Sums of three cubes over a function field

Abstract We use a function field version of the circle method to prove that a positive proportion of elements in F q [ t ] $\mathbb {F}_q[t]$ double struck upper F Subscript q Baseline left bracket t right bracket are representable as a sum of three cubes of minimal degree from F q [ t ] $\mathbb {F}_q[t]$ double struck upper F Subscript q Baseline left bracket t right bracket , assuming a suitable form of the Ratios Conjecture and that char ( F q ) > 3 $\operatorname {\mathrm {char}}(\mathbb {F}_q)>3$ char left parenthesis double struck upper F Subscript q Baseline right parenthesis greater than 3 . The analogue of this conjecture for quadratic Dirichlet L -functions is known for large fixed q , via recent developments in homological stability.