Fibre Products of Supersingular Curves and the Enumeration of Irreducible Polynomials with Prescribed Coefficients

For any positive integers $n\geq 3, r\geq 1$ we present formulae for the number of irreducible polynomials of degree $n$ over the finite field $\mathbb{F}_{2^r}$ where the coefficients of $x^{n-1}$, $x^{n-2}$ 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 $\mathbb{F}_2$ 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 $n$.


Published in:
Finite Fields and Their Applications, 42, 128-164
Year:
2016
Publisher:
San Diego, Elsevier
ISSN:
1071-5797
Keywords:
Laboratories:




 Record created 2016-09-05, last modified 2018-01-28

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