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

@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",

}

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 -