staple
With cedram.org

Search the site

Table of contents for this issue | Previous article | Next article
Todd Cochrane; Jeremy Coffelt; Christopher Pinner
A system of simultaneous congruences arising from trinomial exponential sums
Journal de théorie des nombres de Bordeaux, 18 no. 1 (2006), p. 59-72, doi: 10.5802/jtnb.533
Article PDF | Reviews MR 2245875 | Zbl 05070447

Résumé - Abstract

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

$$\displaystyle M\le 3d^2(p-1)^2+25hk\ell (p-1).$$

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

$$\displaystyle \left|\sum _{x=1}^{p-1}\chi (x) e^{2\pi i f(x)/p}\right| \le 3^{\frac{1}{4}}d^{\frac{1}{2}}p^{\frac{7}{8}} + \sqrt{5} \left(hk\ell \right)^{\frac{1}{4}}p^{\frac{5}{8}},$$

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

Bibliography

[1] N. M. Akuliničev, Estimates for rational trigonometric sums of a special type. Doklady Acad. Sci. USSR 161 (1965), 743–745. English trans in Doklady 161, no. 4 (1965), 480–482.  MR 177956 |  Zbl 0127.02102
[2] T. Cochrane & C. Pinner, An improved Mordell type bound for exponential sums. Proc. Amer. Math. Soc. 133 (2005), 313–320.  MR 2093050 |  Zbl 1068.11053
[3] T. Cochrane, J. Coffelt & C. Pinner, A further refinement of Mordell’s bound on exponential sums. Acta Arith. 116 (2005), 35–41.  MR 2114903 |  Zbl 1082.11050
[4] R. Lidl & H. Niederreiter, Finite Fields. Encyclopedia of mathematics and its applications, Addison-Wesley, 1983.  MR 746963 |  Zbl 0554.12010
[5] L. J. Mordell, On a sum analogous to a Gauss’s sum. Quart. J. Math. 3 (1932), 161–167.  Zbl 0005.24603 |  JFM 58.0191.02
[6] A. Weil, On some exponential sums. Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204–207.  MR 27006 |  Zbl 0032.26102
[7] T. Wooley, A note on simultaneous congruences. J. Number Theory 58 (1996), no. 2, 288–297.  MR 1393617 |  Zbl 0852.11017