On some properties of polynomials over finite fields

We consider polynomials over a finite field. The polynomials of one variables are called transformations. We investigate the polynomials of several variables which do not change under replacement of each variables by some transformation. Such polynomials are called invariant with respect to transformations of variables. We investigate the form of the polynomials invariant with respect to connected transformations. A transformation is called connected if for any two elements $a_1$ and $a_2$ of the field there exist integers $m_1$ and $m_2$ such that the $m_1$-fold iteration of the transformation of $a_1$ coincides with the $m_2$-fold iteration of the transformation of $a_2$.
We consider some integer-valued characteristics of polynomials of several variables, namely, the rank and the weight. We prove the following necessary property of polynomials invariant with respect to connected transformations: if the integers $r$ and $w$ are, respectively, the rank and the weight of a polynomial invariant with respect to connected transformations, then $w^q\ge2^r$, where $q$ is a constant depending on transformations and does not exceed the number of elements of the field.