3 ms·

WTF, this is amazing! I think this could be a great example of experimental mathematics, some examples for which are listed in SE (https://math.stackexchange.co

by Jun8 3y ago

WTF, this is amazing! I think this could be a great example of experimental mathematics, some examples for which are listed in SE (https://math.stackexchange.com/questions/264560/mind-blowing-mathematics-experiments https://math.stackexchange.com/questions/264560/mind-blowing...) but this I believe is much better. What I mean is that you can generate this without any a priori knowledge, then examine it like Galileo examined the moons of Jupiter, to seek interesting phenomena, which then you work to understand. For example: can one prove the empty space in the middle or around +1 and -1? Polynomials of degree <= 5 with integer coefficients in [-4, 4] do not have any roots with nonzero imaginary parts in |r|<r_0, where r_0 seems to be around 0.7.

gjm11 3y ago
The empty space in the middle is pretty easy. These are roots of polynomials of the form +- z^n +- z^(n-1) +- ... +- 1. If |z| is small then the earlier terms are smaller; the absolute value of their sum is at most |z|^n + ... + |z|^1 which is at most |z| / (1-|z|). If this is < 1 then the sum can't be large enough for adding it to +- 1 to give 0. So if |z| < 1-|z|, i.e., if |z| < 1/2, then z can't be a root. That's for the "main" image. The first image in Baez's post is for polynomials whose coefficients are -4,-3,...,+4, and the analysis would be different for those, but it's still true that if |z| is small enough then the sort of calculation in the previous paragraph forces it not to be the root of any nonzero polynomial with small integer coefficients. The holes near +- 1 are more complicated, I think.
abetusk 3y ago
The link references at the bottom have some more detail on the structure. For example, the N-Category Cafe talks about how the dragon curve shows up at the edges [0]. There are other gross level symmetries due to the symmetries of the Littlewood like polynomials involved. For example, if $p(x)$ is a Littlewood-like polynomial, then so is $p(1/x)$ etc. I think the "holes" that show up on the unit disc because of the factors of the degrees of certain polynomials. If I'm not mistaken, the holes follow a Farey sequence [1]. I wrote a little blurb about it but it's still incomplete and haphazard. One thing I'm still curious about is how quickly the holes diminish in size. [0] https://golem.ph.utexas.edu/category/2009/12/this_weeks_finds_in_mathematic_46.html https://golem.ph.utexas.edu/category/2009/12/this_weeks_find... [1] https://en.wikipedia.org/wiki/Farey_sequence https://en.wikipedia.org/wiki/Farey_sequence [2] https://mechaelephant.com/dev/Littlewood-Polynomials-Notes.html https://mechaelephant.com/dev/Littlewood-Polynomials-Notes.h...

by Jun8 3y ago

gjm11 3y ago

The empty space in the middle is pretty easy. These are roots of polynomials of the form +- z^n +- z^(n-1) +- ... +- 1. If |z| is small then the earlier terms are smaller; the absolute value of their sum is at most |z|^n + ... + |z|^1 which is at most |z| / (1-|z|). If this is < 1 then the sum can't be large enough for adding it to +- 1 to give 0. So if |z| < 1-|z|, i.e., if |z| < 1/2, then z can't be a root. That's for the "main" image. The first image in Baez's post is for polynomials whose coefficients are -4,-3,...,+4, and the analysis would be different for those, but it's still true that if |z| is small enough then the sort of calculation in the previous paragraph forces it not to be the root of any nonzero polynomial with small integer coefficients. The holes near +- 1 are more complicated, I think.

abetusk 3y ago

The link references at the bottom have some more detail on the structure. For example, the N-Category Cafe talks about how the dragon curve shows up at the edges [0]. There are other gross level symmetries due to the symmetries of the Littlewood like polynomials involved. For example, if $p(x)$ is a Littlewood-like polynomial, then so is $p(1/x)$ etc. I think the "holes" that show up on the unit disc because of the factors of the degrees of certain polynomials. If I'm not mistaken, the holes follow a Farey sequence [1]. I wrote a little blurb about it but it's still incomplete and haphazard. One thing I'm still curious about is how quickly the holes diminish in size. [0] https://golem.ph.utexas.edu/category/2009/12/this_weeks_finds_in_mathematic_46.html https://golem.ph.utexas.edu/category/2009/12/this_weeks_find... [1] https://en.wikipedia.org/wiki/Farey_sequence https://en.wikipedia.org/wiki/Farey_sequence [2] https://mechaelephant.com/dev/Littlewood-Polynomials-Notes.html https://mechaelephant.com/dev/Littlewood-Polynomials-Notes.h...