nLab unramifiable polynomial

Definition

Given a notion of real number $\mathbb{R}$, let $\sigma_j:\mathbb{R}^n \to \mathbb{R}$ denote the $j$-th elementary symmetric polynomial function in $\mathbb{R}$, and the polynomials functions $e_j:\mathbb{R}^n \to \mathbb{R}$ as

$$e_j(x_1, x_2 \ldots x_n) \coloneqq \prod_{k \neq j} (x_k - x_j)$$

Let us define the polynomial function $f:\mathbb{R}^{n+1} \to \mathbb{R}$ as

$$f(y, x_1, x_2 \ldots x_n) \coloneqq \prod_{j = 0}^n (y + x_j) = \sum_{j = 0}^n \sigma_{n - j}(x_1, x_2, \ldots x_n) y^j$$

Now, $\sigma_j(e_1(x_1, x_2 \ldots x_n), e_2(x_1, x_2 \ldots x_n), \ldots e_n(x_1, x_2 \ldots x_n))$ is by definition also a symmetric polynomial function. Thus, there exists a polynomial function $d_j:\mathbb{R}^n \to \mathbb{R}$ such that

$$d_j(\sigma_1(x_1, x_2, \ldots x_n), \sigma_2(x_1, x_2, \ldots x_n), \ldots \sigma_n(x_1, x_2, \ldots x_n)) = \sigma_j(e_1(x_1, x_2 \ldots x_n), e_2(x_1, x_2 \ldots x_n), \ldots e_n(x_1, x_2 \ldots x_n))$$

The function $f$ is unramifiable if there exists a $j$ such that $d_j(\sigma_1(x_1, x_2, \ldots x_n), \sigma_2(x_1, x_2, \ldots x_n), \ldots \sigma_n(x_1, x_2, \ldots x_n))$ is invertible in $\mathbb{R}$.

Related concepts

• simple root

• elementary symmetric polynomial

• fundamental theorem of algebra

References

• Wim Ruitenburg: Constructing Roots of Polynomials over the Complex Numbers, Computational Aspects of Lie Group Representations and Related Topics, CWI Tract 84 Centre for Mathematics and Computer Science, Amsterdam (1991) 107–-128 [pdf, pdf]