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

Abstract

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.

8 pages