Timeline for How can I "see" that a map is birational?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Oct 10, 2021 at 14:39 | comment | added | Somos | More precisely with infinity, the Randall triple $(t,x,s),$ $t=\frac{4s}{s^2-1}\big(\frac1{x-1}+\frac12+\frac{1-6s^2+s^4}{16s^2}(x-1)+O(x-1)^2\big)$ is a solution and $\lim_{x\to1}(t,x,s)=(\infty,1,s).\,$ | |
| Sep 25, 2021 at 17:53 | comment | added | Hauke Reddmann | @Somos: $(\infty,1,s)$ :-) (It makes sense to treat infinity as a number here...) | |
| Sep 25, 2021 at 17:22 | comment | added | Somos | I can see that $\,(0,1,s)\,$ and $\,(0,-1,s)\,$ and their permutations are valid triples, but that is only 12 triples. What are the other 12 triples? | |
| Sep 25, 2021 at 12:56 | comment | added | Gerry Myerson | "broken rational" is certainly uncommon in my experience – more than uncommon, I can't recall ever seeing it. I don't know what "gebrochen rational" means, either. And I've never seen "factically" before. | |
| Sep 25, 2021 at 11:57 | comment | added | Hauke Reddmann | @GerryMyerson: Is "broken rational" a false friend of "gebrochen rational" and at least uncommon? Methinks "rational function" is the standard term, now that I googled... | |
| Sep 25, 2021 at 11:53 | comment | added | Hauke Reddmann | @alpoge: That was the first thing I tried, but nonlinear equations in the coefficients are factically worthless. Checking the "singular" solutions at least gives only linear equations in the coefficients. For example, $p^2+q^2=5$ gives a parametric solution, and no map (at least not any with the same nice properties as the one given above) sends this solution to $(0,1,s)$, and that I could check by plugging it in and e.g. set $p'=0$ and make a coefficient list of $s$. That computation was quite straightforward and feasible in MATHEMATICA. | |
| Sep 25, 2021 at 11:46 | history | edited | Hauke Reddmann | CC BY-SA 4.0 |
example added, minor clarifications
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| Sep 25, 2021 at 1:42 | comment | added | Gerry Myerson | What does "broken" mean (in "broken rational function")? | |
| S Sep 25, 2021 at 1:22 | history | suggested | CommunityBot | CC BY-SA 4.0 |
x\mapsto y conventionally means if x is the input then y is the output
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| Sep 24, 2021 at 19:36 | comment | added | alpoge | does plugging random values of p q and r in (and maybe even reducing mod a huge prime) not help? I’m thinking of Schwartz-Zippel lemma-like statements, but I have no idea about what is computationally feasible etc. (hence: sorry if this is a useless comment!) | |
| Sep 24, 2021 at 19:01 | comment | added | Will Sawin | One approach would be to look at their action on the set of solutions mod $p$ for $p$ small but not too small - birational maps should be "close to" permutations. | |
| Sep 24, 2021 at 18:14 | review | Suggested edits | |||
| S Sep 25, 2021 at 1:22 | |||||
| Sep 24, 2021 at 17:27 | comment | added | Wolfgang | What do you mean by the dots in your "outrageous" map? | |
| Sep 24, 2021 at 16:57 | history | edited | YCor | CC BY-SA 4.0 |
edited tags, formatting
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| Sep 24, 2021 at 15:42 | history | asked | Hauke Reddmann | CC BY-SA 4.0 |