nLab simple root

Definition

Definition

Given a polynomial function with real or complex coefficients $x \mapsto f(x)$, a simple root is a root of $x \mapsto f(x)$ such that the derivative of $x \mapsto f(x)$ evaluated at the root is invertible.

More generally, let $R$ be a commutative ring. The polynomial ring $R[x]$ is a differential algebra and thus has a derivative function $(-)':R[x] \to R[x]$. There is a canonical ring homomorphism $h:R[x] \to (R \to R)$ from the polynomial ring $R[x]$ to the function algebra $R \to R$ which assigns constant polynomials to constant functions in $R$ and the bare indeterminant to the identity function of $R$, which when uncurried leads to the evaluation of polynomials $(-)((-)):R[x] \times R \to R$. Since the set of polynomial functions in a commutative ring is a polynomial ring generated by the identity function, the situation for polynomial functions is simply a special case of the situation for polynomials. Thus, we have the following definition for general polynomials over fields:

Definition

Given a Heyting field $K$ and a polynomial $f \in K[x]$ with coefficients in $K$, a simple root is a root $a \in K$ of $f$ such that $f'(a)$, the evaluation of the derivative of $f$ at $a$, is invertible.

In classical mathematics (implying acceptance of the law of excluded middle), a simple root $a$ is a root that is not a multiple root: it is not the case that $(x-a)^2$ divides $f(x)$, or “not $f'(a) = 0$”. But for the purposes of constructive mathematics, it is preferable to state the condition as “$f'(a)$ is apart from $0$”, or that $f'(a)$ is invertible. Hence the definitions above.

Related concepts

• root
• elementary symmetric polynomial
• unramifiable polynomial
• fundamental theorem of algebra

References

• Wim Ruitenberg, 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]