Rational trigonometric sums along a curve

S. A. Stepanov

Zapiski Nauchnykh Seminarov POMI, Automorphic functions and number theory. Part II, Tome 134 (1984), pp. 232-251

Résumé

Under certain assumptions on the polynomials $f(x, y)$ and $g(x, y)$ the following estimate $$ \left|\sum_{\substack{x,y=1\\ f(x,y)\equiv0\pmod q}}e^\frac{2\pi ig(x, y)}q\right|\ll q^{1-\frac1{N+1}+\varepsilon},\quad\varepsilon>0 $$ is proved. There $N$ is the maximum over all $p|q$ of the intersection index of the curves $f(x, y)\equiv0\pmod p$ and $\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}-\frac{\partial g}{\partial y}\frac{\partial f}{\partial x}\equiv0\pmod p$ in the finite field of $p$ elements.