# Generalizing the notion of Farey neighbors to the algebraic numbers

"The Beauty of Roots" is about plots of roots of polynomials—specifically, those with degree less than a given number and height less than another given number. As you can see, these plots are really pretty:

Looking on the inside and the outside of that glowing ring, you can see some neat fractal patterns. But today, I'm not interested in those; I'm interested in those holes.

If you look at the roots of unity (and some other places, presumably the roots of other simple polynomials), you'll see that near each one, the algebraic numbers are especially sparse. As the page describes, for any algebraic number that's particularly "simple", its surroundings are relatively vacant of other algebraic numbers, and the "simpler" the algebraic number is, the more its fellow algebraic numbers tend to keep away. (I imagine that every algebraic number has such a "circle of emptiness" surrounding it, but for all but the simplest ones, this circle is tiny.)

I know of one other set that has this property, and that's the set of rational numbers. It's a theorem that given two fully reduced rational numbers a/b and c/d, the closest they can possibly be to each other is 1/bd; thus, if a rational number is "simple" in the sense of having a small denominator, other rational numbers will tend to be far away from it.

Rational numbers a/b and c/d that differ by exactly 1/bd are called Farey neighbors; if two rational numbers are Farey neighbors, then they have exactly one Farey neighbor in common that lies between them, (a+c)/(b+d). For more information, see Farey sequences on Wikipedia.

So, algebraic numbers that are "simple" are never close to each other. The rational numbers exhibit the same phenomenon; here, "simple" refers to the denominator, and you can determine exactly what the minimum distance is (1 over the product of the denominators). Is it possible to extend the notion of denominators and Farey neighbors to the algebraic numbers in general, thereby explaining the "holes" in the picture?

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Farey sequences (or series) are particular case of your construction, corresponding to degree 1. You are right: there should be a generalization to arbitrary degree. The picture is pretty nice (although no information on the degree/height is given); however it takes a while to download it. –  Wadim Zudilin May 10 '10 at 21:05
In this context I would also mention en.wikipedia.org/wiki/Resultant as a useful computable tool for measuring distances between zeros of two polynomials. –  Wadim Zudilin May 10 '10 at 23:31
The website linked to explains that the picture gives all roots of polynomials with degree at most 24 and height 1 (and no zero coefficients). –  Thomas Bloom May 11 '10 at 10:15

Let me make a rather crude remark about the easiest case of the rings, namely the ring around zero. Or even better, the one around infinity.

John Baez mentions the above picture is about integer polynomials of height 1 with degree less than 25. Where by the height of a polynomial I mean the maximum absolute value of the coefficients.

The simplest phenomenon we're seeing in the picture expresses the relation between the height and the Mahler measure.

The Mahler measure of a polynomial is the max of the roots that are outside the unit circle. And there is an elementary bound $M(f) \leq H(f)\sqrt{d+1}$ where H is the height of the polynomial f and M the Mahler measure and d the degree.

In the picture H is always 1 so there can be no roots farther out than 24. So the crudest thing we are seeing is that there are no roots of norm more than 5.

Replacing $x$ by $\frac{1}{x}$ we see that by the same token there can be no root with norm smaller than 1/5 either. So we see a ring of roots, all with modulus between 5 and 1/5.

I suppose one can explain the other rings in a similar way by modifying the polynomial a bit. For example the ring around 1. If f(x) has a root r that is close to one, then g(x) = f(x+1) has a root r-1 very close to zero. So $|r-1| \leq \frac{1}{5}\frac{1}{H(g)}$ The height of f was 1 but the height went up due to the substitution, so H(g) is big and we see a smaller gap around 1.

In terms of Mahler measure, things also get interesting when one asks for polynomials with small Mahler measure, just a tad above 1. Lehmer's conjecture says the minimal Mahler measure is attained at a very specific polynomial, which happens to be the Alexander polynomial of the (-2,3,7) pretzel knot!

