论文标题
多项式一致性的参数化根
Parameterizing roots of polynomial congruences
论文作者
论文摘要
我们使用订单中理想的算术来参数化多项式一致性$ f（μ）\ equiv 0 \ pmod m $，$ f（x）\ in \ mathbb {z} [x] [x] $ monic，norredibible和deg $ d $ $ d $。我们的参数化概括了高斯使用二进制二等式形式对二次元素根的经典参数化，以前仅扩展到立方多项式$ 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 $。最后，我们说明了我们的参数化以及根部和理想之间的对应关系，包括找到近似于$ \fracμ{m} \ in \ mathbb {r}/ \ mathbb {z z} $，找到了Co-Type Zeta Zeta Zeta函数的euler产品$ \ mathbb {z} [2^{\ frac {1} {3}}] $，并根据根来计算立方理想的组成$μ_1\ pmod {m_1} $和$μ_2\ pmod {m_2} $。
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 $μ_2 \pmod {m_2}$.
