Timeline for Can formal power series become polynomial often, when composed with polynomials?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jul 30, 2012 at 23:55 | vote | accept | Henry Yuen | ||
| Jul 30, 2012 at 2:38 | comment | added | David E Speyer | I don't :). In that case, $y^2=1+x$ doesn't have a root in $k[[x]]$. Probably the root of $y^2+y=x$ with $y(0)=0$ would make a good substitute. | |
| Jul 30, 2012 at 1:02 | comment | added | Qiaochu Yuan | @David: what do you mean by $g(x) = \sqrt{1 + x}$ if $F$ has characteristic $2$? | |
| Jul 30, 2012 at 0:07 | answer | added | David E Speyer | timeline score: 8 | |
| Jul 29, 2012 at 18:45 | comment | added | David E Speyer | The easy lower bound is $|F|^{k/2}$, coming from $g(x) = \sqrt{1+x}$. Then $g \circ c$ is a polynomial whenever $c$ is of the form $h(x)^2-1$. My guess is that this is close to optimal. | |
| Jul 29, 2012 at 14:40 | history | edited | Henry Yuen | CC BY-SA 3.0 |
deleted 75 characters in body
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| Jul 27, 2012 at 4:51 | history | edited | Henry Yuen | CC BY-SA 3.0 |
clarification in problem statement.
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| Jul 27, 2012 at 4:50 | comment | added | Henry Yuen | Thanks for pointing this out Ryan. In this case, I have a specific $d$ in mind: if $g(c)$ has degree greater than $kd$, then I do not wish to consider such a $c$. The bound I'm seeking on $|C|$ may depend on $d$. I will edit the problem statement to make this clearer. | |
| Jul 27, 2012 at 4:36 | comment | added | Ryan Reich | I should point out that you don't need $d$, as the set of degree-$k$ polynomials is finite, so $d$ is just the largest value of $\operatorname{deg} g(c) / k$. And $kd$ is better interpreted as just "the highest degree of some $g(c)$". | |
| Jul 27, 2012 at 3:53 | history | asked | Henry Yuen | CC BY-SA 3.0 |