On an iterated construction of irreducible polynomials over finite fields of even characteristic by Kyuregyan

Simone Ugolini

Displaying similar documents to “On an iterated construction of irreducible polynomials over finite fields of even characteristic by Kyuregyan”

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We show explicit forms of the Bertini-Noether reduction theorem and of the Hilbert irreducibility theorem. Our approach recasts in a polynomial context the geometric Grothendieck good reduction criterion and the congruence approach to HIT for covers of the line. A notion of “bad primes” of a polynomial P ∈ ℚ[T,Y] irreducible over ℚ̅ is introduced, which plays a central and unifying role. For such a polynomial P, we deduce a new bound for the least integer t₀ ≥ 0 such that P(t₀,Y) is...

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We describe the polynomials P ∈ ℂ[x,y] such that $P (1 / v ⁿ, A ₁ v ⁿ + A ₂ v^{2 n} + . . . + A_{m - 1} v^{n (m - 1)} + v^{n m - k} w) \in ℂ [v, w]$ . As applications we give new examples of bad field generators and examples of families of polynomials with smooth and irreducible fibers.

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We prove that there exist at least cd⁵ monic irreducible nonreciprocal polynomials with integer coefficients of degree at most d whose Mahler measures are smaller than 2, where c is some absolute positive constant. These polynomials are constructed as nonreciprocal divisors of some Newman hexanomials $1 + x^{r ₁} + \dots + x^{r ₅}$ , where the integers 1 ≤ r₁ < ⋯ < r₅ ≤ d satisfy some restrictions including $2 r_{j} < r_{j + 1}$ for j = 1,2,3,4. This result improves the previous lower bound cd³ and seems to be closer to the correct...

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Turan’s problem asks what is the maximal distance from a polynomial to the set of all irreducible polynomials over Z. It turns out it is sufficient to consider the problem in the setting of F2. Even though it is conjectured that there exists an absolute constant C such that the distance L(f - g) <= C, the problem remains open. Thus it attracts different approaches, one of which belongs to Lee, Ruskey and Williams, who study what the probability is for a set of polynomials ‘resembling’...