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Polynomial equations in Fq[t]

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Polynomial equations in Fq[t]. / Bienvenu, Pierre.

In: Quarterly Journal of Mathematics, Vol. 68, No. 4, hax025, 12.2017, p. 1395-1398.

Research output: Contribution to journalArticle

Harvard

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

APA

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

Vancouver

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

Author

Bienvenu, Pierre. / Polynomial equations in Fq[t]. In: Quarterly Journal of Mathematics. 2017 ; Vol. 68, No. 4. pp. 1395-1398.

Bibtex

@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",

}

RIS - suitable for import to EndNote

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 -