Timeline for $\pm1$-polynomials with a maximal non-real root
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 15, 2014 at 19:18 | comment | added | Lucia | Very interesting! Clearly there's something to think through there. | |
| Nov 15, 2014 at 18:10 | comment | added | Wolfgang | I have looked at the pattern in the Odlyzko-Poonen situation. The maximal root is $-.9299016\pm.9983452\; i $ (indeed $1/|z_0|\approx .73295778$) and the coefficients have a very similar snake pattern! With first line for 1's, 2nd line for 0's, we have group sizes $$\;\text{3 1 1 2 1 1 1 2 1 1 2 1 1 2 2 1 1} \\ \text{ 1 2 1 1 2 2 1 1 2 1 1 2 1 1 1 2}$$ | |
| Nov 12, 2014 at 20:48 | comment | added | Lucia | Interesting! I don't know what it means! You could also look at the Odlyzko-Poonen situation, and see if there is a pattern to the polynomials there that have largest non-real root. Maybe that problem will have a similar feature to what you're seeing here? | |
| Nov 12, 2014 at 20:35 | vote | accept | Wolfgang | ||
| Nov 12, 2014 at 20:32 | comment | added | Wolfgang | Thank you Lucia. I'll accept this answer, probably best possible one can do. I am just wondering: would you see any chance that there is a relationship with the cfrac $\frac1{1+\frac1{1+\frac1{3+\frac1{4+\frac1{3+\frac1{3+...}}}}}}$? Just by curiosity, I did a similar search for a $\pm1$-polynomial where the second largest pair of complex roots has maximal modulus. This turns out to be very irregular, starting with $+-+--+++++++--+---+---++++++--++---------$, and the maximal roots are $1.261231263 \pm0.4399400258\;i$ and $0.06954716993\pm 1.333986117\;i$ (asymptotically same modulus) | |
| Nov 10, 2014 at 1:23 | history | answered | Lucia | CC BY-SA 3.0 |