A system of simultaneous congruences arising from trinomial exponential sums

Cochrane, Todd, Coffelt, Jeremy, and Pinner, Christopher. "A system of simultaneous congruences arising from trinomial exponential sums." Journal de Théorie des Nombres de Bordeaux 18.1 (2006): 59-72. <http://eudml.org/doc/249632>.

@article{Cochrane2006,
abstract = {For a prime $p$ and positive integers $\ell &lt;k&lt;h&lt;p$ 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.},
affiliation = {Department of Mathematics Kansas State University Manhattan, KS 66506, USA; Department of Mathematics Kansas State University Manhattan, KS 66506, USA; Department of Mathematics Kansas State University Manhattan, KS 66506, USA},
author = {Cochrane, Todd, Coffelt, Jeremy, Pinner, Christopher},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {system of simultaneous congruences; trinomial exponential sums},
language = {eng},
number = {1},
pages = {59-72},
publisher = {Université Bordeaux 1},
title = {A system of simultaneous congruences arising from trinomial exponential sums},
url = {http://eudml.org/doc/249632},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Cochrane, Todd
AU - Coffelt, Jeremy
AU - Pinner, Christopher
TI - A system of simultaneous congruences arising from trinomial exponential sums
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 1
SP - 59
EP - 72
AB - For a prime $p$ and positive integers $\ell &lt;k&lt;h&lt;p$ 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.
LA - eng
KW - system of simultaneous congruences; trinomial exponential sums
UR - http://eudml.org/doc/249632
ER -