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Currently displaying 1 – 2 of 2

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The distribution of Mahler's measures of reciprocal polynomials.

Sinclair, Christopher D. — 2004

International Journal of Mathematics and Mathematical Sciences

Patterns and periodicity in a family of resultants

Kevin G. Hare; David McKinnon; Christopher D. Sinclair — 2009

Journal de Théorie des Nombres de Bordeaux

Given a monic degree $N$ polynomial $f (x) \in ℤ [x]$ and a non-negative integer $ℓ$ , we may form a new monic degree $N$ polynomial $f_{ℓ} (x) \in ℤ [x]$ by raising each root of $f$ to the $ℓ$ th power. We generalize a lemma of Dobrowolski to show that if $m < n$ and $p$ is prime then $p^{N (m + 1)}$ divides the resultant of $f_{p^{m}}$ and $f_{p^{n}}$ . We then consider the function $(j, k) \mapsto Res (f_{j}, f_{k}) mod p^{m}$ . We show that for fixed $p$ and $m$ that this function is periodic in both $j$ and $k$ , and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given.

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