Polynomial equations in F_q[t]

The breakthrough paper of Croot et al. on progression-free sets in Zⁿ₄ introduced a polynomial method that has generated a wealth of applications, such as Ellenberg and Gijswijt's solutions to the cap set problem. Using this method, we bound the size of a set of polynomials over F_q of degree less than n that is free of solutions to the equation ∑^k_i=1a_if^r_i=0, where the coefficients ai are polynomials that sum to 0 and the number of variables satisfies k≥2r²+1. The bound we obtain is of the form q^cn for some constant c<1. This is in contrast to the best bounds known for the corresponding problem in the integers, which offer only a logarithmic saving, but work already with as few as k≥r²+1 variables.