Parameterizing roots of polynomial congruences

TL;DR

This paper develops a new method to parameterize roots of polynomial congruences using ideals in orders, extending classical quadratic forms to higher degrees and providing applications in number theory.

Contribution

It generalizes Gauss's quadratic root parameterization to higher degrees, especially cubics, using ideal theory, and demonstrates applications in approximations, zeta functions, and ideal composition.

Findings

Parameterization of roots using special ideals in orders.

Extension of quadratic forms to cubic polynomial roots.

Applications include root approximations, zeta function expressions, and ideal composition.

Abstract

We use the arithmetic of ideals in orders to parameterize the roots $\mu \pmod m$ of the polynomial congruence $F(\mu) \equiv 0 \pmod m$, $F(X) \in \mathbb{Z}[X]$ monic, irreducible and degree $d$. Our parameterization generalizes Gauss's classic parameterization 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 parameterize the roots $\mu \pmod m$, and that in the cubic setting, $d = 3$, general ideals correspond to pairs of roots $\mu_1 \pmod{m_1}$, $\mu_2 \pmod {m_2}$ satisfying $\gcd(m_1, m_2, \mu_1 - \mu_2) = 1$. At the end we illustrate our parameterization and this correspondence between roots and ideals with a few applications, including finding approximations to $\frac{\mu}{m} \in \mathbb{R}/ \mathbb{Z}$, finding an explicit Euler product for the co-type zeta function of $\mathbb{Z}[2^{\frac{1}{3}}]$, and computing the composition of cubic ideals in terms of the roots $\mu_1 \pmod {m_1}$ and $\mu_2 \pmod {m_2}$.