** Authors:**
GAETANA RESTUCCIA, ROSANNA UTANO

** Abstract: **
Let \{D_1,..., D_n\} be a system of derivations of a
k-algebra A, k a field of characteristic p > 0, defined by a coaction
\delta of the Hopf algebra H_c = k[X_1,..., X_n]/(X_1^p,..., X_n^p), c
\in \{0,1\}, the Lie Hopf algebra of the additive group and the
multiplicative group on A, respectively. If there exist x_1, \dots,
x_n \in A, with the Jacobian matrix (D_i(x_j)) invertible, [D_i,D_j] =
0, D_i^p = cD_i, c \in \{0, 1\}, 1 \leq i, j \leq n, we obtain
elements y_1,..., y_n \in A, such that D_i(y_j) = \delta_{ij}(1 +
cy_i), using properties of H_c-Galois extensions. A concrete
structure theorem for a commutative k-algebra A, as a free module on
the subring A^{\delta} of A consisting of the coinvariant elements
with respect to \delta, is proved in the additive case.

** Keywords: **
Hopf algebras, derivations, Jacobian criterion

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