# Some problems for the new year

Part new year resolution and part a birthday present to myself (and those audience members interested), I’ve decided to write up some problems I’ve been thinking about but either don’t have the time or the techniques/knowledge to tackle at the present time. Hopefully they will keep me motivated into 2016, as well as anyone else who’s interested in them. In no particular order:

1) Stewart’s Conjecture: I have already discussed this problem in two earlier posts (here and here). The conjecture is due to my advisor, Professor Cameron Stewart, in a paper from 1991. The conjecture asserts that there exists a positive number $c > 0$ such that for all binary forms $F(x,y)$ of degree $d \geq 3$, integer coefficients, and non-zero discriminant, there exists a positive number $r_F$ which depends on $F$ such that for all integers $h$ with $|h| \geq r_F$, the equation $F(x,y) = h$ has at most $c$ solutions. In particular, the value of $c$ does not depend on $F$ nor $d$. A weaker version of this conjecture asserts the existence of a positive number $c_d$ for every degree $d \geq 3$ for which the above holds.

I suspect that Chabauty’s method, applied to the estimation of integer points on hyperelliptic curves, is close to being able to solve this problem; see this paper by Balakrishnan, Besser, and Muller. However, there may be other tools that may be used without involving a corresponding curve. That said, since a positive answer to Stewart’s conjecture would have significant impact on the theory of rational points on hyperelliptic curves, it seems that the two problems are intrinsically intertwined.

2) Asymptotic Chen’s Theorem: This is related to a problem I’ve been thinking about lately. Chen’s theorem asserts that every sufficiently large even integer $N$ can be written as the sum of a prime and a number which is the product of at most two primes. However, this simple statement hides the nature of the proof. The proof essentially depends on two parts, and (as far as I know) has not been improved on a fundamental level since Chen. The first is the very general Jurkat-Richert theorem, which can handle quite general sequences. Its input is some type of Bombieri-Vinogradov theorem, i.e., some type of positive level of distribution. It essentially churns out semi-primes of some order given a particular level of distribution. We will phrase the result slightly differently, in terms of the twin prime conjecture. Goldbach’s conjecture is quite related, and Chen actually proved the analogous statement for both the twin prime problem and Goldbach’s conjecture. Bombieri-Vinogradov provides the level $1/2$, and with this level, the Jurkat-Richert theorem immediately yields that there exist infinitely many primes $p$ such that $p+2$ is the product of at most three primes. Using this basic sieve mechanism and the Bombieri-Vinogradov theorem, it is impossible to breach the ‘three prime’ barrier. A higher level of distribution would do the trick, but so far, Bombieri-Vinogradov has not been improved in general (although Yitang Zhang‘s seminal work on bounded gaps between primes does provide an improvement in a special case). Thus, we require the second piece of the proof of Chen’s theorem, the most novel part of his proof. He was able to show that there aren’t too many primes $p$ such that $p+2$ has exactly three prime factors, so few that the difference in number between those primes $p$ where $p+2$ has at most three prime factors and those with exactly three prime factors can be detected. However, the estimation of these two quantities using sieves (Chen’s theorem does not introduce any technology that’s not directly related to sieves) produce terms with the same order of magnitude, so Chen’s approach destroys any hope of establishing an asymptotic formula for the number of primes $p$ for which $p+2$ is the product of at most two primes. It would be a significant achievement to prove such an asymptotic formula, because it means there has been a significant improvement to the underlying sieve mechanism, or some other non-sieve technology has been brought in successfully to tackle the problem. Either case, it would be quite the thing to behold.

3) An interpolation between geometrically irreducible forms and decomposable forms: A celebrated theorem of Axel Thue is the statement that for any binary form $F(x,y)$ with integer coefficients, degree $d \geq 3$, and non-zero discriminant and for any non-zero integer $h$, the equation $F(x,y) = h$ has only finitely many solutions in integers $x,y$.  Thue’s theorem is ineffective, meaning one cannot actually find an upper bound for the number of solutions except to know that it must be finite. Thue’s theorem has been refined by many authors over the past century, with some of the sharpest results known today due to my advisor Cam Stewart and Shabnam Akhtari.

If one wishes to generalize Thue’s theorem to higher dimensions, then there are two obvious candidates. The more obvious one is to consider general homogeneous polynomials $F(x_1, \cdots, x_n)$ in many variables. However, in this case Thue’s techniques do not generalize in an obvious way. Thue’s original argument reduced the problem to a diophantine approximation problem, i.e., to show that there are only finitely many rational numbers which are `very close’ to a given root of $F$. This exploits the fact that all binary forms can be factored into linear forms, a feature which is absent for general homogeneous polynomials in $n \geq 3$ variables. Thus, one needs to narrow the scope and instead consider decomposable forms, meaning homogeneous polynomials $F(x_1, \cdots, x_n)$ which can be factored into linear forms over $\mathbb{C}$, say. To this end, significant progress has been made. Most notably, Schmidt’s subspace theorem was motivated by this precise question. Schmidt, Evertse, and several others have worked over the years to establish results which are quite close to the case of Thue equations, though significant gaps remain, but that’s a separate issue and we omit further discussion.

The question I have is whether there is a way to close the gap between what can be proved about decomposable forms and for general forms. The forms which are the most different from decomposable forms, which are essentially as degenerate as possible geometrically, are the ones that are the least degenerate; i.e., the geometrically irreducible forms. These are the forms that cannot be factored at all. Specifically, their lack of factorization is not because its factorability is hidden by some arithmetic or algebraic obstruction but because it is geometrically not reducible. Precisely, geometrically irreducible forms are those forms $F(x_1, \cdots, x_n)$ which do not have factors of positive degree even over an algebraically closed field, say $\mathbb{C}$. For decomposable forms, a necessary condition is to ensure that the degree $d$ exceeds the number of variables $n$; much like the condition $d \geq 3$ in the case of Thue’s theorem. However, absent from the case when $n = 2$ is the possibility that there are forms of degree exceeding one which behave `almost’ like linear forms, in a concrete sense. By this I mean we can show that as long as basic local conditions are satisfied, the form represents all integers. This has shown to be the case for forms whose degree is very small compared to the number of variables; the first such result is due to Birch, and has been improved steadily since then. Thus the interpolation I am wondering about is the following: let $F(x_1, \cdots, x_n)$ be a homogeneous polynomial with integer coefficients and degree $d \geq n+1$, with no repeated factors of positive degree. Suppose that $F$ factors, over $\mathbb{C}$, into forms of very small degree, say $d' \ll \log n$. Can we hope to establish finiteness results like we can for decomposable forms? This seems like a very interesting question.

If you are interested in any of these problems or if you have an idea as to how to approach any of them, please let me know!