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Counting the powers of a companion matrix that possess nonzero leading principal minors

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1

$begingroup$

Let $p$ be prime, let $A$ be the $ntimes n$ companion matrix associated to a primitive polynomial $f(x)$ of degree $n$ with coefficients in $mathbb Z_p$, and let $T=A^jmid 1leq jleq p^n-1subseteq mathbb Z_p^ntimes n$. Note that $T$ can be identified with the nonzero elements of $GF(p^n)$. Suppose $S$ is the subset of $T$ consisting of matrices possessing the property that every leading principal minor is nonzero. A couple of questions: (1) What is the cardinality of $S$? and (2) Does $S$ have some special meaning as a subset of $GF(p^n)$ or otherwise? Perhaps (1) depends on $f(x)$.

Thanks for your thoughts!

linear-algebra matrices polynomials finite-fields

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1

$begingroup$

Let $p$ be prime, let $A$ be the $ntimes n$ companion matrix associated to a primitive polynomial $f(x)$ of degree $n$ with coefficients in $mathbb Z_p$, and let $T=A^jmid 1leq jleq p^n-1subseteq mathbb Z_p^ntimes n$. Note that $T$ can be identified with the nonzero elements of $GF(p^n)$. Suppose $S$ is the subset of $T$ consisting of matrices possessing the property that every leading principal minor is nonzero. A couple of questions: (1) What is the cardinality of $S$? and (2) Does $S$ have some special meaning as a subset of $GF(p^n)$ or otherwise? Perhaps (1) depends on $f(x)$.

Thanks for your thoughts!

linear-algebra matrices polynomials finite-fields

share|cite|improve this question

asked Mar 13 at 18:11

MurphyMurphy

61

New contributor

Murphy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

$begingroup$

Let $p$ be prime, let $A$ be the $ntimes n$ companion matrix associated to a primitive polynomial $f(x)$ of degree $n$ with coefficients in $mathbb Z_p$, and let $T=A^jmid 1leq jleq p^n-1subseteq mathbb Z_p^ntimes n$. Note that $T$ can be identified with the nonzero elements of $GF(p^n)$. Suppose $S$ is the subset of $T$ consisting of matrices possessing the property that every leading principal minor is nonzero. A couple of questions: (1) What is the cardinality of $S$? and (2) Does $S$ have some special meaning as a subset of $GF(p^n)$ or otherwise? Perhaps (1) depends on $f(x)$.

Thanks for your thoughts!

linear-algebra matrices polynomials finite-fields

share|cite|improve this question

asked Mar 13 at 18:11

MurphyMurphy

61

New contributor

Murphy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

asked Mar 13 at 18:11

MurphyMurphy

61

New contributor

Murphy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

asked Mar 13 at 18:11

MurphyMurphy

61

New contributor

Murphy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 13 at 18:11

MurphyMurphy

61

New contributor

Murphy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 13 at 18:11

MurphyMurphy

61