Are there some nice results characterizing which polynomial functions (reals to reals) are injective (for example a necessary and sufficient condition)? Obviously all polynomials of degree $1$ are injective, and for degree ${}>1$ such a polynomial must have odd degree and at at most one root, but that is hardly sufficient.
Characterizing Injective Polynomials – Functions and Polynomials
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Best Answer
A continuous function $f:\mathbb{R}\to \mathbb{R}$ is injective (1-1) if and only if it is monotone (strictly increasing or strictly decreasing). If $f$ is a (real) polynomial, this means the derivative $f'$ is nonnegative or nonpositive, respectively, as well as not identically zero.
This is only possible if $f$ has odd degree (equiv. $f'$ has even degree), but of course this is only a necessary condition. A sufficient condition would be (as Meelo comments) that $f'$ has no real roots. A necessary and sufficient condition would be that any real roots of $f'$ have even multiplicity.
In exact arithmetic these conditions can be investigated using Sturm sequences. In particular the derivative $f'$ would factor into $P(x)Q^2(x)$, where $P(x)$ is square-free and has no real roots, if and only if $f$ is monotone.