All Questions
14 questions
4
votes
0
answers
263
views
Cosine Modulo $p$?
Consider the integers modulo a prime $p$. I'm looking for a nice polynomial function that acts as a sort of "cosine" on the integers modulo $p$. Specifically, I'm looking for solutions to ...
- 591
1
vote
1
answer
133
views
Constant term of a power modulo a polynomial
I'm interested in the constant term of $$(x+k)^m \in F_p[x]$$
modulo a polynomial $q(x)$ over the field $F_p$. The polynomial $q(x)$ is relatively simple in practice, take $q(x) = x^6 -2x^3+3$, for an ...
- 591
1
vote
0
answers
96
views
Polynomial composition utilizing polynomials in two different finite fields
At every $n\in\mathbb N$ (all polynomials are of degree $O(1)$) is there $g_{3,1}^{(n)},\dots,g_{3,k}^{(n)}\in\mathbb F_3[x_1,\dots,x_n]$ at $k=\mathsf{poly}(n)$ and $g_2^{(n)}\in\mathbb F_2[x_1,\dots,...
- 13.9k
5
votes
1
answer
163
views
Polynomials vanishing on prescribed layers
Given a prime $p$ and an integer $n\ge p$, what is the smallest possible degree of a polynomial $Q\in\mathbb F_p[x_1,\dotsc, x_n]$ such that $Q$ vanishes on every vector $x\in\{0,1\}^n$ of weight $w(x)...
- 23k
2
votes
1
answer
192
views
A Vandermonde-type system
For a prime $p$ and $a_1,\dotsc,a_n\in\mathbb F_p^\times$, consider the system of equations
$$ \begin{cases}
\begin{align}
a_1 + \dotsb + a_n &= 0 \\
a_1x_1 + \dotsb + a_nx_n &...
- 23k
8
votes
3
answers
385
views
Self-reciprocal polynomials over finite fields
Let $SRMI_q(2n)$ denote the number of self-reciprocal
irreducible monic polynomials of even degree $2n$ over the finite
field $\mathbf{F}_q$ with $q$ elements. Recall that
a polynomial $p(x) \in \...
- 481
1
vote
1
answer
82
views
question about sets of polynomials with a special agreement guarantee
Let $\mathbb{F}$ be a finite field and $S\subset\mathbb{F}_{\leq d}[x,y]$, a set of bivariate polynomials over $\mathbb{F}$ of degree at most $d\ll|\mathbb{F}|$. Assume the linear span of $S$ is all ...
- 125
5
votes
0
answers
205
views
Polynomials representing locally constant functions
Let $K$ be a finite field with $p$ elements.
(a) Let $f\in K\lbrack x\rbrack$ be such that (i) $\deg(f)<p$ and (ii) $f(2x) = f(x)$ for $\geq (1-\epsilon) p$ values of $x$ in $K$. What can we say ...
- 20.2k
4
votes
1
answer
382
views
Counting couples of square-free polynomials over finite fields
I have a curve defined by the following equations over the finite field $\mathbb{F}_q$ with $q=p^r$ with $p \geq 3$:
$$C_{h_1,h_2}:\begin{cases} y_1^2=h_1(t) & \\
y_2^2=h_2(t) &...
- 41
1
vote
0
answers
207
views
Polynomial existence over finite field
Denote $\mathcal{F_n}$ as collection of multiaffine polynomials $f\in\Bbb F_2[x_1,\dots,x_n]$.
Denote total degree of $f\in\mathcal{F_n}$ as $deg(f)$ (note $deg(f)\leq n$).
Denote $e_i=(0,\dots,0,\...
- 13.9k
2
votes
0
answers
337
views
Enumerating certain types of permutation polynomials
Given a prime power $q$, I would like to enumerate (preferably up to isomorphism*) all the permutation polynomials $f(x)$ on $K = GF(q^3)$ satisfying the following conditions:
$f(ax) = af(x)$ for all ...
- 1,197
17
votes
4
answers
1k
views
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 ...
- 23k
67
votes
6
answers
7k
views
How to recognise that the polynomial method might work
A couple of days ago I was at a nice seminar given by Christian Reiher, during which he told us about a short proof of the following special case of a theorem of Olson.
Theorem. Let $(a_1,b_1),\dots,(...
- 29k
16
votes
3
answers
2k
views
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" f:[0,1]→[...
- 60.5k