On the circle and never home

A root of unity is a point of the unit circle whose powers come back to 1: a seventh root after seven steps, a twelfth root after twelve. The powers of any point on the circle stay on the circle, since multiplying by it is a pure turn, so for such a point the only question is whether the turning ever lands on the start again.

For $(3 + 4i)/5$ it never does. The point is on the circle, because $3^2 + 4^2 = 5^2$, and it is a perfectly respectable number — a ratio of two whole numbers with $i$ in one of them. Its powers run round the circle for ever without once landing on 1.

So lying on the circle is not enough to make a point a root of unity. What else is needed is the subject of this essay, and the answer, found by Kronecker in 1857, is not about the circle at all. It is about the other roots of the polynomial the point satisfies, and whether every one of them stays on or inside the circle too.

Why the Pythagorean point never returns #

The claim is that no power of $3 + 4i$ equals the same power of 5, and the proof is a factorisation.

In the Gaussian integers — the numbers $a + bi$ with $a$ and $b$ whole — the prime 5 splits as $5 = (2 + i)(2 - i)$, and $3 + 4i$ is a square: $(2 + i)^2 = 4 + 4i - 1 = 3 + 4i$. So if $(3 + 4i)^k$ were $5^k$, then $(2 + i)^{2k}$ would equal $(2 + i)^k (2 - i)^k$, and cancelling would leave $(2 + i)^k = (2 - i)^k$.

That cannot happen. Factorisation into primes is unique in the Gaussian integers up to the four units $\pm 1$ and $\pm i$ — the fact that decides which primes are sums of two squares — and $2 + i$ and $2 - i$ are different primes, neither a unit multiple of the other. The two sides of the equation have different factorisations, so they are different numbers. The same argument rules out $(3 + 4i)^k$ being $-5^k$ or any other real number, since squaring would bring it back to the equation above.

So the angle of $(3 + 4i)/5$, 53.1301 degrees, is not a rational fraction of a turn. If it were $p/q$ of a turn, the $q$-th power would be 1. Its powers therefore behave like an irrational rotation: they never repeat, they come arbitrarily close to every point of the circle, and at every stage they cut the circle into arcs of at most three different lengths.

The same holds for every Pythagorean triple. A point $(a + bi)/c$ with $a^2 + b^2 = c^2$ and neither $a$ nor $b$ zero lies on the circle, has rational coordinates, and never returns. The only roots of unity with rational coordinates are $\pm1$ and $\pm i$, so every triple on the circle other than those four is a point that wanders.

How close the wandering comes #

A point that never returns still comes back as close as anybody likes, and the rate at which it does is forced by counting.

Take the first $N$ powers. They are $N$ points on a circle of circumference $2\pi$, so two of them are within $2\pi/N$ of each other along the circle. If those are the $j$-th and $k$-th powers with $j < k$, then turning the $j$-th back to 1 turns the $k$-th to the $(k - j)$-th power, and that power is within $2\pi/N$ of 1. Among the first $N$ powers of any point on the circle, one is within $2\pi/N$ of home — the argument of more things than boxes, and the same one that gives the good fractions for an irrational number.

For the Pythagorean point and $N = 60$ the bound is about 0.105, and the figure finds the 27th power at 0.0957: a near-return guaranteed by counting and achieved a little better than guaranteed.

The seventh root of unity is the contrast, and its turn of 51.43 degrees is nearly the same as the Pythagorean point’s. Seven steps bring it home and fourteen bring it home twice. Nothing about the two pictures up to the sixth power tells them apart — both are six points, each about a seventh of a turn further round than the last — and the difference only shows at the seventh, where one lands exactly on 1 and the other overshoots by twelve degrees and keeps missing.

The polynomials the powers satisfy #

The difference between the two points is invisible on the circle and plain in their polynomials.

The seventh root satisfies $x^7 - 1 = 0$, a polynomial whose leading coefficient is 1 and whose other coefficients are whole numbers. A number satisfying a polynomial of that kind is called an algebraic integer. The Pythagorean point satisfies $5x^2 - 6x + 5 = 0$, whose roots are $(3 \pm 4i)/5$, and the 5 in front cannot be divided away without leaving fractions: its monic form is $x^2 - \tfrac65 x + 1$. It is algebraic, and it is not an integer.

Kronecker’s idea is to follow not the point but the polynomial whose roots are the $k$-th powers of the point and of its partners — the other roots of its minimal polynomial, its conjugates. There are as many of them as the degree of the number over the rationals, and no rational arithmetic can tell a number from its conjugates: any identity with rational coefficients that one of them satisfies, all of them satisfy.

For the Pythagorean point the polynomial of the $k$-th powers is $x^2 - 2\,\mathrm{Re}(\alpha^k)\,x + 1$, and its middle coefficient is a fraction whose denominator is exactly $5^k$, because the real part of $(3 + 4i)^k$ is never a multiple of 5. The coefficients never repeat and nothing forces them to.

