An important paper by Vesselin Dimitrov (Dimitrov 2019), first posted online in 2019, has now been accepted for publication in the Annals of Mathematics. The paper proves the Schinzel–Zassenhaus conjecture, a long-standing and remarkably simple-to-state conjecture about the location of the roots of integer polynomials.
Let denote the ring of polynomials in one variable with integer coefficients. A polynomial is irreducible if it cannot be written as a product
with both non-constant. An irreducible polynomial is called cyclotomic if it divides for some integer . Thus all roots of a cyclotomic polynomial are roots of unity, and in particular they all lie on the unit circle.
The Schinzel–Zassenhaus conjecture concerns what happens for irreducible polynomials that are not cyclotomic. Let be a monic, irreducible, non-cyclotomic polynomial of degree . Since the roots of a cyclotomic polynomial all have absolute value , one expects a genuinely non-cyclotomic polynomial to have a root that moves away from the unit circle. The conjecture asserts that this movement cannot be arbitrarily small compared with the degree: there should exist an absolute constant such that has a root satisfying
The question goes back to the work of A. Schinzel and H. Zassenhaus in 1965 (Schinzel and Zassenhaus 1965). Their results gave an explicit, but much weaker, bound. They also observed that the natural scale of the problem should be of order . In other words, the central issue was to prove a lower bound that remains uniformly of size as the degree grows.
There has been substantial progress on this problem in special families. For example, in 2007, Peter Borwein, Edward Dobrowolski, and Michael J. Mossinghoff proved the conjecture for polynomials with odd coefficients, and more generally for certain congruence classes of coefficients (Borwein et al. 2007). The general case, however, remained open.
In 2019, Dimitrov posted a preprint giving a proof in full generality (Dimitrov 2019). The result is completely explicit:
This is stronger than merely asserting the existence of some universal constant . Indeed,
and hence one may take
Thus the conjectured -scale separation from the unit circle holds uniformly for every degree.
The theorem can also be viewed as a particularly clean dichotomy: cyclotomic polynomials have all their roots exactly on the unit circle, while every other monic irreducible integer polynomial of degree must have a root outside the circle of radius by the explicit amount encoded in .
The acceptance of Dimitrov’s paper by the Annals of Mathematics marks the culmination of a problem that originated more than sixty years ago and illustrates how a very elementary-looking question about polynomial roots can lead to deep arithmetic ideas.