Parametrizing roots of polynomial congruences

Welsh, Matthew

doi:10.2140/ant.2022.16.881

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Parametrizing roots of polynomial congruences

Matthew Welsh

Vol. 16 (2022), No. 4, 881–918

DOI: 10.2140/ant.2022.16.881

Abstract

We use the arithmetic of ideals in orders to parametrize the roots $μ (\mod m)$ of the polynomial congruence $F (μ) \equiv 0 (\mod m)$ , $F (X) \in ℤ [X]$ monic, irreducible and degree $d$ . Our parametrization generalizes Gauss’s classic parametrization of the roots of quadratic congruences using binary quadratic forms, which had previously only been extended to the cubic polynomial $F (X) = X^{3} - 2$ . We show that only a special class of ideals are needed to parametrize the roots $μ (\mod m)$ , and that in the cubic setting, $d = 3$ , general ideals correspond to pairs of roots $μ_{1} (\mod m_{1})$ , $μ_{2} (\mod m_{2})$ satisfying $\gcd (m_{1}, m_{2}, μ_{1} - μ_{2}) = 1$ . At the end we illustrate our parametrization and this correspondence between roots and ideals with a few applications, including finding approximations to $\frac{μ}{m} \in ℝ ∕ ℤ$ , finding an explicit Euler product for the cotype zeta function of $ℤ [2^{1 ∕ 3}]$ , and computing the composition of cubic ideals in terms of the roots $μ_{1} (\mod m_{1})$ and $μ_{2} (\mod m_{2})$ .

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Keywords

roots of congruences, parametrization, polynomial congruences, cubic congruences, ideals

Mathematical Subject Classification

Primary: 11A07

Secondary: 11C08, 11F99, 11R47

Milestones

Received: 5 August 2020

Revised: 6 July 2021

Accepted: 5 August 2021

Published: 5 August 2022

Authors

Matthew Welsh
School of Mathematics University of Bristol Bristol United Kingdom

Vol. 16, No. 4, 2022