Vesselin Dimitrov on Schinzel–Zassenhaus

Suppose that \(P(x) \in \mathbf{Z}[x]\) is a monic polynomial. A well-known argument of Kronecker proves that if every complex root of \(P(x)\) has absolute value at most 1, then \(P(x)\) is cyclotomic. It trivially follows that, for a non-cyclotomic polynomial, the largest root \(\alpha\) in absolute value satisfies \(|\alpha| > 1\). Elementary considerations imply that this can be improved to

\(|\alpha| > 1 + c_n\)

for some real constant \(c_n > 0\) that only depends on the degree. What is the true rate of decay of this parameter as the degree increases? By considering the example \(x^n – 2\), the best one can hope for is that \(c_n\) can be taken to have the form \(c/n\) for some constant \(c\). This is exactly what is predicted by the Schinzel-Zassenhaus conjecture:

Conjecture [Schinzel-Zassenhaus] there is an absolute constant \(c\) and a bound

\(\displaystyle{|\alpha| > 1 + \frac{c}{n}} \)

for the largest root of all non-cyclotomic polynomials.

In fact, Schinzel-Zassenhaus don’t actually make this conjecture. Rather, they first prove a bound where \(c_n\) has the form \(2^{-n}\) up to a constant, and then go on to say that they “cannot disprove” the claim. And of course, this then gets turned into a conjecture named after them! The best bounds were rapidly improved from exponential to something much better, but the original conjecture remained open. That is, until Vesselin Dimitrov in this paper proved the following:

Theorem [Vesselin Dimitrov] The Schinzel-Zassenhaus conjecture is true.

Vesselin’s result is completely explicit, and gives the effective bound \(|\alpha| \ge 2^{1/4n}\), or

\(\displaystyle{c_n = 2^{1/4n} – 1 \sim \frac{\log(2)}{4n}.}\)

The actual proof is very short. Step 0 is to assume the polynomial is reciprocal, which is a quite reasonable assumption because the conjecture (and much more, including Lehmer’s conjecture) was already known by work of Smyth the non-reciprocal case (MR0289451). I’m not sure this step is even needed, since the conjecture is certainly true for polynomials whose constant term is not plus or minus one, and so one can simply replace the polynomial by the reciprocal polynomial in what comes below. Step 1 is to show the inclusion

\( \displaystyle{\sqrt{\prod (1 – \alpha^2_i/X)(1 – \alpha^4_i/X)} \in \mathbf{Z}[[1/X]].}\)

The argument here is elementary (the only prime to worry about is \(p = 2\)). If the original polynomial is cyclotomic, then this squareroot is actually a polynomial, but otherwise it is a power series which is not rational. But now one has a power series which has an analytical continuation outside a very specific region in the plane, namely the “hedgehog” (I would have called it a spider) consisting of rays from \(0\) to \(\alpha^4\) and \(\alpha^2\) in \(\mathbf{C}\). These rays may overlap, but that only improves the final bound. The complement of the Hedgehog is a simply connected region, and know one wants to say that any power series with integer coefficients that has an analytic continuation to such a region with sufficiently large transfinite diameter has to be rational. Step 2 is then to note that such theorems exist! The transfinite diameter of the region in question can be computed from results already computed in the literature, and the consequent bounds are enough to prove the main theorem, all in no more than a couple of pages! It is very nice argument indeed. As a comparison, to orient the reader not familiar with Bertrandias’ theorem (which is used to deduce rationality of the power series in question), it might be useful to give the following elementary variation. Suppose that instead of the hedgehog, one instead took the complement of the entire disc or radius \(r\) for some \(r < 1\). (Importantly, this does not contain the hedgehog above which has spikes outside the unit circle.) Replacing \(X\) by \(1/X\), one ends up with a power series on the complement which is a disc of radius greater than one. Now one can apply the following Trivial Theorem: A power series in \(\mathbf{Z}[X]\) with radius of convergence bigger than one is a polynomial.

Plugging this into Dimitrov’s setup, one deduces a new proof of Kronecker’s theorem! So the main technical point is that the “trivial theorem” above can be replaced by a more sophisticated version (to due Bertrandias and many others) where the region of analytic continuation can be taken to be something other than a disc. (For an exposition of some of these rationalization/algebraization theorems, a good point to start is this post of Matt Baker.)