Spectrally Arbitrary Patterns Over Various Rings

Jillian Louise Glassett

A pattern $\\mathcal{A}$ is a matrix where the location, but not the magnitude, of the nonzero entries are known. A subpattern of $\\mathcal{A}$, say $\\mathcal{B}$, is pattern where a nonzero entry from $\\mathcal{A}$ may be zero in $\\mathcal{B}$. Pattern $\\mathcal{A}$ is spectrally arbitrary over $\\mathscr{R}$, a commutative ring with unity, if for each $n$-th degree monic polynomial $f(x)\\in\\mathscr{R}[x]$, there exists a matrix $A$ over $\\mathscr{R}$ with pattern $\\mathcal{A}$ where the characteristic polynomial $p_A(x)=f(x)$. Similarly, a pattern $\\mathcal{A}$ is relaxed spectrally arbitrary over $\\mathscr{R}$ if for each $n$-th degree monic polynomial $f(x)\\in\\mathscr{R}[x]$, there exists a matrix $A$ over $\\mathscr{R}$ with either pattern $\\mathcal{A}$ or a subpattern of $\\mathcal{A}$ where the characteristic polynomial $p_A(x)=f(x)$. We evaluate how the structure of rings, compared to the structure of fields, affects how we determine if a pattern is spectrally arbitrary. Using these results, we consider whether a pattern $\\mathcal{A}$ that is spectrally arbitrary over $\\mathscr{R}$ is spectrally arbitrary or relaxed spectrally arbitrary over another commutative ring $\\mathscr{S}$ with unity, establishing some results using ring homomorphisms. Our results imply that a spectrally arbitrary pattern over the integers, $\\mathbb{Z}$, is relaxed spectrally arbitrary over the integers modulo $m$, $\\mathbb{Z}/m\\mathbb{Z}$, and spectrally arbitrary over $\\mathbb{Q}$. Similarly, a spectrally arbitrary pattern over the $p$-adic integers, $\\mathbb{Z}_p$ for some prime $p$, is spectrally arbitrary over the $p$-adic numbers, $\\mathbb{Q}_p$, and the finite field of order $p$, $\\mathbb{F}_p$. We establish the minimum number of nonzero entries necessary for $n\\times n$ pattern to be relaxed spectrally arbitrary over a ring when $n\\leq4$. We also obtain some necessary conditions on the digraph of a relaxed spectrally arbitrary pattern over $\\mathbb{F}_2$ and $\\mathbb{Z}$. The combination of all our results provide us with some insight on how mappings preserve spectrally arbitrariness. Additionally, they provide a library of $3\\times 3$ irreducible spectrally arbitrary and relaxed spectrally arbitrary patterns over various rings, along with a list of $2\\times 2$ and $3\\times 3$ patterns that are not relaxed spectrally arbitrary over $\\mathbb{Z}$. Finally, we consider numerous open questions related to finite rings, relaxed spectrally arbitrary patterns, and potential techniques for finding spectrally arbitrary patterns over $\\mathbb{Q}_p$.

Spectrally Arbitrary Patterns Over Various Rings

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