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The distance to an irreducible polynomial, II

Authors: Michael Filaseta and Michael J. Mossinghoff
Journal: Math. Comp. 81 (2012), 1571-1585
MSC (2010): Primary 11C08; Secondary 11R09, 11Y40
Published electronically: December 19, 2011
MathSciNet review: 2904591
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Abstract: P. Turán asked if there exists an absolute constant $ C$ such that for every polynomial $ f\in \mathbb{Z}[x]$ there exists an irreducible polynomial $ g\in \mathbb{Z}[x]$ with $ \deg (g)\leq \deg (f)$ and $ L(f-g)\leq C$, where $ L(\cdot )$ denotes the sum of the absolute values of the coefficients. We show that $ C=5$ suffices for all integer polynomials of degree at most $ 40$ by investigating analogous questions in $ \mathbb{F}_p[x]$ for small primes $ p$. We also prove that a positive proportion of the polynomials in $ \mathbb{F}_2[x]$ have distance at least $ 4$ to an arbitrary irreducible polynomial.

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Additional Information

Michael Filaseta
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

Michael J. Mossinghoff
Affiliation: Department of Mathematics, Davidson College, Davidson, North Carolina 28035-6996

Keywords: Turán’s problem, irreducible polynomial, distance
Received by editor(s): July 28, 2010
Received by editor(s) in revised form: March 12, 2011
Published electronically: December 19, 2011
Additional Notes: Research of the second author supported in part by NSA grant number H98230-08-1-0052.
Article copyright: © Copyright 2011 American Mathematical Society

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