A system of simultaneous congruences arising from trinomial exponential sums
Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 1, pp. 59-72.

Pour p un nombre premier et <k<h<p des entiers positifs avec d=(h,k,,p-1), nous montrons que M, le nombre de solutions simultanées x,y,z,w dans p * de x h +y h =z h +w h , x k +y k =z k +w k , x +y =z +w , satisfait à

M3d2(p-1)2+25hk(p-1).

Quand hk=o(pd 2 ), nous obtenons un comptage asymptotique précis de M. Cela conduit à une nouvelle borne explicite pour des sommes d’exponentielles tordues

x=1p-1χ(x)e2πif(x)/p314d12p78+5hk14p58,

pour des trinômes f=ax h +bx k +cx , et à des résultats sur la valeur moyenne de telles sommes.

For a prime p and positive integers <k<h<p with d=(h,k,,p-1), we show that M, the number of simultaneous solutions x,y,z,w in p * to x h +y h =z h +w h , x k +y k =z k +w k , x +y =z +w , satisfies

M3d2(p-1)2+25hk(p-1).

When hk=o(pd 2 ) we obtain a precise asymptotic count on M. This leads to the new twisted exponential sum bound

x=1p-1χ(x)e2πif(x)/p314d12p78+5hk14p58,

for trinomials f=ax h +bx k +cx , and to results on the average size of such sums.

@article{JTNB_2006__18_1_59_0,
     author = {Cochrane, Todd and Coffelt, Jeremy and Pinner, Christopher},
     title = {A system of simultaneous congruences arising from trinomial exponential sums},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {59--72},
     publisher = {Universit\'e Bordeaux 1},
     volume = {18},
     number = {1},
     year = {2006},
     doi = {10.5802/jtnb.533},
     mrnumber = {2245875},
     zbl = {05070447},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.533/}
}
Cochrane, Todd; Coffelt, Jeremy; Pinner, Christopher. A system of simultaneous congruences arising from trinomial exponential sums. Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 1, pp. 59-72. doi : 10.5802/jtnb.533. http://www.numdam.org/articles/10.5802/jtnb.533/

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