Smyth’s conjecture and a probabilistic local-to-global principle

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• Jordan Ellenberg (University of Wisconsin-Madison)
• Wednesday 10 April 2024, 15:30-16:30
• Seminar Room 1, Newton Institute.

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OOEW04 - Structure and Randomness - a celebration of the mathematics of Timothy Gowers

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.  

This talk is part of the Isaac Newton Institute Seminar Series series.

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