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 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]