Research output: Contribution to journal › Article

**Polynomial equations in F_{q}[t].** / Bienvenu, Pierre.

Research output: Contribution to journal › Article

Bienvenu, P 2017, 'Polynomial equations in **F**_{q}[*t*]' *Quarterly Journal of Mathematics*, vol. 68, no. 4, hax025, pp. 1395-1398. https://doi.org/10.1093/qmath/hax025

Bienvenu, P. (2017). Polynomial equations in **F**_{q}[*t*]. *Quarterly Journal of Mathematics*, *68*(4), 1395-1398. [hax025]. https://doi.org/10.1093/qmath/hax025

Bienvenu P. Polynomial equations in **F**_{q}[*t*]. Quarterly Journal of Mathematics. 2017 Dec;68(4):1395-1398. hax025. https://doi.org/10.1093/qmath/hax025

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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",

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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.

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