Abstract

Let a,b,c be integers.  Are there algebraic numbers x,y,z which are Galois conjugate to each other over Q and which satisfy the equation ax bx cy = 0? In 1986, Chris Smyth proposed an appealingly simple conjecture that a set of plainly necessary local conditions on a,b,c guarantees the existence of such an x,y,z.  I will present a provisional proof of this conjecture, joint with Will Hardt.  As Smyth observed, this problem, which appears on its face to be a question about algebraic number theory, is really a question in combinatorics (related to a problem solved by David Speyer: what can the eigenvalues of the sum of two permutation matrices be?)  It turns out, though, that this combinatorics problem can be interpreted in number-theoretic terms, as a question about whether certain kinds of equations in probability distributions (I’ll explain what’s meant by this!)  have solutions over Q whenever they have solutions over every completion of Q.  Solving this number theoretic problem, in the end, comes down to a bit of additive combinatorics.