Is there a way of computing the genus of a parametrized curve over a finite field? For instance I am interested in the genus of the following space curve in the m-dimensional space over $F_{q^k}$ given parametrically as: $(x, x^h, x^{h^2}, \dots , x^{h^{m-1}})$ Here h and m are some fixed constants, say.
I basically work in theoretical computer science, more specifically in algebraic coding theory, so am not that familiar with advanced algebraic geometry, so I apologize in advance if the question is vague. Any help or references in this regard would be appreciated.
The map you describe is also defined over the algebraic closure of $\mathbb F_q$ and is a morphism of algebraic varieties from the affine line to the affine space of dimension $m$. The projection to the first coordinate, when restricted to the image, is the inverse map. So the image is isomorphic to the affine line. You can say it has genus $0$ (usually, the genus is defined for projective curves).