Irreducible Polynomials with Coefficients in an Affine Algebraic Set

In this paper, we give error bounds on the number of monic irreducible polynomials $a_0+a_{1}x+\cdots +a_{n-1}{x}^{n-1}+x^n$ over a finite field ${\mathbb{F}}_q$ of degree $n$ with $(a_0,a_1,\dots ,a_{n-1},1)$ lying in a fixed affine algebraic set $V$ of points in ${\mathbb{F}}_q^{n+1}$.