I am interested in the following problem: I have a finite field $F\_q$, two positive integers
$n>m$ and elements $a\_1,...,a\_m\in F\_q$. How many of the polynomials
$x^n+a\_1x^{n-1}+...+a\_mx^{n-m}+c\_{m+1}x^{n-m-1}+...+c\_n,c\_i\in F\_q$ are irreducible?
What are the best known estimates, esp. for $q$ fixed and $m,n\to\infty$?



Number of irreducible polynomials with some coefficients fixed over a finite field
This is similar to counting irreducibles in arithmetic progressions modulo $x^m$ (once you replace $x$ by $1/x$). You can turn the problem into counting rational points on a curve (coming from a "cyclotomic function field" in the sense of Carlitz) over $F\_{q^n}$ and get an estimate $q^n/n + O(gq^{n/2})$, where $g$ is the genus of the curve. Unfortunately $g$ grows like $mq^m$ so you only get good estimates for $m$ small and nothing when $m$ gets close to $n/2$. There are plenty of papers on this (e.g. by S. Cohen). There is also some experimental work by Panario et al.. Mathscinet should help you locate these.


