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arXiv · 1605.07229

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

Omran Ahmadi·Faruk Gologlu·Robert Granger·Gary McGuire·Emrah Sercan Yilmaz
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Abstract

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 $n$.

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Omran Ahmadi, Faruk Gologlu, Robert Granger, Gary McGuire, Emrah Sercan Yilmaz. 2016-08-09. Fibre Products of Supersingular Curves and the Enumeration of Irreducible Polynomials with Prescribed Coefficients. https://arxiv.org/abs/1605.07229

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