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@Roland: This is related to the question, but not well enough. The gaps around zero and infinity are very special, and I don't see any obvious links to the other "sparsities". Is the last paragraph some kind of arguing that the original question is not a right one? ;) –  Wadim Zudilin May 10 '10 at 22:08
There is a version of Lehmer's problem, due to Schinzel and Zassenhaus, in which it is asked whether there exists a positive absolute constant $c$ such that the maximum of the absolute values of the conjugates of an algebraic nonzero integer $\alpha$ of degree $d$, denoted by $house{\alpha}$, always satisfies $house{\alpha}\ge1+c/d$ when $\alpha$ is not a root of unity. This somehow explains the rooms around roots of unity. The best known quantitative results in this direction are due to Paul Voutier [Acta Arithmetica LXXIV.1 (1996) 81-95]. –  Wadim Zudilin May 10 '10 at 22:40
@Wadim, well I sketched also how I think one can explain the sparsity around 1. I think all sparsities have the same explanation. Am I wrong? By the way I'm not suggesting the original question is not right. I just had to mention the (-2,3,7) pretzel knot because I'm a knot theorist :). –  Roland van der Veen May 10 '10 at 22:47
@Roland, as a number theorist I enjoy the link to knots (why knot?). I don't think that Lehmer or Schinzel-Zassenhaus can really explain the gaps around roots other than roots of unity, or their lattice shifts. I would say that if a Farey-type approach exists, it could produce some novelties in Lehmer's problem. –  Wadim Zudilin May 10 '10 at 23:05
The Mahler measure is not the max of the roots that are outside the unit circle. Rather it is the modulus of the product of the leading coefficient and all the roots that are outside the unit circle. –  Gerry Myerson May 11 '10 at 0:16

A natural generalization of the Farey sequences was defined by Brown and Mahler in 1971 (http://oldweb.cecm.sfu.ca/Mahler/174.pdf ) as follows:

The $m$-th degree Farey sequence of order $n$ is the sequence of all real roots of the set of integral polynomials $$a_m x^m + a_{m-1} x^{m-1} + \cdots + a_0,$$ where $|a_i|\leq n$.

They made some conjectures about the properties of this sequence, but proved no results. Your suggestion seems eminently plausible, though, and this definition might give you a starting point for formalizing it.

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So, the original problem is actually due to Brown and Mahler. They already notified that this is a generalization of Farey sequences. Thanks for the link, Eben! Do you know whether there was some further research in this direction? –  Wadim Zudilin May 10 '10 at 23:13
Alas, those Farey sequences only include real numbers, so I don't think they can explain these holes. –  Tanner Swett May 10 '10 at 23:42
@Tanner: Aren't holes between real numbers possible? A straightforward investigation brought me to 2 articles by F.Delmer and J.-M.Deshouillers [J. Number Theory 55:1 (1995) 60-67; Zbl 0905.11012 & MR1361559 (97d:11036), and Contemp. Math. 143 (1993) 243-246; Zbl 0791.11009 & MR1210504 (93j:11002)]. –  Wadim Zudilin May 10 '10 at 23:52
A somewhat different generalization due to Mahler is the subject of F27 in Guy's Unsolved Problems In Number Theory (page 400 of the 3rd edition). He lists quadratics $ax^2+bx+c$, $a\ge0$, gcd(a, b, c)=1, $b^2\ge4ac$, $\max(a,|b|,|c|)\le n$ which have positive real roots in order of the size of the roots. –  Gerry Myerson May 11 '10 at 0:26
The Brown-Mahler paper discusses the quadratic case only. Delmer and Deshouillers disprove, in their 1st article, a conjecture from the original paper, while in the 2nd one they show that it is true "in average". It seems that the problem of gaps between zeros of polynomials is already too hard in the real case, so I doubt whether something is really known for complex zeros. Already Lehmer's problem of isolating roots of unity is extremely hard! –  Wadim Zudilin May 11 '10 at 0:50

Let $f$ and $g$ be polynomials of degree at most $n$ with integer coefficients of absolute value at most M, and with no common zeros. Then the resultant of $f$ and $g$ (which Wadim pointed to) is at least 1 in absolute value. On the other hand, the resultant is $f_0^rg_0^s\prod(a-b)$, where $f_0$ and $g_0$ are the leading coefficients of $f$ and $g$, respectively, and $r$ and $s$ are the degrees of $g$ and $f$, respectively, and $a$ and $b$ run through the roots of $f$ and $g$, respectively. Now you can find some trivial upper bound for $|a|$, e.g., I think $|a|\lt M+1$ works, so $|a-b|\lt2M+2$, so $|a-b|\gt f_0^{-r}g_0^{-s}(2M+2)^{-(n^2-1)}$.

This is probably far from best possible, but it does reduce to $1/bd$ when $n=1$.

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Yes, this gives a clear bound between roots a la Liouville. +1 –  Wadim Zudilin May 11 '10 at 5:00

Inversion with respect to a circle of radius $\sqrt 2$ centered at $i$ exchanges the unit circle and the real numbers (union infinity). One can apply this inversion to the set of roots of a palindromic polynomial (palindromic: $\xi$ and $1/\xi$ are simultaneous roots) getting the roots of another palindromic polynomial. This transformation yields an involution on the set of palindromic polynomials with rational coefficients (and roots in $\mathbb C^*\setminus\lbrace i,-i\rbrace$) which exchanges the role of the unit circle and the real numbers. This is thus a sort of geometrical explanation of the phenomenon.

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