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Polynomial equations in Fq[t]

Research output: Contribution to journalArticle

  • Pierre Bienvenu
Original languageEnglish
Article numberhax025
Pages (from-to)1395-1398
Number of pages4
JournalQuarterly Journal of Mathematics
Issue number4
Early online date23 Jun 2017
DateAccepted/In press - 12 May 2017
DateE-pub ahead of print - 23 Jun 2017
DatePublished (current) - Dec 2017


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 kr2+1 variables.

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