Timeline for Multiple roots of polynomials with coefficients $\pm 1$
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Jun 13, 2022 at 19:47 | vote | accept | Taras Banakh | ||
| Jun 13, 2022 at 19:20 | comment | added | Peter Taylor | I can go one better, with a root that's actually in the bounds you asked about. | |
| Jun 13, 2022 at 19:19 | answer | added | Peter Taylor | timeline score: 15 | |
| Jun 13, 2022 at 17:40 | comment | added | Taras Banakh | @PeterTaylor Thank you very much for your comment. You can write down it as an answer and I will accept it, which will allow me to close this question. | |
| Jun 13, 2022 at 15:56 | comment | added | Peter Taylor | @TimothyChow, $(z^{18} + 2z^{15} + 2z^{13} + z^{12} + 2z^{11} + 3z^{10} + 3z^8 + 2z^7 + z^6 + 2z^5 + 2z^3 + z^2 + 1)(z^2 + 1)(z - 1)(z^3 - z - 1)^2$ and $(z^{18} + 2z^{15} + 2z^{13} + z^{12} + 3z^{10} + z^8 + 2z^7 + z^6 + 2z^5 + 2z^3 + z^2 + 1)(z^2 + 1)(z - 1)(z^3 - z - 1)^2$ are Littlewood polynomials with a non-cyclotomic repeated cubic factor. | |
| Jun 11, 2022 at 4:41 | comment | added | Fedor Petrov | @AlexRavsky ah, then this is true not only for real roots: for even degree, a Littlewood polynomial $P$ has no multiple root even over $\mathbb{F}_2$ (because the derivative of $(x-1)P$ has only zero roots). | |
| Jun 10, 2022 at 22:34 | answer | added | მამუკა ჯიბლაძე | timeline score: 6 | |
| Jun 10, 2022 at 22:17 | comment | added | Alex Ravsky | @FedorPetrov Oops, I meant "has no multiple real roots". Sorry. | |
| Jun 10, 2022 at 21:17 | comment | added | Timothy Chow | Double roots of random Littlewood polynomials by Peled et al. seems relevant, although at first glance it does not directly answer the question. Near the end of the paper, they say they do not know if there exists a Littlewood polynomial with at least one non-cyclotomic double root. | |
| Jun 10, 2022 at 20:39 | comment | added | Fedor Petrov | @AlexRavsky Wait, but $x^2+x-1$ has a real root | |
| Jun 10, 2022 at 18:35 | comment | added | Alex Ravsky | Moreover, in Appendix C was proposed the Mathematica document “which creates all combinations of Littlewood polynomials up until a give degree $l$ and evaluates for each of these polynomials whether there exist values for which both the function and its derivative equals 0. Having run this program for $l = 13$ (it takes much more computation time when $l$ gets bigger), we see that the only values that appear to sometimes be a double zero, are $-1$ and $1$. The above strongly suggests that double zero's simply do not exist, except $-1$ and $1$”. | |
| Jun 10, 2022 at 18:35 | comment | added | Alex Ravsky | Such polynomials (series) are called Littlewood polynomials (series). Vader in Theorem 7.1 of “Real roots of Littlewood polynomials” showed that a Littlewood polynomial of even degree has no real roots. | |
| Jun 10, 2022 at 17:48 | comment | added | მამუკა ჯიბლაძე | Another thing about P: for each odd $m=2n-1$ there are two $P$'s with multiple roots, namely $\frac{(1-x^n)^2}{1-x}=1+x+...+x^{n-1}-x^n-x^{n+1}-...-x^{2n-1}$ and $\frac{(1-(-x)^n)^2}{1+x}=1-x+...+(-x)^{n-1}-(-x)^n-(-x)^{n+1}-...-x^{2n-1}$, and for $n$ prime these seem to be the only ones (again, checked experimentally up to $m=21$); but for $n$ not prime there are very many. Numbers of such polynomials go like $0,0,0,2,0,2,0,6,0,2,0,62,0,2,0,518,0,134,0,5452,0,2,...$ | |
| Jun 10, 2022 at 16:02 | comment | added | მამუკა ჯიბლაძე | Regarding P: even without the $P(1)=0$ restriction, up to $m=20$, there are no multiple roots at all for $m$ even, while for $m$ odd all multiple roots that may occur are $m+1$st roots of unity. | |
| Jun 10, 2022 at 12:18 | comment | added | Fedor Petrov | If we had an example in question P without extra condition $P(1)=0$, we can reach it by multiplying $P$ by $1-x^{m+1}$. Also this yields a positive answer to question $A$ by a similar reasoning | |
| Jun 10, 2022 at 12:05 | history | edited | Taras Banakh | CC BY-SA 4.0 |
added 20 characters in body
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| Jun 10, 2022 at 11:48 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Added Question P.
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| Jun 10, 2022 at 7:48 | history | asked | Taras Banakh | CC BY-SA 4.0 |