Timeline for cyclic polynomials and their solutions
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 23, 2012 at 20:01 | comment | added | Charles Matthews | For n = 4 I'm getting 6 independent monomials in linear forms. | |
| Feb 23, 2012 at 19:14 | comment | added | David Epstein | Ben Webster says that 3 cyclically invariant polynomials cannot solve the problem for n=3. Suppose we look at $L_0$, $L_1^3$, $L_2^3$ and $L_1L_2$. Then the 3 choices of $L_1$ are linked to the 3 choices of $L_2$. This gives 3 choices of $(z_1,z_2,z_3)$, which is EXACTLY right, isn't it? Furthermore we get singularities exactly when $L_1=0$ or $L_2=0$, though I don't really understand what the singularity looks like.If this is right, then it's very nice, and indicates that Ben Webster knew what he was talking about. Any thoughts on $n=4$? | |
| Feb 23, 2012 at 15:49 | comment | added | Charles Matthews | Sorry for the "thinking aloud". The idea I have now added seems to solve the problem "up to a finite number of possibilities". But perhaps it will serve as a better basis for discussion. | |
| Feb 23, 2012 at 15:47 | history | edited | Charles Matthews | CC BY-SA 3.0 |
typo
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| Feb 23, 2012 at 11:24 | comment | added | David Epstein | In the application I have in mind (finite approximations to smooth simple closed curves in the plane), singularities in the quotient space will be avoided by insisting that the polygonal approximation has no self-intersections. The open subset of the quotient that survives may not be too easy to work with, but it should be much less ad hoc than what is in the biology literature---one possible important application is to shapes of cells as they appear in images taken with a microscope. | |
| Feb 23, 2012 at 11:09 | comment | added | David Epstein | What does the quotient look like? For the full symmetric group, one gets complex projective space, the correspondence being between the roots of a polynomial and its coefficients. But I suppose that in the cyclic case there will be some branching. (Don't fully understand what I'm saying here, but it feels right.) | |
| Feb 23, 2012 at 11:04 | comment | added | David Epstein | Could you spell this out for someone who has never studied Kummer theory, nor Chevalley-Shepard-Todd? By "rational functions", do you mean the field of rational functions over the complex numbers with one variable? Or are you taking the quotient field of the polynomial ring in N variables? I don't understand the connection between Galois theory and my problem, though I did have the vague feeling they were connected when I posed the problem. I vaguely remember from 50 years ago a correspondence between points and maximal ideals, but it's very vague. A readable source? "A kind of obvious way"??? | |
| Feb 22, 2012 at 16:02 | history | edited | Charles Matthews | CC BY-SA 3.0 |
add thought
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| Feb 22, 2012 at 13:17 | history | answered | Charles Matthews | CC BY-SA 3.0 |