### Further results on derived sequences.

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Let M be a d × d real contracting matrix. We consider the self-affine iterated function system Mv-u, Mv+u, where u is a cyclic vector. Our main result is as follows: if $\left|detM\right|\ge {2}^{-1/d}$, then the attractor ${A}_{M}$ has non-empty interior. We also consider the set ${}_{M}$ of points in ${A}_{M}$ which have a unique address. We show that unless M belongs to a very special (non-generic) class, the Hausdorff dimension of ${}_{M}$ is positive. For this special class the full description of ${}_{M}$ is given as well. This paper continues our work begun...

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.

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