BIVARIATE TRINOMIALS OVER FINITE FIELDS

We study the number of points in the family of plane curves defined by a trinomial with fixed exponents 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, the exponents, and the field, with a small error term for which we provide an upper bound in terms of an analog of the genus and the size of the field. We obtain these upper bounds from some linear and quadratic identities that the error terms satisfy. These identities are, in some cases, strong enough to determine the error terms completely.