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

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

M3d 2 (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=1 p-1 χ(x)e 2πif(x)/p 3 1 4 d 1 2 p 7 8 +5hk 1 4 p 5 8 ,

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

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

M3d 2 (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=1 p-1 χ(x)e 2πif(x)/p 3 1 4 d 1 2 p 7 8 +5hk 1 4 p 5 8 ,

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

DOI: 10.5802/jtnb.533
Cochrane, Todd 1; Coffelt, Jeremy 1; Pinner, Christopher 1

1 Department of Mathematics Kansas State University Manhattan, KS 66506, USA
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Cochrane, Todd; Coffelt, Jeremy; Pinner, Christopher. A system of simultaneous congruences arising from trinomial exponential sums. Journal de théorie des nombres de Bordeaux, Volume 18 (2006) no. 1, pp. 59-72. doi : 10.5802/jtnb.533. http://archive.numdam.org/articles/10.5802/jtnb.533/

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