7 ms·

Mathematician Measures the Repulsive Force Within Polynomials

43 points by sandwall 6y ago

jefftk 6y ago
The title reads like a joke, but apparently it's not
lisper 6y 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 6y 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.
  - LolWolf 6y 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 6y 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?
- ordu 6y ago
  > In fact, it's actively harmful. How is it harmful?
  - naringas 6y 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
    - lisper 6y ago
      No, I meant exactly what I said. See https://news.ycombinator.com/item?id=23195709 https://news.ycombinator.com/item?id=23195709
      - deleted 6y ago
        [deleted]
  - lisper 6y 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/ https://www.ramtha.com/
    - ordu 6y ago
      > people [...] might [...] come to all manner of false conclusions. Let them. It is their fundamental right to be wrong.
    - klyrs 6y ago
      This is like saying that geometry is harmful because flat-earthers misapply it to convince themselves and others of falsehoods. Yawn.
    - 6y ago
- Gollapalli 6y 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.
  - jfkebwjsbx 6y ago
    > especially in mathematics What do you mean?
    - Gollapalli 6y ago
      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.
      - jfkebwjsbx 6y ago
        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.
  - lisper 6y 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 6y ago
  >Forces are physical things Me thinks thou doth protest too much https://en.m.wikipedia.org/wiki/Virtual_work#Principle_of_virtual_forces https://en.m.wikipedia.org/wiki/Virtual_work#Principle_of_vi... Me also thinks you have no clue what you're talking about
- klyrs 6y 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.
  - lisper 6y 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 6y 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 6y 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.
LolWolf 6y 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 6y ago
Does anyone know how this idea of repulsive forces in root spacing might relate to filter analysis?
- LolWolf 6y ago
  What do you mean by filter analysis? As in classical linear filtering? (as in, you're finding the roots of the transfer function?)
deleted 6y ago
[deleted]
syockit 6y 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 6y ago
  Use R: > plot(polyroot(c(1,-1,1,-1,1,-1)))
danharaj 6y 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 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/45/1/80/297678 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 https://arxiv.org/abs/1301.4116
- LolWolf 6y 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.

43 points by sandwall 6y ago

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

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 6y 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.
- LolWolf 6y 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 6y 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?
ordu 6y ago
> In fact, it's actively harmful. How is it harmful?
- naringas 6y 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
  - lisper 6y ago
    No, I meant exactly what I said. See https://news.ycombinator.com/item?id=23195709 https://news.ycombinator.com/item?id=23195709
    - deleted 6y ago
      [deleted]
- lisper 6y 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/ https://www.ramtha.com/
  - ordu 6y ago
    > people [...] might [...] come to all manner of false conclusions. Let them. It is their fundamental right to be wrong.
  - klyrs 6y ago
    This is like saying that geometry is harmful because flat-earthers misapply it to convince themselves and others of falsehoods. Yawn.
  - 6y ago
Gollapalli 6y 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.
- jfkebwjsbx 6y ago
  > especially in mathematics What do you mean?
  - Gollapalli 6y ago
    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.
    - jfkebwjsbx 6y ago
      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.
- lisper 6y 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 6y ago
>Forces are physical things Me thinks thou doth protest too much https://en.m.wikipedia.org/wiki/Virtual_work#Principle_of_virtual_forces https://en.m.wikipedia.org/wiki/Virtual_work#Principle_of_vi... Me also thinks you have no clue what you're talking about
klyrs 6y 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.
- lisper 6y 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 6y 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 6y 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.

LolWolf 6y 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 6y ago

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

LolWolf 6y ago
What do you mean by filter analysis? As in classical linear filtering? (as in, you're finding the roots of the transfer function?)

deleted 6y ago

[deleted]

syockit 6y 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 6y ago
Use R: > plot(polyroot(c(1,-1,1,-1,1,-1)))

danharaj 6y 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 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/45/1/80/297678 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 https://arxiv.org/abs/1301.4116

LolWolf 6y 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.