3 Took advantage of the term "height" (of a polynomial)

"The Beauty of Roots" is about plots of roots of polynomials—specifically, those with degree less than a given number , and whose coefficients are integers whose absolute value is height less than a 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?

2 Turned link to image into image

"The Beauty of Roots" is about plots of roots of polynomials—specifically, those with degree less than a given number, and whose coefficients are integers whose absolute value is less than a given number. As you can see, these plots are really pretty. (image):

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?

1

# 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 whose coefficients are integers whose absolute value is less than a given number.

As you can see, these plots are really pretty. (image)

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?