An exponential sum estimate for systems with linear polynomials
Shuntaro Yamagishi1
1 Department of Mathematics & Statistics Queen’s University Kingston, ON K7L 3N6, Canada
Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 2, pp. 485-499.

Dans son article [5], W. M. Schmidt a obtenu une estimation de somme exponentielle pour les systèmes de polynômes sans polynômes linéaires, qui a ensuite été utilisée pour appliquer la méthode du cercle de Hardy–Littlewood. Nous démontrons une estimation analogue pour les systèmes qui incluent des polynômes linéaires.

In his paper [5], W. M. Schmidt obtained an exponential sum estimate for systems of polynomials without linear polynomials, which was then used to apply the Hardy–Littlewood circle method. We prove an analogous estimate for systems which include linear polynomials.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1035
Classification : 11L07, 11P55
Mots clés : Hardy–Littlewood circle method, exponential sum estimate
Shuntaro Yamagishi 1

1 Department of Mathematics & Statistics Queen’s University Kingston, ON K7L 3N6, Canada
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{JTNB_2018__30_2_485_0,
     author = {Shuntaro Yamagishi},
     title = {An exponential sum estimate for systems  with linear polynomials},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {485--499},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {2},
     year = {2018},
     doi = {10.5802/jtnb.1035},
     mrnumber = {3891323},
     zbl = {1443.11160},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1035/}
}
TY  - JOUR
AU  - Shuntaro Yamagishi
TI  - An exponential sum estimate for systems  with linear polynomials
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2018
SP  - 485
EP  - 499
VL  - 30
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1035/
DO  - 10.5802/jtnb.1035
LA  - en
ID  - JTNB_2018__30_2_485_0
ER  - 
%0 Journal Article
%A Shuntaro Yamagishi
%T An exponential sum estimate for systems  with linear polynomials
%J Journal de théorie des nombres de Bordeaux
%D 2018
%P 485-499
%V 30
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1035/
%R 10.5802/jtnb.1035
%G en
%F JTNB_2018__30_2_485_0
Shuntaro Yamagishi. An exponential sum estimate for systems  with linear polynomials. Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 2, pp. 485-499. doi : 10.5802/jtnb.1035. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1035/

[1] Bryan J. Birch Forms in many variables, Proc. R. Soc. Lond., Ser. A, Volume 265 (1962), pp. 245-263 | Zbl

[2] Timothy D. Browning; Roger Heath-Brown Fooms in many variables and differing degrees, J. Eur. Math. Soc., Volume 19 (2017) no. 2, pp. 357-394 | DOI | Zbl

[3] Timothy D. Browning; Sean M. Prendiville Improvements in Birch’s theorem on forms in many variables, J. Reine Angew. Math., Volume 731 (2017), pp. 203-234 | MR | Zbl

[4] Brian Cook; Ákos Magyar Diophantine equations in the primes, Invent. Math., Volume 198 (2014) no. 3, pp. 701-737 | DOI | MR | Zbl

[5] Wolfgang M. Schmidt The density of integer points on homogeneous varieties, Acta Math., Volume 154 (1985) no. 3-4, pp. 243-296 | DOI | MR | Zbl

[6] Stanley Yao Xiao; Shuntaro Yamagishi Zeroes of polynomials in many variables with prime inputs (2015) (https://arxiv.org/abs/1512.01258) | Zbl

[7] Shuntaro Yamagishi Prime solutions to polynomial equations in many variables and differing degrees, Forum Math. Sigma, Volume 6 (2018), e19, 89 pages (Art. ID. e19, 89 p.) | DOI | MR | Zbl

Cité par Sources :