For a fifth root of unity the four conjugates are the four primitive fifth roots, and their $k$-th powers are again the four primitive fifth roots — or, when $k$ is a multiple of five, four copies of 1. So the polynomial is always $x^4 + x^3 + x^2 + x + 1$ or $(x - 1)^4$. Whole coefficients from a short list. That is the whole of the difference, and the next section makes it a proof.

Why the list has to repeat #

Suppose $\alpha$ is an algebraic integer of degree $d$ and every one of its conjugates lies on or inside the unit circle. The polynomial whose roots are the $k$-th powers of the conjugates has coefficients that are symmetric combinations of those roots, so they are whole numbers — a fact about symmetric polynomials, which turns the whole coefficients of the minimal polynomial into whole coefficients of every later one. And each coefficient is a sum of products of roots of length at most one, so the coefficient of $x^{d-j}$ is at most the binomial coefficient $\binom{d}{j}$ in size.

Whole numbers in a bounded range form a finite list. There are only finitely many monic polynomials of degree $d$ with coefficients in those ranges, so all the powers of $\alpha$ are among their finitely many roots. An infinite sequence taking finitely many values repeats: some $\alpha^a = \alpha^b$ with $a < b$, so $\alpha^{b - a} = 1$, and $\alpha$ is a root of unity — unless it is 0.

That is Kronecker’s theorem. A monic whole-number polynomial whose roots all lie in the closed unit disc has only roots of unity and zero as roots, and so it is a product of cyclotomic polynomials and a power of $x$.

Every hypothesis did work. The whole coefficients made the list finite, and the Pythagorean point, whose coefficients are fractions, escapes there. The bound on the size of every conjugate made the list bounded, and the next point to be drawn escapes there.

Every polynomial in the box, searched #

The theorem can be watched as well as proved, because the box it describes is finite and small degrees can be searched completely.

Each polynomial’s roots are located numerically, and the search is arranged so that no verdict rests on a digit. Every polynomial whose largest root reads within a thousandth of the circle is then divided, in whole numbers, by cyclotomic polynomials until nothing is left; the cyclotomic products of each degree are also multiplied out independently and required to be exactly the list found; and the nearest polynomial outside must sit well clear of the tolerance, which it does — its largest root has length 1.1510, while repeated roots on the circle read no further out than 1.0002.

The orders that appear are exactly those $m$ for which $\varphi(m)$, the number of primitive $m$-th roots, is at most 4 — the degrees of the cyclotomic polynomials that fit. The same totient decides which regular polygons can be drawn, and here it decides which corners of which polygons a polynomial of degree four can reach. The polynomial nearest the circle from outside is worth a second look. Its two complex roots have length 1.1510 each and its real root has length 0.7549, and the product of the two outside is 1.3247, which is the smallest number that will reappear below as a floor.

An integer on the circle that still wanders #

Kronecker’s hypothesis bounds every conjugate, and it is tempting to shorten it to “an algebraic integer on the unit circle is a root of unity”. That shortened statement is false, and the counterexample is the most famous polynomial in the subject.

The polynomial is Lehmer’s, from 1933. It is monic, its coefficients are $1, 1, 0, -1, -1, -1, -1, -1, 0, 1, 1$, and eight of its ten roots lie exactly on the unit circle. Each of those eight is an algebraic integer of length one. None is a root of unity, because the polynomial does not factor over the rationals and is not cyclotomic.

What lets them escape is visible in the proof. Their conjugates include the other two roots, one outside the circle at 1.17628 and one inside at its reciprocal. The powers of the outside root grow without limit, so the polynomials of the $k$-th powers have coefficients that grow too, the finite list is lost, and the powers of the eight roots on the circle are free to wander. The condition is on the whole family of conjugates, and a single member outside the circle releases all the others.

How far outside a root must go #

The size of the escape is measured by the Mahler measure of the polynomial: the product of the lengths of its roots outside the unit circle. Kronecker’s theorem says that a monic whole-number polynomial has measure exactly 1 only when it is cyclotomic, apart from powers of $x$. Lehmer asked in 1933 whether the measure of any other such polynomial can be as close to 1 as desired.

The search table shows the measures falling as the degree rises, which is the reason the question is hard: allowing more roots allows the escape to be shared out, and nothing in the table suggests where the fall stops. Lehmer’s own polynomial remains the smallest measure above 1 known at any degree. Computer searches through far higher degrees have found nothing smaller, and nobody has shown that nothing smaller exists.

