All Questions
9
questions
1
vote
0
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54
views
Does there exists something like the BKK Theorem for polynomials over finite fields?
I'm trying to count the number of common $\mathbb{F}_q^\times$-zeros of some polynomials $f,g \in \mathbb{F}_q[x,y]$. I thought I had a solution using the BKK theorem but I reread the theorem ...
- 41
1
vote
0
answers
31
views
Cardinality of the zero locus of a degree 2 homogeneous polynomial on Z/2Z: avoiding Chevalley-Warning
I have never developed sufficient knowledge in algebraic geometry but I ran into an apparently easy problem, so I apologise in advance if my question sounds naive.
Suppose to have a degree $2$ ...
- 123
1
vote
0
answers
72
views
Is $\{x^3, x+b\}$ a generating set of $\mathrm{Sym}(\mathbb F_q)$?
Let $q=2^n$ where $n$ is a sufficiently large odd number. Consider the fintie field $\mathbb F_q$ and the symmetric group $\mathrm{Sym}(\mathbb F_q)$ over it.
I use $x^3$ to denote the permutation $x \...
- 21
2
votes
2
answers
125
views
Application of the study of equations over finite fields
I’m currently studying equations over finite fields (in particular solutions over $F_{q^k}$ to equations of the form $y^q-y=f(X)$ for a polynomial $f(X) \in F_q[X]$) and approximations of the number ...
- 21
1
vote
1
answer
78
views
Showing no solution exists for small system of multivariate quadratics over a finite field
I would like to show for the system of polynomials
$$
f_j(\vec{a})=\sum_{i=1}^8 a_i a_{i+j \pmod{8}} = 0, \quad j=1,2,3,4
$$
with $a_i \in \{-1,+1\}$, no common solution can exist. It is easy to ...
- 129
4
votes
1
answer
69
views
Low-degree polynomial $T\in\mathbb F[x,y]$ with $T(P(z),Q(z))=0$
Given polynomials $P,Q\in\mathbb F[z]$ over a finite field $\mathbb F$, one can find a non-zero polynomial $T\in\mathbb F[x,y]$ such that $T(P(z),Q(z))=0$ for any $z\in\mathbb F$. Is there a way to ...
- 4,649
0
votes
1
answer
48
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Families of quadratic polynomials over $\mathbb{F}_p$
Consider the set $\mathcal{P}$ of polynomials $f \in \mathbb{F}_p[x, y]$ in two variable over the field with $p$ elements, that are quadratic in $x$ and of unrestricted degree in $y$. For each fixed $...
- 10.7k
5
votes
1
answer
1k
views
Rank of a Polynomial function over Finite Fields
Consider a function $f:\mathbb{R}^k \to \mathbb{R}^n$ where $f=(f_1,f_2,\dots,f_n)$ with each $f_i$ being a polynomial function of variables $t_1,t_2,\dots,t_k$.
Now if the polynomials $f_1,f_2,\dots,...
- 2,517
2
votes
1
answer
144
views
Polynomials in $\mathbb F_3[X_1,\ldots,X_N]$ that Vanish in a Particular Way
I'm trying to apply the polynomial method to a combinatorial problem. In one special case of the problem, I have an integer $N\geq 4$ and a set $D\subseteq\mathbb F_3^N$ with $|D|=3^{N-1}+1$. I have ...
- 1,297