Bivariate trinomials over finite fields

Abstract

We study the number of points in the family of plane curves defined by a trinomial \[ \mathcal{C}(\alpha,\beta)= \{(x,y)\in\mathbb{F}_q^2\,:\,\alpha x^{a_{11}}y^{a_{12}}+\beta x^{a_{21}}y^{a_{22}}=x^{a_{31}}y^{a_{32}}\} \] with fixed exponents (not collinear) and varying coefficients over finite fields. We prove that each of these curves has an almost predictable number of points, given by a closed formula that depends on the coefficients, exponents, and the field, with a small error term $N(\alpha,\beta)$ that is bounded in absolute value by $2\tilde{g}q^{1/2}$, where $\tilde{g}$ is a constant that depends only on the exponents and the field. A formula for $\tilde{g}$ is provided, as well as a comparison of $\tilde{g}$ with the genus $g$ of the projective closure of the curve over $\overline{\mathbb{F}_q}$. We also give several linear and quadratic identities for the numbers $N(\alpha,\beta)$ that are strong enough to prove the estimate above, and in some cases, to characterize them completely.