Birch's theorem over function fields with quadratically many variables

Abstract

For smooth hypersurfaces over rational function fields of characteristic greater than the degree and with sufficiently large constant field, we improve the number of variables required in Birch's theorem from an exponential function of the degree to a quadratic one. This agrees, up to constants, with the sharp quadratic threshold for the unconditional existence of rational points on smooth hypersurfaces. Drawing on an idea of Pugin developed by Sawin for Waring's problem, we treat the minor arcs using complete exponential sums over finite fields; a result of Katz reduces the required cancellation to obtaining lower bounds for the codimensions of certain singular loci. Our main innovation is a new method for proving these bounds: we introduce the notion of multiplication rank for the linear functionals indexing these exponential sums and combine the resulting rank stratification with a weighted degeneration of the Jacobian equations to obtain a codimension estimate that grows linearly with multiplication rank.