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Practical Aurifeuillian factorization

Bill Allombert, Karim Belabas (2008)

Journal de Théorie des Nombres de Bordeaux

We describe a simple procedure to find Aurifeuillian factors of values of cyclotomic polynomials Φ d ( a ) for integers a and d > 0 . Assuming a suitable Riemann Hypothesis, the algorithm runs in deterministic time O ˜ ( d 2 L ) , using O ( d L ) space, where L log ( a + 1 ) .

Preface

J. Berstel, T. Harju, J. Karhumäki (2008)

RAIRO - Theoretical Informatics and Applications

Preservation of log-concavity on summation

Oliver Johnson, Christina Goldschmidt (2006)

ESAIM: Probability and Statistics

We extend Hoggar's theorem that the sum of two independent discrete-valued log-concave random variables is itself log-concave. We introduce conditions under which the result still holds for dependent variables. We argue that these conditions are natural by giving some applications. Firstly, we use our main theorem to give simple proofs of the log-concavity of the Stirling numbers of the second kind and of the Eulerian numbers. Secondly, we prove results concerning the log-concavity of the sum of...

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