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.

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