Timeline for Levi-Civita field in unusual basis
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 27, 2022 at 0:23 | vote | accept | Anixx | ||
| Jan 27, 2022 at 0:16 | answer | added | Wojowu | timeline score: 3 | |
| Jan 27, 2022 at 0:03 | comment | added | Alec Rhea | It is a member of the obvious group algebra in play here; all ‘formal power series’ constructions are members of a group algebra of one sort or another. | |
| Jan 26, 2022 at 23:49 | comment | added | Wojowu | No, it is not a Hahn series in $p$ either. The set of exponents appearing in a Hahn series has to be well-ordered, and $\{-n\mid n\in\mathbb N\}$ is not well-ordered. | |
| Jan 26, 2022 at 23:48 | comment | added | Anixx | @Wojowu but it is Hahn series, no? Yes, I think, this is satisfactory. | |
| Jan 26, 2022 at 23:46 | comment | added | Wojowu | No, a Laurent series in $p^{-1}$ need not be a power series in $p$. Consider for instance $\sum_{n=1}^\infty p^{-n}$. If you find this to be satisfactory I can post it as an answer. | |
| Jan 26, 2022 at 23:45 | comment | added | Anixx | @Wojowu but is not Laurent series in $p^{-1}$ is also a power series in $p$ (say, Hahn series)? What you said is very interesting and looks like an answer! | |
| Jan 26, 2022 at 23:41 | comment | added | Wojowu | Infinite Laurent series in $p$ are not going to converge either, but Laurent series in $p^{-1}$ converge and they give all Laurent series in $\varepsilon$. They can't possibly give any elements outside it - a Laurent series in a power series can't possibly involve any fractional powers of the indeterminate. | |
| Jan 26, 2022 at 23:38 | comment | added | Anixx | @LSpice well, I see now that this would not converge, but let's consider generalizations of Levi-Civita field (transseries or Hardy fields). I meant the coefficients to be real. I meant Laurent series in p. | |
| Jan 26, 2022 at 23:29 | comment | added | LSpice | What are the coefficients in the power series? The only power series in $p$ with real coefficients that converge in the Levi-Civita field are polynomials, and, of course, all polynomials in $p$ "converge"; so we would obtain the Levi-Civita field as a quotient of $\mathbb R[p]$ (a one-variable polynomial ring in the now-formal variable $p$), but the only such field quotients are $\mathbb R$ and $\mathbb C$. (Ah, on reflection, I guess you must mean LC coefficients anyway, since otherwise obviously we can't get any non-integral powers of $\varepsilon$.) | |
| Jan 26, 2022 at 23:21 | history | edited | LSpice | CC BY-SA 4.0 |
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| Jan 26, 2022 at 22:34 | history | asked | Anixx | CC BY-SA 4.0 |