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Kevin G. Hare; David McKinnon; Christopher D. Sinclair
Patterns and periodicity in a family of resultants
Journal de théorie des nombres de Bordeaux, 21 no. 1 (2009), p. 215-234, doi: 10.5802/jtnb.667
Article PDF | Reviews MR 2537713 | Zbl pre05620678

Résumé - Abstract

Given a monic degree $N$ polynomial $f(x) \in \mathbb{Z}[x]$ and a non-negative integer $\ell $, we may form a new monic degree $N$ polynomial $f_{\ell }(x) \in \mathbb{Z}[x]$ by raising each root of $f$ to the $\ell $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 $(j,k) \mapsto \operatorname{Res}(f_j, f_k) \;\@mod \;p^m$. 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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