
Mathematician Measures the Repulsive Force Within Polynomials - sandwall
https://www.quantamagazine.org/new-math-measures-the-repulsive-force-within-polynomials-20200514/
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jefftk
The title reads like a joke, but apparently it's not

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LolWolf
Wow, this is an absolutely lovely presentation of that result. Huge props to
Hartnett for writing this piece! It's a perfect mix of intuitive and well-
explained without being too hand-wavy and it's quite an interesting subject,
too.

Again, big props to Hartnett (and Dimitrov, of course)!

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syockit
Off-topic but can someone recommend me a software for drawing diagrams as
shown in this article? Something easier to use than matplotlib, TikZ?

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fxj
Use R:

> plot(polyroot(c(1,-1,1,-1,1,-1)))

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aDfbrtVt
Does anyone know how this idea of repulsive forces in root spacing might
relate to filter analysis?

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LolWolf
What do you mean by filter analysis? As in classical linear filtering? (as in,
you're finding the roots of the transfer function?)

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danharaj
Not to shit on all of your middlebrow dismissals but mathematicians are known
to steal language from other fields as metaphors for mathematical phenomena.
Sometimes these metaphors are very rigorous and precise and sometimes they're
fast and loose. Here, look, I found all these papers which use "repulsion" as
a mathematical metaphor for distances between zeroes, eigenvalues, and other
special values of a geometric object:

Random matrices: tail bounds for gaps between eigenvalues

Gaps (or spacings) between consecutive eigenvalues are a central topic in
random matrix theory. The goal of this paper is to study the tail distribution
of these gaps in various random matrix models. We give the first repulsion
bound for random matrices with discrete entries and the first super-polynomial
bound on the probability that a random graph has simple spectrum, along with
several applications."

[https://arxiv.org/abs/1504.00396](https://arxiv.org/abs/1504.00396)

Real roots of random polynomials: expectation and repulsion

[https://arxiv.org/abs/1409.4128](https://arxiv.org/abs/1409.4128)

Zero repulsion in families of elliptic curve L-functions and an observation of
Miller

[https://academic.oup.com/blms/article-
abstract/45/1/80/29767...](https://academic.oup.com/blms/article-
abstract/45/1/80/297678)

Integral Points on Elliptic Curves and the Bombieri-Pila Bounds

Let C be an affine, plane, algebraic curve of degree d with integer
coefficients. In 1989, Bombieri and Pila showed that if one takes a box with
sides of length N then C can obtain no more than
O_{d,\epsilon}(N^{1/d+\epsilon}) integer points within the box. Importantly,
the implied constant makes no reference to the coefficients of the curve.
Examples of certain rational curves show that this bound is tight but it has
long been thought that when restricted to non-rational curves an improvement
should be possible whilst maintaining the uniformity of the bound. In this
paper we consider this problem restricted to elliptic curves and show that for
a large family of these curves the Bombieri-Pila bounds can be improved. The
techniques involved include repulsion of integer points, the theory of heights
and the large sieve. As an application we prove a uniform bound for the number
of rational points of bounded height on a general del Pezzo surface of degree
1.

[https://arxiv.org/abs/1301.4116](https://arxiv.org/abs/1301.4116)

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LolWolf
Absolutely agreed—it's a little funny to see dismissals of the article (which,
as I mentioned elsewhere in the thread is actually quite good), even though,
as a mathematician [0] we use these analogies (and sometimes really the whole
idea) all the time.

To add to your list (for less number-theory specific topics):

\- Lyapunov functions? Energy in physics (just a mathematical surrogate)

\- Exponential families? Canonical ensemble in physics

\- Convex duality? Lagrange duality in classical mechanics

and the list continues.

\-----

[0] And, admittedly, a member of a physics lab, even though I don't really do
any physics.

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danharaj
Yea! And here's a metaphor to... baking!! The moment this metaphor clicked for
me reading a musty textbook with yellowing pages was delightful, I can go back
to that moment just by thinking about it.

[https://en.wikipedia.org/wiki/Milnor%E2%80%93Thurston_kneadi...](https://en.wikipedia.org/wiki/Milnor%E2%80%93Thurston_kneading_theory)

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LolWolf
I love it! I had heard of some of these results in relation to Persi
Diaconis's work ( _e.g._ , [0]). I'm curious, what is the textbook? Would love
to take a quick glance at it at least! :)

EDIT: I also really like the fact that even the original paper mentioned in
the quanta article references other (quite pictorially fun) things, such as
the aptly named Hedgehog spaces [1] (see Theorem 3, for example).

Perhaps it's violating the prime directive to be mentioning this, but it's a
little bit of a shame to see what I think is a good article so quickly
dismissed by what is essentially just bikeshedding (that isn't even
justified!).

\----

[0]
[https://link.springer.com/article/10.1007/s10955-011-0284-x](https://link.springer.com/article/10.1007/s10955-011-0284-x)

[1]
[https://en.wikipedia.org/wiki/Hedgehog_space](https://en.wikipedia.org/wiki/Hedgehog_space)

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danharaj
Thought about it, and it was definitely the original paper in some journal
volume. I remember the pictures, and the ones I drew myself :) Found it here:
[http://www-personal.umich.edu/~kochsc/MilnorThurston.pdf](http://www-
personal.umich.edu/~kochsc/MilnorThurston.pdf)

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LolWolf
Awesome, thank you!

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danharaj
the paper by Persi Diaconis you linked to was so fun that I want to learn
about markov chain mixing now.

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LolWolf
Oh it’s an absolutely lovely overview. Honestly I would highly recommend most
stuff from Diaconis, he’s got a fantastic expository style (and all of his
papers are quite readable!)

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lisper
This is an interesting and important result, but framing it in terms of a
"repulsive force" is beyond ridiculous. In fact, it's actively harmful. Forces
are physical things and this result has nothing to do with anything physical.
It's pure number theory.

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Gollapalli
This is silly, and I wish I had enough points to downvote you.

Mathematicians borrow the terminology that is most helpful for explaining an
idea. such a thing cannot be seen as harmful. Readers are not infants, and
should not be treated like infants. They are responsible for the conclusions
they come to, especially in mathematics.

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jfkebwjsbx
> especially in mathematics

What do you mean?

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Gollapalli
Mathematics is one of those fields where you can actually, really verify the
truth or falseness of a statement, if you're willing to do the work involved.

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jfkebwjsbx
That does not make sense within the context of the discussion.

In any science, engineering, etc. field you are responsible for the
conclusions you come in your papers/projects/etc.

Even in sub-fields of those with an empirical component (if that is your
angle) there are standards you have to reach to claim a discovery/success.

