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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 more 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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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 get more 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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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 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.

Things

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 get more interesting when one asks for polynomials with small Mahler measure, say 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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