Two results bracket the question. For polynomials that are not reciprocal — whose coefficients do not read the same backwards — Smyth proved in 1971 that the measure is at least 1.3247, the real root of $x^3 - x - 1$, which is exactly the measure met in the search above. And for polynomials of degree $d$ in general, Dobrowolski proved in 1979 that the measure exceeds 1 by at least a constant times $(\log\log d/\log d)^3$ — a floor that sinks towards 1 as the degree grows, but so slowly that it never reaches it at any finite degree.

The measure as an average round the circle #

The Mahler measure was defined from the roots, and it has a second description that never mentions them. Walk a point once round the unit circle, record the logarithm of the polynomial’s size at every point, and average. By Jensen’s formula from complex analysis, that average is exactly the logarithm of the measure: each root inside the circle contributes nothing on average, and each root outside contributes the logarithm of its length.

For a cyclotomic polynomial the average is therefore zero. The polynomial is large at some points of the circle and small at others — $x^7 - 1$ is 2 at $-1$ and vanishes at the seven roots — and the peaks and troughs of its logarithm balance exactly. For Lehmer’s polynomial the average is the logarithm of 1.17628, about 0.1624, and for the root-of-unity filter’s $(1 + x)^n$ it is zero again, since its only root, $-1$, lies on the circle.

Read that way, Kronecker’s theorem is a statement about sizes rather than roots. A monic whole-number polynomial whose logarithm averages to zero round the unit circle is a product of cyclotomic polynomials, and Lehmer’s question asks whether a non-cyclotomic one can average as little as desired above zero. The reformulation is why the measure turns up far from polynomials — the entropy of certain dynamical systems built from a polynomial is exactly this average, so the question of how small an arithmetic average can be is also a question about the least disorder such a system can have.

One root outside and the rest inside #

The largest root of Lehmer’s polynomial, 1.17628, has a name of its own: it is a Salem number, a real algebraic integer above 1 whose other conjugates lie in the closed disc with at least one on the circle. Its cousins with every other conjugate strictly inside are Pisot numbers, and the smallest of those is the same 1.3247, as Siegel proved in 1944.

Pisot numbers explain an old curiosity. The golden ratio $\varphi = 1.618\ldots$ is one, since its only conjugate is $-0.618\ldots$, inside the circle. The sum $\varphi^n + (-0.618\ldots)^n$ is a whole number for every $n$ — a symmetric combination of the two roots of $x^2 - x - 1$, exactly as in Kronecker’s argument — and the second term shrinks to nothing. So the powers of $\varphi$ come ever closer to whole numbers: $\varphi^{10}$ is 122.99187, less than a hundredth short of 123, and the rectangle built from the golden ratio sits behind that near-miss.

The same bookkeeping sorts all three cases. Every conjugate in the disc forces the powers to cycle. One conjugate outside and the rest strictly inside forces them towards whole numbers. One outside and some on the circle, as for Lehmer’s number, leaves the powers on the circle wandering for ever.

Where the searches stop #

Every search here is finite. The box of polynomials is searched to degree four, where it holds 2106 polynomials; at degree five it holds more than a hundred thousand, and the theorem is about every degree. The search confirms Kronecker’s statement where it looks and proves nothing beyond.

The roots are found numerically, and the verdicts are not. Repeated roots on the circle are located to about four decimal places and simple ones to many more, and every decision that matters — a product of cyclotomic factors, the denominators of the powers, the imaginary part of $(3 + 4i)^k$ — is made in whole numbers.

And the Lehmer search is restricted twice. It looks only at reciprocal polynomials, which is where Smyth’s theorem says small measures must live, and only at coefficients $-1$, $0$ and $1$. That is where every small measure yet found has turned up, and it is a hunting ground rather than a proof.

Still open: whether the measure can creep down to one #

Lehmer’s question is still unanswered. Either there is a gap above 1 that the Mahler measure of a non-cyclotomic polynomial can never enter — and the width of the gap is then at most 0.17628 — or there are polynomials whose measures come arbitrarily close to 1 without reaching it.

Every approach so far meets the same obstacle: the measure is a product over all the roots, and a polynomial of high degree can hide a small excess among many roots that each stray only slightly. Dobrowolski’s floor controls that hiding but not completely. A proof that the gap exists would say that the arithmetic of whole-number polynomials holds every root away from the circle by a fixed margin unless it pins them all to it; a counterexample would say that Kronecker’s theorem is sharp in the worst possible way.

A condition on the whole family #

Lying on the circle is a condition on one number. Being a root of unity is a condition on a number together with every conjugate its polynomial ties to it, and the gap between the two statements is where all the interest lies.

The Pythagorean point fails the second condition by not being an integer, so that its polynomials acquire denominators and the finite list never forms. Lehmer’s roots fail it by having a partner outside the circle, so that the list grows. When a property of a single number seems to follow from where it sits, check what its conjugates are doing — the polynomial it satisfies sees all of them at once, and the picture of the circle shows only one.