TY - JOUR
T1 - Polynomial equations in Fq[t]
AU - Bienvenu, Pierre
PY - 2017/12
Y1 - 2017/12
N2 - 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.
AB - 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.
U2 - 10.1093/qmath/hax025
DO - 10.1093/qmath/hax025
M3 - Article
VL - 68
SP - 1395
EP - 1398
JO - Quarterly Journal of Mathematics
JF - Quarterly Journal of Mathematics
SN - 0033-5606
IS - 4
M1 - hax025
ER -