Hacker News Reader

Comment by Aardwolf | Hacker News Reader

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

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.

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.

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.

And how random sparse polynomials behave would also be an interesting question!

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.

Aren't roots continuous functions of the coefficients?

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.

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).

The first line of the question says "real".

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

Reply on news.ycombinator.com