/10/27 by Shubhrajit Bhattacharya, Bhattacharya, Shubhrajit, Andrew O’Desky +1 · 1 citation
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2310.17831

Abstract

We compute the asymptotic number of monic trace-one integral polynomials with Galois group C₃ and bounded height. For such polynomials we compute a height function coming from toric geometry and introduce a parametrization using the quadratic cyclotomic field \mathbb Q(√(-3)). We also give a formula for the number of polynomials of the form t³ -t² + at + b ∈ \mathbb Z[t] with Galois group C₃ for a fixed integer a.

Cited by

• Counting polynomials with positive roots

Related

• On trace-one generators of abelian cubic fields
• On monic abelian cubics
• Integral points on singular toric varieties and cyclic normal polynomials
• Upper bounds on polynomials with small Galois group
• Counting polynomials with positive roots