Fibre products of supersingular curves and the enumeration of irreducible polynomials with prescribed coefficients
2016
Abstrakt
For any positive integers n >= 3, r >= 1 we present formulae for the number of irreducible polynomials of degree n over the finite field F(2)r where the coefficients of x(n-1), z(n-2) and x(n-3) are zero. Our proofs involve counting the number of points on certain algebraic curves over finite fields, a technique which arose from Fourier-analysing the known formulae for the F-2 base field cases, reverse-engineering an economical new proof and then extending it.
This approach gives rise to fibre products of supersingular curves and makes explicit why the formulae have period 24 in n.