Around Smyth’s conjecture

Abstract: Are there algebraic numbers $x,y,z$ which are Galois conjugate to each other over $\mathbb{Q}$ and which satisfy the equation $5x + 6y + 7z = 0$? In 1986, Chris Smyth proposed an appealingly simple conjecture about linear relations between Galois conjugates, which would provide answers to the above questions and all questions of the same form, and which has remained unsolved. My experience is that most people, upon seeing Smyth’s conjecture, immediately think it must be false (I certainly did!), but I have come to think it’s true, and I’ll talk about a provisional solution, joint with Will Hardt, in the case of three conjugates. I’ll explain why (as Smyth observed) this is really a conjecture about linear combinations of permutation matrices (related question, solved by Speyer: which algebraic numbers can be eigenvalues of the sum of two permutation matrices?), and why our approach can be thought of as proving a “Hasse principle for probability distributions” in a particular case, plus a bit of additive number theory. Much of this talk, maybe all, will be suitable for undergraduates.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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