All Questions
34
questions
1
vote
0
answers
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
0
votes
0
answers
19
views
Explicit bivariate resultant example
I have two polynomials modulo a prime $p$ with a common root $(x_0,y_0)$ with $|x_0|,|y_0|<\sqrt{p}$ and there are no other roots in $\mathbb F_p$ or no other roots of this size in $\mathbb F_p$ (...
- 6,245
1
vote
3
answers
99
views
Number of roots of $f=x^5-x-1$ in $\mathbb{F}_4$
According to David Cox’s ‘Galois Theory’ (Proposition 11.1.5.),
If $f \in \mathbb{F}_p[x]$ is nonconstant and $n \geq 1$, then the
number of roots of $f$ in $\mathbb{F}_{p^n}$ is the degree of the
...
- 1,788
0
votes
0
answers
54
views
What happens to the degree of the quotient polynomial if the divison is not clear?
Let's say that I have one polynomial $a(x)$ of degree $n$ with coefficients over $\mathbb{F}_p$, where $\mathbb{F}_p$ is a finite field of size $p$.
Asuming that $(x - r) \mid a(x)$ (i.e., that $r \in ...
- 321
2
votes
2
answers
146
views
How to find the n-th root of a polynomial in a binary field?
The question states the following:
Let us consider the field $𝐺𝐹(2^4)$ with multiplication modulo $𝑥^4+𝑥^3+1$, find all $y$ such that $y^{33} = 0101$.
My first approach was finding another way to ...
- 155
1
vote
5
answers
622
views
Polynomial with no roots over the field $ \mathbb{F}_p $.
How can I show that for any prime $p$ and any $d\ge 2$ there exists a polynomial of degree $d$ in $\mathbb{F}_p[X]$ with no roots? ($\mathbb{F}_p$ is the finite field with $p$ elements).
Thanks in ...
- 1,435
3
votes
1
answer
353
views
Degree of a multivariate polynomial over a finite field with many roots
Question
Let $q$ be a prime power, $k\in\{1,\ldots,q-1\}$ and $f$ be a multivariate polynomial in $\mathbb{F}_q[x_1,\ldots,x_n]$ having $q^n - k$ roots.
Show that $\deg(f) \geq (q-1)n - k + 1$.
(The ...
- 23.1k
0
votes
1
answer
49
views
Is there an analog of Sturm sequences for finite fields?
In finite fields, is there anything analogous to Sturm sequences for counting the number of roots of a polynomial in a given interval? Alternatively, showing that there are zero roots in a given ...
1
vote
1
answer
38
views
Quadratic equation in $\mathbb{F}_q$ for an even $q$ and $u \neq 0$
I have to show for an even $q$ and $u \neq 0$ that the equation $X^2 + ux + v = 0, u,v \in \mathbb{F}_q$ is solvable over $\mathbb{F}_q$ iff $v/u^2$ is of the form $z^2+z$ for a $z \in \mathbb{F}_q$.
...
user695468
0
votes
2
answers
103
views
Inverse of a matrix in $\mathbb{F}_5^{4\times4}$
Let
$f, \, g, \, h \in \mathbb{F}_5[X]$ where
$$f=X^9+X^8+ \cdots +X^2+X+1,\\ g=X^4+X-2 = X^4+X+3,
\\ h = 3X^2+4X+3.$$
$h$ is the greatest common divisor of $f$ and $g$.
It holds that
...
- 635
2
votes
4
answers
144
views
Polynomial of degree 4 over $\mathbb{F}_2$ has a root in $\mathbb{F}_{16}$
In my textbook it says that it is clear that polynomials of degree 4 over $\mathbb{F}_2$ have always roots in $\mathbb{F}_{16}$. How does this work, since it is not true for example for polynomials of ...
- 468
1
vote
1
answer
94
views
Find prime fields over which a polynomial has roots.
Suppose we have a polynomial
$$h(x) = a_n x^n + \dots + a_1 x + 1$$
Given the values $a_1,\ldots,a_n$, how to determine whether there exists such prime $p$ that $h(x)$ has roots over the field $\...
- 3,212
1
vote
1
answer
49
views
Number of roots of $f(x,y)\equiv0\bmod p$?
Given a prime $p$ and degree $d$ is it possible to define polynomials $f(x,y)\in\mathbb Z[x,y]$ with total degree $d$ and number of roots at most $t$ where $t$ is any integer in $[0,B]$ for some upper ...
- 6,245
-1
votes
1
answer
135
views
Prove that $f(x)$ and $g(x)$ do not have any roots in common.
Suppose that $a(x)f(x) +b(x)g(x) = 135$ where $a(x), b(x), f(x)$ and $g(x)$ are polynomials over $F$. Prove that $f(x)$ and $g(x)$ do not have any roots in common.
Any help is appreciated; thanks!
- 103