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Single polynomials that correspond to pairs of cyclotomic polynomials with interlacing zeros

James McKeeChris Smyth — 2013

Open Mathematics

We give a complete classification of all pairs of cyclotomic polynomials whose zeros interlace on the unit circle, making explicit a result essentially contained in work of Beukers and Heckman. We show that each such pair corresponds to a single polynomial from a certain special class of integer polynomials, the 2-reciprocal discbionic polynomials. We also show that each such pair also corresponds (in four different ways) to a single Pisot polynomial from a certain restricted class, the cyclogenic...

The Lehmer constants of an annulus

Artūras DubickasChris J. Smyth — 2001

Journal de théorie des nombres de Bordeaux

Let M ( α ) be the Mahler measure of an algebraic number α , and V be an open subset of . Then its L ( V ) is inf M ( α ) 1 / deg ( α ) , the infimum being over all non-zero non-cyclotomic α lying with its conjugates outside V . We evaluate L ( V ) when V is any annulus centered at 0 . We do the same for a variant of L ( V ) , which we call the transfinite Lehmer constant L ( V ) .Also, we prove the converse to Langevin’s Theorem, which states that L ( V ) > 1 if V contains a point of modulus 1 . We prove the corresponding result for L ( V ) .

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