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On the Distribution of Irreducible Trinomials

Open Access article
 Printed: Dec 2011
  • Igor E. Shparlinski,
    Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
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We obtain new results about the number of trinomials $t^n + at + b$ with integer coefficients in a box $(a,b) \in [C, C+A] \times [D, D+B]$ that are irreducible modulo a prime $p$. As a by-product we show that for any $p$ there are irreducible polynomials of height at most $p^{1/2+o(1)}$, improving on the previous estimate of $p^{2/3+o(1)}$ obtained by the author in 1989.
Keywords: irreducible trinomials, character sums irreducible trinomials, character sums
MSC Classifications: 11L40, 11T06 show english descriptions Estimates on character sums
11L40 - Estimates on character sums
11T06 - Polynomials

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