Abstract:
We consider monic (with higher coefficient 1) polynomials of fixed degree $n$ over an arbitrary finite field $GF(q)$, where $q\ge2$ is a prime number or a power of a prime number. It is assumed that on the set $\mathscr F_n=\{f_n\}$ of all $q^n$ such polynomials the uniform measure is defined which assigns the probability $q^{-n}$ to each polynomial. For an arbitrary polynomial $f_n\in\mathscr F_n$, its local structure $\mathscr K_n=\mathscr K(f_n)$ is defined as the set of multiplicities of all irreducible factors in the canonical decomposition of $f_n$, and various structural characteristics of a polynomial (its exact and asymptotic as $n\to\infty$ distributions) which are functionals of $\mathscr K_n$ are studied. Directions of possible further research are suggested.
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