Timeline for Which continuous functions are polynomials?
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14 events
| when toggle format | what | by | license | comment | |
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| Aug 3, 2010 at 22:21 | comment | added | Pietro Majer | that's a good question too. | |
| Aug 3, 2010 at 22:09 | comment | added | Tom Goodwillie | It might be good to begin with a local question: Which germs of real-valued functions are topologically conjugate to polynomials. | |
| Aug 3, 2010 at 21:52 | comment | added | Tom Goodwillie | And (I meant to say) from the special case when only infinity goes to infinity you can read off a characterization of polynomials $\mathbb C\to \mathbb C$ up to homeomorphism of the domain. | |
| Aug 3, 2010 at 21:46 | comment | added | Tom Goodwillie | I suppose you can say something nice in the complex $n=1$ case: If a continuous map from the Riemann sphere to itself looks locally topologically like a candidate to be a rational map, in the sense that for every point in the codomain there is a punctured neighborhood over which the thing is a covering space, then up to homeomorphism of the domain it is a rational map. (Use the fact that a complex $1$-manifold homeomorphic to the Riemann sphere is isomorphic to the Riemann sphere, plus the fact that a complex-analytic map from the Riemann sphere to itself is a rational map.) | |
| Aug 3, 2010 at 21:42 | comment | added | Pietro Majer | I think so. There exists $n\in\mathbb{N}$ and an infinite $S\subset \mathbb{R}$, such that $\deg P(x,\cdot)\leq n$ for all $x\in S.$ Therefore for all $x\in S$ these $P(x,\cdot)$ are determined by the $n+1$ values at $y=0,1\dot,n$. By the Lagrange interpolation formula, $P$ coincides with a polynomial $p\in\mathbb{R}[x,y]$ on the whole set $S\times\mathbb{R}$. But then, for any $y\in\mathbb{R}$, $P(x,y)=p(x,y)$ for all $x$, because they are two polynomials in $x$ coinciding on the infinite set $S$. Is this right? | |
| Aug 3, 2010 at 21:36 | comment | added | Tom Goodwillie | $n=1$, smooth: A necessary condition is that there are only finitely many points where $f'=0$ and that each of these has finite multiplicity in the sense that some higher derivative is nonzero at that point. | |
| Aug 3, 2010 at 20:55 | comment | added | Piero D'Ancona | Is it true that $P(x,y)$ is a polynomial iff all its sections at $x=const$ and $y=const.$ are polynomials? | |
| Aug 3, 2010 at 20:45 | comment | added | Pietro Majer | It would be nice a characterization like this: a continuous $f:\mathbb{R}^n\to\mathbb{R}$ is topologically conjugated to a polynomial iff it has finitely many "topological" critical points, and they have finite rank critical groups (to be defined in terms of the relative singular homology in the standard way). | |
| Aug 3, 2010 at 20:35 | comment | added | Piero D'Ancona | In the real valued case, $n=1$, a necessary and sufficient condition might be that $f$ makes a finite number of oscillations and is unbounded as $x\to\pm\infty$. | |
| Aug 3, 2010 at 19:57 | answer | added | Greg Kuperberg | timeline score: 4 | |
| Aug 3, 2010 at 18:46 | comment | added | Dylan Wilson | A theorem of Whitney says that every continuous function is homotopic to a smooth one, perhaps one could use this to reduce the question to the case where the original function is smooth? Though I'm not at all sure how you would get a homeomorphism of R^n from the homotopy... | |
| Aug 3, 2010 at 18:42 | comment | added | Victor Protsak | Topological classification of continuous real functions ($n=1$) may be obtained by considering their intervals of monotonicity. In general, differentiable and analytic classifications are harder than topological. A keyword is "singularity theory". | |
| Aug 3, 2010 at 18:34 | comment | added | Tom Goodwillie | In the case $n=1$ a necessary (and I would guess sufficient) condition for $f$ to be expressible as a nonconstant polynomial composed with a homeomorphism is: (1) there are only finite many values of $x$ at which $f(x)$ has a local maximum or minimum and (2) $f$ is proper (i.e. tends to plus or minus infinity at each end of the line). A function like $x+2 sin x$ has finite point preimages but infinitely many local maxima. The case $n=2$ seems much more interesting/harder. | |
| Aug 3, 2010 at 18:21 | history | asked | Eric O. Korman | CC BY-SA 2.5 |