In this question I have asked if every odd polynomial $f \in \mathbb{R}[x]$ (= all monomials in $f$ are of odd degrees) is a one-one map on $\mathbb{R}$, and of course the answer is no, as $x^3-x$ shows.
In the answers it was proved that $f$ is one-one if and only if $f'$ never changes sign.
On the one hand, one of the answers said: ''Also requiring the coefficient of the top power monomial to be large (or small) enough with respect to the other coefficients would be sufficient. Here the idea is the following: We know that if we go away from zero then any polynomial will be 1-1. Therefore forcing the polynomial to speed up around zero too (to make it 1-1) requires some dancing with the coefficient of the top power monomial".
On the other hand, another answer said: ''If the first and last non-zero coefficients have different signs then the function is not one to one because for sufficiently large $x$ the largest power dominates and for sufficiently small $x$ the smallest (possibly zero) power dominates".
Can one please explain which argument is correct?
Thank you very much!
If I'm reading the answers correctly, they are not incompatible. Notice the first answer says "large (or small) enough." I think the sign condition in the second answer was obvious to this person, and so this means "of large enough magnitude (with the obvious sign)."
With that said, the second answer is certainly correct. As came up in the comments, it is important that we are considering an odd polynomial, so that we can make the dominated-by-highest-term and dominated-by-lowest-term arguments on the derivative (whose highest and lowest non-zero coefficients will still have opposite signs) and not just the original polynomial.
The first answer seems right to me too, although I'm not coming up with any clean proof of it. The idea is that the derivative will be dominated by the lowest term near zero, and by the highest term "eventually," but that we can make that "eventually" start at any positive distance from $0$ by making the leading coefficient big enough (again, in absolute value). So we just have to make it big enough that "near zero" and "eventually" overlap. As the original poster (who seems to know a lot more about this kind of thing than I do) suggested, it doesn't seem particularly easy to write down how big is big enough.