nLab simple root




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

More generally, let RR be a commutative ring. The polynomial ring R[x]R[x] is a differential algebra and thus has a derivative function ():R[x]R[x](-)':R[x] \to R[x]. There is a canonical ring homomorphism h:R[x](RR)h:R[x] \to (R \to R) from the polynomial ring R[x]R[x] to the function algebra RRR \to R which assigns constant polynomials to constant functions in RR and the bare indeterminant to the identity function of RR, which when uncurried leads to the evaluation of polynomials ()(()):R[x]×RR(-)((-)):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:


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

In classical mathematics (implying acceptance of the law of excluded middle), a simple root aa is a root that is not a multiple root: it is not the case that (xa) 2(x-a)^2 divides f(x)f(x), or “not f(a)=0f'(a) = 0”. But for the purposes of constructive mathematics, it is preferable to state the condition as “f(a)f'(a) is apart from 00”, or that f(a)f'(a) is invertible. Hence the definitions above.


  • 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]

Last revised on May 20, 2023 at 15:04:41. See the history of this page for a list of all contributions to it.