# fixedpoint.jp

## What does (no) existence of fixed points of a polynomial tell? (2018-08-26)

Does your favorite polynomial have fixed points? The answer definitely helps us guess what they are. Herein we consider only polynomials of a single indeterminate, say $$x$$, with real coefficients. More formally, let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be a univariate, real polynomial function.

If every real number is a fixed point of $$f$$, we are done: $$f(x) = x.$$ In other words $$f$$ is the identity function.

If $$f$$ has two or more fixed points and is not the identity function, then it cannot be a constant nor linear, so $$\mathrm{deg}(f) \geq 2.$$

If there is no fixed point of $$f$$ at all (H) and $$f$$ is non-linear, then $$\mathrm{deg}(f)$$ is even. Its proof immediately follows from the intermediate value theorem.

Moreover, if the above (H) holds and $$\mathrm{deg}(f) = 1$$, then $$f(x) = x + b$$ with $$b \neq 0.$$ If (H) holds and $$\mathrm{deg}(f) = 2$$ i.e. $$f(x)$$ is of form $$a x^2 + b x + c$$, then the signs of $$a$$ and $$c$$ are the same.