Survey on Counting Special Types of Polynomials

Abstract

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gauß count the remaining ones, approximately and exactly. For polynomials in two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. This survey presents counting results for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the $s$-powerful ones (divisible by the $s$th power of a nonconstant polynomial), the relatively irreducible ones (irreducible but reducible over an extension field), the decomposable ones, and also for reducible space curves. These come as exact formulas and as approximations with relative errors that essentially decrease exponentially in the input size.

Furthermore, a univariate polynomial $f$ is decomposable if $f = g\,\circ \,h$ for some nonlinear polynomials $g$ and $h$. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials.

The tame case, where the characteristic $p$ of ${\mathbb {F}}_{q}$ does not divide $n = \deg f$, is fairly well-understood, and we obtain closely matching upper and lower bounds on the number of decomposable polynomials. In the wild case, where $p$ does divide $n$, the bounds are less satisfactory, in particular when $p$ is the smallest prime divisor of $n$ and divides $n$ exactly twice. The crux of the matter is to count the number of collisions, where essentially different $(g, h)$ yield the same $f$. We present a classification of all collisions at degree $n = p^{2}$ which yields an exact count of those decomposable polynomials.