Fibre products of supersingular curves and the enumeration of irreducible polynomials with prescribed coefficients
| Autor: | Faruk Göloğlu, Emrah Sercan Yılmaz, Robert Granger, Omran Ahmadi, Gary McGuire |
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| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
Pure mathematics
0102 computer and information sciences Base field Mathematical proof irreducible polynomials 01 natural sciences Theoretical Computer Science Binary fields 12Y05 14H99 prescribed coefficients Enumeration FOS: Mathematics Number Theory (math.NT) 0101 mathematics binary fields Mathematics Algebra and Number Theory Degree (graph theory) Mathematics - Number Theory Applied Mathematics 010102 general mathematics General Engineering Zero (complex analysis) Finite field characteristic polynomial of Frobenius 010201 computation theory & mathematics Algebraic curve Supersingular curves |
| Popis: | 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 $n$. Comment: 29 pages. Final version. To appear in Finite Fields and Their Applications |
| Databáze: | OpenAIRE |
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