3 ms·

Does random polynomial mean with real or complex coefficients, though?

deleted 2y ago
[deleted]
cool_dude85 2y ago
The question posed specifies real coefficients, looks like experimentally tested on random coefficients selected from [-1, 1] from uniform distribution and a sort of scaled normal distribution.
- VyseofArcadia 2y ago
  This stuck out to me, and might be a place where our intuition breaks down. As a human if you tell me, "give me a polynomial with degree n with coefficients drawn uniformly from [-1, 1]," I will likely produce one with a lot of zero coefficients, just because a lot of the polynomials you encounter in the wild have a lot of zero coefficients. In reality, if you grab a coefficient uniformly from [-1,1], the probability that it is 0 is 0.
  - madcaptenor 2y ago
    And how random sparse polynomials behave would also be an interesting question!
- Someone 2y ago
  Likely (can’t find the test code) tested on coefficients selected from the IEEE doubles in [-1, +1], so from a subset of numbers of the form m/2^n with m and n integers and m between 0 and 2^53 or thereabouts. For this problem, I don’t see how that would give different results as picking any number in [-1, +1], but that’s not a priori guaranteed. Mathematicians are well-trained as nitpickers.
  - eternauta3k 2y ago
    Aren't roots continuous functions of the coefficients?
    - Someone 2y ago
      Why would that matter? I was thinking “If you’d use 4-bit ‘IEEE’ floats, it wouldn’t be very surprising to see an anomaly like this. Would it be for 5-bit ones? 6-bit? What’s a large enough number of bits?” I also said “For this problem, I don’t see how that would give different results as picking any number in [-1, +1]” because I think 64 is (more than) large enough, but still, I think you’d need an argument if you want to draw a mathematically sound conclusion “this is worth looking into” from such a simulation.
      - eternauta3k 2y ago
        True, the function could in principle have a structure such that sampling it at m/2*n gives a misleading impression of its behavior outside that set. I can't think of an actual function which behaves like that, though (IANAM).
Smaug123 2y ago
The first line of the question says "real".
- Aardwolf 2y ago
  But that first line is a statement about something related but different (the question is about largest root, this statement is about any root), and the term "random polynomial" without specifying anything is used throughout the rest of the content
outop 2y ago
If you allow complex coefficients then the distribution is rotationally symmetric in the complex plane. So a root is no more likely to be in the real line than in any other line through the origin. There would be vanishingly few polynomials with largest root real.

deleted 2y ago

[deleted]

The question posed specifies real coefficients, looks like experimentally tested on random coefficients selected from [-1, 1] from uniform distribution and a sort of scaled normal distribution.

VyseofArcadia 2y ago
This stuck out to me, and might be a place where our intuition breaks down. As a human if you tell me, "give me a polynomial with degree n with coefficients drawn uniformly from [-1, 1]," I will likely produce one with a lot of zero coefficients, just because a lot of the polynomials you encounter in the wild have a lot of zero coefficients. In reality, if you grab a coefficient uniformly from [-1,1], the probability that it is 0 is 0.
- madcaptenor 2y ago
  And how random sparse polynomials behave would also be an interesting question!
Someone 2y ago
Likely (can’t find the test code) tested on coefficients selected from the IEEE doubles in [-1, +1], so from a subset of numbers of the form m/2^n with m and n integers and m between 0 and 2^53 or thereabouts. For this problem, I don’t see how that would give different results as picking any number in [-1, +1], but that’s not a priori guaranteed. Mathematicians are well-trained as nitpickers.
- eternauta3k 2y ago
  Aren't roots continuous functions of the coefficients?
  - Someone 2y ago
    Why would that matter? I was thinking “If you’d use 4-bit ‘IEEE’ floats, it wouldn’t be very surprising to see an anomaly like this. Would it be for 5-bit ones? 6-bit? What’s a large enough number of bits?” I also said “For this problem, I don’t see how that would give different results as picking any number in [-1, +1]” because I think 64 is (more than) large enough, but still, I think you’d need an argument if you want to draw a mathematically sound conclusion “this is worth looking into” from such a simulation.
    - eternauta3k 2y ago
      True, the function could in principle have a structure such that sampling it at m/2*n gives a misleading impression of its behavior outside that set. I can't think of an actual function which behaves like that, though (IANAM).

Smaug123 2y ago

The first line of the question says "real".

Aardwolf 2y ago
But that first line is a statement about something related but different (the question is about largest root, this statement is about any root), and the term "random polynomial" without specifying anything is used throughout the rest of the content

outop 2y ago

If you allow complex coefficients then the distribution is rotationally symmetric in the complex plane. So a root is no more likely to be in the real line than in any other line through the origin. There would be vanishingly few polynomials with largest root real.