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### Further results on derived sequences.

Journal of Integer Sequences [electronic only]

### Multidimensional self-affine sets: non-empty interior and the set of uniqueness

Studia Mathematica

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 $|detM|\ge {2}^{-1/d}$, 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...

### Patterns and periodicity in a family of resultants

Journal de Théorie des Nombres de Bordeaux

Given a monic degree $N$ polynomial $f\left(x\right)\in ℤ\left[x\right]$ and a non-negative integer $\ell$, we may form a new monic degree $N$ polynomial ${f}_{\ell }\left(x\right)\in ℤ\left[x\right]$ 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 ${p}^{N\left(m+1\right)}$ divides the resultant of ${f}_{{p}^{m}}$ and ${f}_{{p}^{n}}$. We then consider the function $\left(j,k\right)↦Res\left({f}_{j},{f}_{k}\right)\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}{p}^{m}$. 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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