Linear Relations Among Galois Conjugates Over $\mathbb{F}_q(t)$

TL;DR

This paper classifies linear relations among Galois conjugates over function fields, confirming a conjecture by Smyth in the context of $\, ext{F}_q(t)$, and extends the discussion to arbitrary number fields.

Contribution

It provides a complete characterization of Smyth tuples over $\, ext{F}_q(t)$, solving a function field analogue of Smyth's 1986 conjecture.

Findings

Necessary and sufficient local conditions for Smyth tuples over $\, ext{F}_q(t)$

A combinatorial characterization of Smyth tuples

Formulation of a generalized Smyth's Conjecture for arbitrary number fields

Abstract

We classify the coefficients $(a_1,...,a_n) \in \mathbb{F}_q[t]^n$ that can appear in a linear relation $\sum_{i=1}^n a_i \gamma_i =0$ among Galois conjugates $\gamma_i \in \overline{\mathbb{F}_q(t)}$. We call such an $n$-tuple a Smyth tuple. Our main theorem gives an affirmative answer to a function field analogue of a 1986 conjecture of Smyth over $\mathbb{Q}$. Smyth showed that certain local conditions on the $a_i$ are necessary and conjectured that they are sufficient. Our main result is that the analogous conditions are necessary and sufficient over $\mathbb{F}_q(t)$, which we show using a combinatorial characterization of Smyth tuples due to Smyth. We also formulate a generalization of Smyth's Conjecture in an arbitrary number field that is not a straightforward generalization of the conjecture over $\mathbb{Q}$ due to a subtlety occurring at the archimedean places.