Linear relations among Galois conjugates over ${\mathbb {F}}_q(t)$

Abstract

We classify the coefficients $(a_1, \ldots , a_n) \in {\mathbb {F}}_q[t]^n$ that 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 (J Numb Theory 23, 243–254, 1986) 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 from Smyth (J Numb Theory 23, 243–254, 1986). 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.