## Abstract

The breakthrough paper of Croot et al. on progression-free sets in

**Z**^{n}_{4}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***of degree less than n that is free of solutions to the equation*_{q}**∑**^{k}_{i=1}**a**_{i}f^{r}_{i}=**0**, where the coefficients ai are polynomials that sum to 0 and the number of variables satisfies**≥2***k**r*^{2}+**1**. The bound we obtain is of the form*q*for some constant^{cn}*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**≥*r*^{2}+1 variables.Original language | English |
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Article number | hax025 |

Pages (from-to) | 1395-1398 |

Number of pages | 4 |

Journal | Quarterly Journal of Mathematics |

Volume | 68 |

Issue number | 4 |

Early online date | 23 Jun 2017 |

DOIs | |

Publication status | Published - Dec 2017 |