  A non-singular complete irreducible algebraic curve $F_{k,n}$, defined over
an algebraically closed field $K$, is called a generalized Fermat curve of type
$(k,n)$, where $n, k \geq 2$ are integers and $k$ is relatively prime to the
characteristic $p$ of $K$, if it admits a group $H \cong {\mathbb Z}_{k}^{n}$
of automorphisms such that $F_{k,n}/H$ is isomorphic to ${\mathbb P}_{K}^{1}$
and it has exactly $(n+1)$ cone points, each one of order $k$. By the
Riemann-Hurwitz-Hasse formula, $F_{k,n}$ has genus at least one if and only if
$(k-1)(n-1) >1$. In such a situation, we construct a basis, called an standard
basis, of its space $H^{1,0}(F_{k,n})$ of regular forms, containing a subset of
cardinality $n+1$ that provides an embedding of $F_{k,n}$ into ${\mathbb
P}_{K}^{n}$ whose image is the fiber product of $(n-1)$ classical Fermat curves
of degree $k$. For $p=2$, we obtain a lower bound (which is sharp for $n=2,3$)
for the dimension of the space of the exact one-forms, that is, the kernel of
the Cartier operator. Also, we done this for $p=3$, $k=2$ and $n=4$.
