Mathematician Measures the Repulsive Force Within Polynomials

https://www.quantamagazine.org/new-math-measures-the-repulsive-force-within-polynomials-20200514/

by sandwall • 6 years ago

43 43 6 years ago

jefftk• 6 years ago

The title reads like a joke, but apparently it's not

lisper9• 6 years ago

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.

Nasrudith• 6 years ago

Yeah - divergence would be a far better term rather than the analogy. They go apart but specifically don't interact.

Instead it gets people to overlook it as more new age nonsense woo that abuses appropeiated quantum physics terminology.

LolWolf7• 6 years ago

that abuses appropeiated quantum physics terminology

I would certainly not only specifically not say this is borrowed from quantum physics (perhaps classical EM?), but in fact the whole point of the theorem is to show that such roots do interact in a specific way. From Theorem 1 of the paper:

Let P ∈ Z[X] be a monic integer irreducible polynomial of degree n > 1. If P is not cyclotomic, then [the largest root of P is bounded away from 0 with norm at least 2^(1/4n)].

In particular, it is a statement about a specific root (the largest one) lying outside of the unit disk, whereas the remaining roots could potentially lie within it (and specific examples of this 'tightness' condition are given in applications in section 5).

pas• 6 years ago

From the article:

"""

Schinzel and Zassenhaus predicted that every non-cyclotomic polynomial must have at least one root that’s outside the unit circle and at least some minimum distance away.

Or, to put the Schinzel-Zassenhaus conjecture in terms of repulsion, it predicted that the smallest roots of a non-cyclotomic polynomial — which might fall within the unit circle — effectively push other roots outside the unit circle, like magnets pushing each other away.

You can think of the roots of a polynomial as negatively charged particles that repel each other with a force that decays when the distance increases,

"""

So if they can write a formula for distance between these roots, and it's always positive and has a similar dynamic to Newton's (or Maxwell's, or Ampere's) force, then why not call it repulsive force?

ordu2• 6 years ago

In fact, it's actively harmful.

How is it harmful?

naringas• 6 years ago

I think they meant to say it can lead to confusion or be misinterpreted

ironically, their misuse of the term harmful can also lead to confusion or be misinterpreted

all in all, we should use more precise words (and thinking), but this requires more effort from both the writer and the reader

lisper9• 6 years ago

Because people who don't already understand what is going on might think that the words are literally true and come to all manner of false conclusions. Allowing such misconceptions to persist and fester can result in the rise of cults. For example: https://www.ramtha.com/

Gollapalli2• 6 years ago

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.

jfkebwjsbx2• 6 years ago

especially in mathematics

What do you mean?

lisper9• 6 years ago

Quanta is targeted at a lay audience so many of the people who read it can reasonably be called "infants" in terms of their mathematical and technical knowledge. Furthermore, Quanta often has articles about quantum mechanics, a field where misinformation is rampant and leads in some cases to actual substantial harm.

Also, there is nothing wrong with appropriating terminology -- when it is appropriate (no pun intended). But, to quote Tom Stoppard, "If there is any point to using language at all, it is that a word is taken to stand for a particular fact or idea and not for other facts or ideas." The phrase "measure a repulsive force" has an established meaning in English, and that meaning is intimately bound to the physical world. There is nothing wrong with using that physical phenomenon as a metaphor, but that's not what the headline does. The headline says "Mathematician Measures the Repulsive Force Within Polynomials" and that is simply false under the well-established English semantics of the phrase "measure a ... force." No mathematician has ever measured a force, at least not in the course of conducting the business of being a mathematician.

throwlaplace• 6 years ago

Forces are physical things

Me thinks thou doth protest too much

https://en.m.wikipedia.org/wiki/Virtual_work#Principle_of_vi...

Me also thinks you have no clue what you're talking about

klyrs5• 6 years ago

The Gibbs phenomenon is an example of this intuition being useful. Wanna make a wacky function with weird features? Fine. Wanna make sharp features? Sure... but the more points you specify, the wilder the behavior you'll see! Is there an electric force pushing the unspecified regions into the undulations? Hell no. Is it useful to think about there being an energy level that depends on how close the specified points are? Yes.

When you say "actively harmful" -- do you mean that it's doing physical harm to something, or are you using an incomplete metaphor of the kind you're railing against? Because I'm speaking as somebody with some training in number theory and teaching advanced math and I think you're way off-base.

lisper9• 6 years ago

Physical harm is not the only kind of harm there is. The harm that this causes is to further undermine trust in science and technology. In case you hadn't noticed, the validity of science is being seriously called into question nowadays. The result is, in many cases, serious physical harm. People are dying right now from covid-19 in no small measure because science denialism is having a significant influence on U.S. government policy. Articles like this reinforce the science denialist position because it allows them to validly criticize the advocates of science for advancing claims that are manifestly absurd on their face, like that they have "measured the repulsive force within polynomials."

juped• 6 years ago

Connecting fields of study by discovering common underlying structures and principles is among the most rewarding methods of inquiry. You may as well say Noether's theorem is actively harmful and causing millions to die of coronavirus every second because physical laws aren't even shapes so how can they be "symmetries"?

contravariant• 6 years ago

Similarly it is absolutely unacceptable that framers keep appropriating the term sheaf for something as silly as a bundle of cereal. Sheafs are proper mathematical objects that don't have anything at all to do with agriculture.

LolWolf7• 6 years ago

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)!

aDfbrtVt• 6 years ago

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

LolWolf7• 6 years ago

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

syockit• 6 years ago

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

fxj• 6 years ago

Use R:

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

danharaj4• 6 years ago

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

Real roots of random polynomials: expectation and repulsion

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...

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

LolWolf7• 6 years ago

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.

danharaj4• 6 years ago

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...

Mathematician Measures the Repulsive Force Within Polynomials

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