A system of simultaneous congruences arising from trinomial exponential sums

Todd Cochrane; Jeremy Coffelt; Christopher Pinner

doi:10.5802/jtnb.533

Todd Cochrane ¹; Jeremy Coffelt ¹; Christopher Pinner ¹

¹ Department of Mathematics Kansas State University Manhattan, KS 66506, USA

Journal de Théorie des Nombres de Bordeaux, Volume 18 (2006) no. 1, pp. 59-72.

Abstract
Résumé

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

M \leq 3 d^{2} {(p - 1)}^{2} + 25 h k ℓ (p - 1) .

When $h k ℓ = o (p d^{2})$ we obtain a precise asymptotic count on $M$ . This leads to the new twisted exponential sum bound

|\sum_{x = 1}^{p - 1} χ (x) e^{2 π i f (x) / p}| \leq 3^{\frac{1}{4}} d^{\frac{1}{2}} p^{\frac{7}{8}} + \sqrt{5} {(h k ℓ)}^{\frac{1}{4}} p^{\frac{5}{8}},

for trinomials $f = a x^{h} + b x^{k} + c x^{ℓ}$ , and to results on the average size of such sums.

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

M \leq 3 d^{2} {(p - 1)}^{2} + 25 h k ℓ (p - 1) .

Quand $h k ℓ = o (p d^{2})$ , nous obtenons un comptage asymptotique précis de $M$ . Cela conduit à une nouvelle borne explicite pour des sommes d’exponentielles tordues

|\sum_{x = 1}^{p - 1} χ (x) e^{2 π i f (x) / p}| \leq 3^{\frac{1}{4}} d^{\frac{1}{2}} p^{\frac{7}{8}} + \sqrt{5} {(h k ℓ)}^{\frac{1}{4}} p^{\frac{5}{8}},

pour des trinômes $f = a x^{h} + b x^{k} + c x^{ℓ}$ , et à des résultats sur la valeur moyenne de telles sommes.

Received: 2004-03-29
Published online: 2008-12-02

MR: 2245875 | Zbl: 05070447

DOI: 10.5802/jtnb.533
Author's affiliations:

Todd Cochrane ¹; Jeremy Coffelt ¹; Christopher Pinner ¹

¹ Department of Mathematics Kansas State University Manhattan, KS 66506, USA

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Todd Cochrane; Jeremy Coffelt; Christopher Pinner. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.533/

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