Around Smyth's Conjecture

Author / Creator

Hardt, Will, author

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Online

Physical

Summary

In 1986, Smyth conjectured an elegant classification of the tuples of integer coefficients that appear in a linear relation among Galois conjugates over Q. Thirty-seven years later, the conjecture ...

In 1986, Smyth conjectured an elegant classification of the tuples of integer coefficients that appear in a linear relation among Galois conjugates over Q. Thirty-seven years later, the conjecture remains open with little direct progress made on it. This thesis compiles evidence in favor of the conjecture, including a proof of a function field analogue and a proposed number field generalization. Additionally, we establish a surprising connection between the conjecture and the recent notion of slice rank from additive combinatorics, via Strassen's asymptotic spectrum. We show that Smyth's Conjecture would be implied by certain "Smyth tensors'' having full asymptotic slice rank, and we prove that the Smyth tensors do have full slice rank. We discuss the obstacle to extending this argument to asymptotic slice rank. Finally, we provide a counterexample to an old conjecture of Brualdi and Csima regarding the support patterns of stochastic tensors.

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