Geodesic
Smooth Polynomial Solutions to aTernary Additive Equation
Canadian journal of mathematics, Tome 70 (2018) no. 1, pp. 117-141

Voir la notice de l'article provenant de la source Cambridge University Press

Let ${{\mathbf{F}}_{q}}[T]$ be the ring of polynomials over the finite field of $q$ elements and $Y$ a large integer. We say a polynomial in ${{\mathbf{F}}_{q}}[T]$ is $Y$ -smooth if all of its irreducible factors are of degree at most $Y$ . We show that a ternary additive equation $a\,+\,b\,=\,c$ over $Y$ -smooth polynomials has many solutions. As an application, if $S$ is the set of first $s$ primes in ${{\mathbf{F}}_{q}}[T]$ and $s$ is large, we prove that the $S$ -unit equation $u\,+\,v\,=\,1$ has at least $\text{exp}\left( {{s}^{1/6-\in }}\,\text{log}\,\text{q} \right)$ solutions.
DOI : 10.4153/CJM-2017-023-x
Mots-clés : 11T55, 11D04, 11L07, 11T23, smooth number, polynomial over a finite field, circle method 
Ha, Junsoo. Smooth Polynomial Solutions to aTernary Additive Equation. Canadian journal of mathematics, Tome 70 (2018) no. 1, pp. 117-141. doi: 10.4153/CJM-2017-023-x
@article{10_4153_CJM_2017_023_x,
     author = {Ha, Junsoo},
     title = {Smooth {Polynomial} {Solutions} to {aTernary} {Additive} {Equation}},
     journal = {Canadian journal of mathematics},
     pages = {117--141},
     year = {2018},
     volume = {70},
     number = {1},
     doi = {10.4153/CJM-2017-023-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2017-023-x/}
}
TY  - JOUR
AU  - Ha, Junsoo
TI  - Smooth Polynomial Solutions to aTernary Additive Equation
JO  - Canadian journal of mathematics
PY  - 2018
SP  - 117
EP  - 141
VL  - 70
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2017-023-x/
DO  - 10.4153/CJM-2017-023-x
ID  - 10_4153_CJM_2017_023_x
ER  - 
%0 Journal Article
%A Ha, Junsoo
%T Smooth Polynomial Solutions to aTernary Additive Equation
%J Canadian journal of mathematics
%D 2018
%P 117-141
%V 70
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2017-023-x/
%R 10.4153/CJM-2017-023-x
%F 10_4153_CJM_2017_023_x

[1] [1] Drappeau, Sary, Sur les solutions friables de l'équation a + b = c. Math. Proc. Cambridge Philos. Soc. 154(2013), no. 3, 439–463. http://dx.doi.Org/10.1017/S0305004112000643 Google Scholar

[2] [2] Drappeau, Sary, Théorèmes de type Fouvry-Iwaniec pour les entiers friables. Compos. Math. 151(2015), no. 5, 828–862. http://dx.doi.Org/10.1112/S0010437X14007933 Google Scholar

[3] [3] Erdös, P., Stewart, C. L., and Tijdeman, R., Some Diophantine equations with many solutions. Compositio Math. 66(1988), no. 1, 37–56. Google Scholar

[4] [4] Evertse, J.-H., On equations in S-units and the Thue-Mahler equation. Invent. Math. 75(1984), no. 3, 561–584. Google Scholar | DOI

[5] [5] Ha, Junsoo, Some problems in multiplicative number theory. Ph.D. thesis, Stanford University, 2014. Google Scholar

[6] [6] Harper, A. J., On finding many solutions to S-unit equations by solving linear equations on average. arxiv:1108.3819 Google Scholar

[7] [7] Harper, A. J., Bombieri-Vinogradov and Barban-Davenport-Halberstam type theorems for smooth numbers. arxiv:1208.5992 Google Scholar

[8] [8] Harper, A. J., Minor arcs, mean values, and restriction theory for exponential sums over smooth numbers. Compos. Math. 152(2016), no. 6,1121–1158. http://dx.doi.Org/10.1112/S0010437X1 5007782 Google Scholar

[9] [9] Hayes, David R., The distribution of irreducibles in GF[q, x]. Trans. Amer. Math. Soc. 117(1965),101–127. Google Scholar

[10] [10] Hayes, David R., The expression of a polynomial as a sum of three irreducibles. Acta Arith. 11(1966), 461–488. Google Scholar

[11] [11] Hayes, David R., Explicit class field theory for rational function fields. Trans. Amer. Math. Soc. 189(1974), 77–91. http://dx.doi.Org/10.1090/S0002-9947-1 974-0330106-6 Google Scholar

[12] [12] Hsu, Chih-Nung, The distribution of irreducible polynomials in F9 [t]. J. Number Theory 61(1996),no. 1, 85–96. Google Scholar | DOI

[13] [13] Konyagin, S. and Soundararajan, Kannan, Two S-unit equations with many solutions. J. Number Theory 124(2007), no. 1, 193–199. http://dx.doi.Org/10.1016/j.jnt.2006.07.017 Google Scholar

[14] [14] Kubota, R. M., Waring's problem for F[x]. Dissertationes Math. (Rozprawy Mat.) 117(1974), 60. Google Scholar

[15] [15] Lagarias, Jeffrey C. and Soundararajan, Kannan, Smooth solutions to the abc equation: the xyz conjecture. J. Théor. Nombres Bordeaux 23(2011), no. 1, 209–234. Google Scholar | DOI

[16] [16] Lagarias, Jeffrey C., Counting smooth solutions to the equation A + B = C. Proc. Lond. Math. Soc. 104(2012), no. 4, 770–798. http://dx.doi.Org/10.1112/plms/pdr037 Google Scholar

[17] [17] Liu, Yu-Ru and Wooley, Trevor D., Waring's problem in function fields. J. Reine Angew. Math. 638(2010), 1–67. Google Scholar | DOI

[18] [18] Manstavicius, E., Remarks on elements of semigroups that are free of large prime factors. Liet. Mat. Rink. 32(1992), no. 4, 512–525. Google Scholar

[19] [19] Manstavicius, E., Semigroup elements free of large prime factors. In: New trends in probability and statistics. Vol. 2 (Palanga, 1991), pages 135–153. VSP, Utrecht, 1992. Google Scholar

[20] [20] Ramakrishnan, Dinakar and Valenza, Robert J., Fourier analysis on number fields, Graduate Texts in Mathematics, 186. Springer-Verlag, New York, 1999. Google Scholar

[21] [21] Rosen, Michael, Number theory in function fields. Graduate Texts in Mathematics, 210. Springer-Verlag, New York, 2002. Google Scholar

Cité par Sources :