# Tagged Questions

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### probability of having linearly independent sparse vectors over finite fields

Suppose that there are $i< N$ linearly independent $N$-dimensional vectors $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_i$ over a finite field $\mathbb{F}_q$, whose elements are denoted as ...
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### Dimension of incomplete matrix over finite fields.

Hi, Suppose one has an incompletely specified $2^n \times 2^n$ matrix over some fixed finite field $\mathbb{F}_{p^k}$. In fact, one knows that the diagonal entries are zero and all other entries are ...
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### More expanders?

Having received several exhausting answers to my recent question about the expansion properties of a certain graph, I now wonder whether anything is known on the following graphs of a similar nature: ...
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### An expander (?) graph

For a prime $p$, consider the graph on the vertex set ${\mathbb F}_p$, in which every vertex $z$ is adjacent to $z\pm 1$ and also to $z^{-1}$ (unless $z=0$). I was told that this graph is known to be ...
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### A mixing property for finite fields of characteristic $2$

In connection with this MO post, here is a question somewhat implicitly contained in a joint paper of S. Kopparty, S. Saraf, M. Sudan, and myself. Let ${\mathbb F}$ be a finite field, and suppose ...
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### Periodic orbits and polynomials

There are two simple and classic enumerations that still I'm puzzled about. Let's start with a simple counting problem from a well-known dynamical system. fact 1 Consider the "tent map" ...
Let $q$ be a power of 2. Let $P$ be the set of polynomials in $F_q [x]$ of degree d or less. Let $\mathbb{Z}$ be the ring of integers. For any $f \in P$, let $\psi(f)$ be the number of distinct ...
Let $q$ be a power of a prime. It's well-known that the function $B(n, q) = \frac{1}{n} \sum_{d | n} \mu \left( \frac{n}{d} \right) q^d$ counts both the number of irreducible polynomials of degree ...