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

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 $$\gamma \overline{{\mathbb {F}}_q(t)}$$ . call such an n-tuple Smyth tuple. Our main theorem gives affirmative answer to function field analogue of 1986 conjecture (J Numb Theory 23, 243–254, 1986) over $${\mathbb {Q}}$$ showed certain local conditions on $$a_i$$ are necessary and conjectured they sufficient. result is analogous sufficient {F}}_q(t)$$ which we show using combinatorial characterization tuples from 1986). also formulate generalization Smyth’s Conjecture arbitrary number not straightforward due subtlety occurring at archimedean places.