Timeline for Polynomials over $\mathbb F_2$ without zeros in $\mathbb F_2$ having an inverse series with support of large density.
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 14, 2013 at 21:09 | comment | added | François Brunault | Further computations show that this observation breaks down for $n=23$. However this is true when we restrict to primitive irreducible polynomials, as explained by Peter Müller in his answer. | |
| Feb 14, 2013 at 18:01 | answer | added | Peter Mueller | timeline score: 6 | |
| Feb 14, 2013 at 14:59 | comment | added | François Brunault | Here is an empirical observation. I computed the density for the power series $1/P$ where $P$ runs through the irreducible polynomials of degree $n$ over $\mathbf{F}_2$. When $n$ is prime, the maximal density seems to be exactly $2^{n-1}/(2^n-1)$ (up to $n=17$). I don't know how one would go to prove that. | |
| Feb 13, 2013 at 22:14 | comment | added | Gerry Myerson | I know of one paper where these reciprocals over ${\bf F}_2$ are studied, MR2281861 (2007h:11015) Cooper, Joshua N.; Eichhorn, Dennis; O'Bryant, Kevin; Reciprocals of binary series, Int. J. Number Theory 2 (2006), no. 4, 499–522. | |
| Feb 13, 2013 at 22:07 | answer | added | AnswerMan | timeline score: 2 | |
| Feb 13, 2013 at 21:30 | comment | added | Peter Mueller | @Joel: Because the expansion is periodic. | |
| Feb 13, 2013 at 20:01 | comment | added | Joël | Nice question. How de we know that the density $\delta_n$ exists ? | |
| Feb 13, 2013 at 17:38 | history | asked | Roland Bacher | CC BY-SA 3.0 |