• Corpus ID: 14301968

Ranks of Quotients, Remainders and $p$-Adic Digits of Matrices

@article{Elsheikh2014RanksOQ,
  title={Ranks of Quotients, Remainders and \$p\$-Adic Digits of Matrices},
  author={Mustafa Elsheikh and Andrew Novocin and Mark Giesbrecht},
  journal={ArXiv},
  year={2014},
  volume={abs/1401.6667}
}

• M. Elsheikh, A. Novocin, M. Giesbrecht
• Published 26 January 2014
• Mathematics
• ArXiv

For a prime $p$ and a matrix $A \in \mathbb{Z}^{n \times n}$, write $A$ as $A = p (A \,\mathrm{quo}\, p) + (A \,\mathrm{rem}\, p)$ where the remainder and quotient operations are applied element-wise. Write the $p$-adic expansion of $A$ as $A = A^{[0]} + p A^{[1]} + p^2 A^{[2]} + \cdots$ where each $A^{[i]} \in \mathbb{Z}^{n \times n}$ has entries between $[0, p-1]$. Upper bounds are proven for the $\mathbb{Z}$-ranks of $A \,\mathrm{rem}\, p$, and $A \,\mathrm{quo}\, p$. Also, upper bounds are proven for the $\mathbb{Z}/p\mathbb{Z}$-rank of $A^{[i]}$ for all $i \ge 0$ when $p = 2$, and a conjecture is presented for odd primes.