An exponential sum estimate for systems with linear polynomials
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.

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DOI : https://doi.org/10.5802/jtnb.1035
Classification : 11L07,  11P55
Mots clés : Hardy–Littlewood circle method, exponential sum estimate
@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},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1035/}
}
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

[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 | Zbl 1383.11039

[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 | Zbl 06790208

[4] Brian Cook; Ákos Magyar Diophantine equations in the primes, Invent. Math., Volume 198 (2014) no. 3, pp. 701-737 | Zbl 1360.11063

[5] Wolfgang M. Schmidt The density of integer points on homogeneous varieties, Acta Math., Volume 154 (1985) no. 3-4, pp. 243-296 | Zbl 0561.10010

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

[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.) | Article | Zbl 06958602