Timeline for Simultaneous Equations Involving Power Sums
Current License: CC BY-SA 2.5
5 events
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| Nov 17, 2009 at 16:17 | comment | added | David E Speyer | Right. And it is convenient to multiply one polynomial by x, so that they have the same degree. So another version of the problem is: do there exist r and c such that, for every n, there are polynomials f and g, of degree n, such that f(1)=0, all the roots of f(z)/(z-1) and of g(z) lie in |z| < 1, and f-g has degree cn. | |
| Nov 17, 2009 at 16:13 | comment | added | Kaveh Khodjasteh | David, if you take the original power sum setting for n and n+1 variables, then if you have all the equations for 1<k<\ell then you can reduce them to equalities about roots of two polynomials of degrees n and n+1 [as people pointed out above, using Newton's identities] and subsequently two polynomials with common coefficients except one having one degree higher with roots that are all larger in modulus than 1 ... | |
| Nov 17, 2009 at 15:59 | history | edited | David E Speyer | CC BY-SA 2.5 |
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| Nov 17, 2009 at 15:00 | comment | added | Kaveh Khodjasteh | Thanks David for the new problem! It does seem like a tight problem now that I have spent about a week on it. I might try tweaking my setup to see either I can get something more general to disprove or something more specific to get the solutions on. In general, I just need x_i,y_i, and their sum [and thus their number] to be bounded from below independent of \ell and from above by a polynomial in \ell. This will still give me the semi-super-exponential improvements that I am looking for in my original problem. I might look at alternative solution to the original as well. | |
| Nov 17, 2009 at 12:13 | history | answered | David E Speyer | CC BY-SA 2.5 |