Wikipedia:Reference desk/Archives/Mathematics/2014 November 7

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November 7

Roots(mod n) => Root in Z

Let N be an infinite set of integers, what can we say about those functions of integers f so that f has a root mod n for all n in N implies that f has an integer root? I'm generally interested in the case of N the naturals( > 1) or the primes. I'd also be interested in the case of replacing the naturals with a ring and N with a collection of ideals. I'm mainly interested in characterizing the set of such f, what algebraic properties it might have, and what/when families of functions are part of it (like polynomials, exponentials, etc.). Thanks for any assistance:-) --obviously any function with a root in Z will have roots in all mod's, so, it may make more sense to talk in terms of f's with modular roots sans integer ones.Phoenixia1177 (talk) 07:48, 7 November 2014 (UTC)

Are you familiar with Swinnerton-Dyer polynomials? (sorry I was being stupid and confusing roots with factors) --catslash (talk) 19:45, 7 November 2014 (UTC)

Let's go about this the other way:

For any given odd prime p, at least one of x² + 1, x² − 2, and x² + 2 have roots modulo all prime powers pⁿ. x² + 7 has roots modulo all powers of 2. Hence:

$\left(x^{2}+1\right)\left(x^{2}-2\right)\left(x^{2}+2\right)\left(x^{2}+7\right)$

has (by the chinese remainder theorem) roots modulo all positive integers n, but does not have an integer root. — Arthur Rubin (talk) 21:10, 8 November 2014 (UTC)

Trying to find an specific sample of data (which are about botany) that it's commonly used in machine learning

A set of data that it's used commonly to show clustering in machine learning/pattern recognition. I think there were 4 plants in it, and it's relatively old, but used frequently in statistics, r programming courses.--Senteni (talk) 19:02, 7 November 2014 (UTC)

Is it the Iris_flower_data_set? Note there are three species, and four attributes. But other than that it seems to fit your description rather well. SemanticMantis (talk) 20:40, 7 November 2014 (UTC)

Yes, it was exactly that.--Senteni (talk) 19:39, 8 November 2014 (UTC)

Swinnerton-Dyer polynomials

I understand why these have many roots mod any prime, but not why they can't have roots in the integers. The expositions I've come across prove the former property but take it that the latter fact is obvious; it's not obvious to me, so could somebody explain please? Thanks --catslash (talk) 20:00, 7 November 2014 (UTC)

When you make a polynomial of the form $p(x)=\prod _{i}(x-a_{i})$, the roots are precisely the $a_{i}$. This is a consequence of the ring of polynomials being a division ring: fix any $b$ not occurring in the $a_{i}$, and divide $p(x)$ by $(x-b)$. This gets us $p(x)=q(x)(x-b)+C$ for some non-zero constant $C$. Now it's easy to see that $p(b)=C$.--80.109.80.31 (talk) 22:26, 7 November 2014 (UTC)

Sorry, I was being stupid and confusing roots and factors. Why is it obvious that the Swinnerton-Dyer polynomials have no proper factors with integer coefficients? On reflection, 80.109.80.31's answer may have given me the clue: any factor over the integers will be one of the factors over the reals - could the coefficient of the second-highest term of a factor over the reals be an integer? Seems unlikely. --catslash (talk) 22:53, 7 November 2014 (UTC)

I believe what you're looking for follows from Q[√p₁, √p₂, ... , √p_k] having degree 2^k over Q. This does not appear to be obvious but is well known, see the math exchange article [1]. --RDBury (talk) 22:11, 10 November 2014 (UTC)