### Patterns and periodicity in a family of resultants

Kevin G. Hare, David McKinnon, Christopher D. Sinclair (2009)

Journal de Théorie des Nombres de Bordeaux

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Given a monic degree $N$ polynomial $f\left(x\right)\in \mathbb{Z}\left[x\right]$ and a non-negative integer $\ell $, we may form a new monic degree $N$ polynomial ${f}_{\ell}\left(x\right)\in \mathbb{Z}\left[x\right]$ 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 $(j,k)\mapsto Res({f}_{j},{f}_{k})\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}{p}^{m}$. 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. ...