Asymptotic estimates for rational spaces on hypersurfaces in function fields

Let q [ t ] denote the polynomial ring over the finite field q . For c 1 , …, c s ∈ q [ t ]\{0}, we consider the hypersurface H : c 1 z 1 k + … + c s z s k = 0. Let I P denote the subset of q [ t ] containing all polynomials of degree strictly less than P . Let N s , k , d , c ( P ) denote the number of d ‐tuples (x 1 ,… ,x d ) ∈ ( I P s ) d such that z = 1 x 1 +…+ d x d lies on H for all 1 , …, d ∈ q ( t ). In this paper, we apply a variant of the Hardy–Littlewood circle method to establish an asymptotic formula for N s , k , d , c ( P ). As a consequence, we see that under certain conditions there are infinitely many d ‐dimensional rational spaces on H .