Paper detail

Parameterizing roots of polynomial congruences

We use the arithmetic of ideals in orders to parameterize the roots $μ\pmod m$ of the polynomial congruence $F(μ) \equiv 0 \pmod m$, $F(X) \in \mathbb{Z}[X]$ 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 $μ\pmod m$, and that in the cubic setting, $d = 3$, general ideals correspond to pairs of roots $μ_1 \pmod{m_1}$, $μ_2 \pmod {m_2}$ satisfying $\gcd(m_1, m_2, μ_1 - μ_2) = 1$. At the end we illustrate our parameterization and this correspondence between roots and ideals with a few applications, including finding approximations to $\fracμ{m} \in \mathbb{R}/ \mathbb{Z}$, 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 $μ_1 \pmod {m_1}$ and $μ_2 \pmod {m_2}$.