First stated in 1936 by the Dutch algebraist Bartel van der Waerden, this result (see below) had previously been approached only partially.
The conjecture was already known to hold for polynomials of degree n ≤ 4, thanks first to van der Waerden himself and then to Sam Chow and Rainer Dietmann. In 2010, David Zywina obtained an elegant general result for "large" n, notably improving estimates obtained by Patrick Ximenes Gallagher. Mathematicians had thus been pursuing the conjecture for many years! In his paper, Manjul Bhargava also notes that other teams were also close to a solution.
The Princeton mathematician beat all his colleagues to the result. His proof runs to some thirty pages; what is striking is that it divides all these polynomials into three classes, then bounds the number in each class using methods tailored to that class.
A remarkable insight on the author's part, though he readily admits: "I had been thinking about it from time to time for at least seven or eight years. An idea might occur to me at any moment, even in connection with another problem, and I would think: "Oh, could that have an application here?" "
