@article{d55154491ac646328a4fa9129cdb38e0, title = "Polynomial equations in Fq[t]", abstract = "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 Fq of degree less than n that is free of solutions to the equation ∑ki=1aifri=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 qcn 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.", author = "Pierre Bienvenu", year = "2017", month = "12", doi = "10.1093/qmath/hax025", language = "English", volume = "68", pages = "1395--1398", journal = "Quarterly Journal of Mathematics", issn = "0033-5606", publisher = "Oxford University Press", number = "4", }