All Questions
11
questions
0
votes
0
answers
96
views
Algebraic independence and substitution for quadratics
Let $f_{1},...,f_{n-1} \in \mathbb{F}[x_1,...,x_n]$ such that $\{ f_1,..., f_{n-1},x_n \}$ is algebraically independent over $\mathbb{F}$. Let $G \in \mathbb{F}[x_1,...,x_n,y_1,...,y_{n-1}]\...
- 181
0
votes
0
answers
112
views
Relation between minimality and algebraic independence for binomials?
$\DeclareMathOperator\supp{supp}$Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ such that
$f_1 = x_1 + q_1$
$f_2 = x_2 + q_2$
$\cdot \cdot \cdot$
$f_{n-1} = x_{n-1} + q_{n-1}$
$f_{n} = q_n$
such that ...
- 181
2
votes
1
answer
200
views
Minimality implies algebraic independence?
$\DeclareMathOperator\supp{supp}$Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ such that
$f_1 = x_1 + q_1$
$f_2 = x_2 + q_2$
$\cdot \cdot \cdot$
$f_{n-1} = x_{n-1} + q_{n-1}$
$f_{n} = q_n$
such that ...
- 181
1
vote
0
answers
82
views
When does sum of algebraically independent polynomial become dependent?
Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ where $f_n = g + h$. Suppose the sets $\{ f_1,...,f_{n-1},g \}$ and $\{ f_1,...,f_{n-1},h \}$ are algebraically independent then is there a ...
- 181
8
votes
1
answer
617
views
A question on algebraic independence
Let $f_1,f_2,\ldots,f_n, g \in \mathbb{F}_q[x_1,...,x_m]$. Assume that $f_1,\ldots,f_n$ vanish at $0$, so that $\mathbb{F}_q[[f_1,...,f_n]]$ is a subring of $\mathbb{F}_q[[x_1,...,x_n]]$. Suppose that ...
- 181
4
votes
0
answers
775
views
When spreading out a scheme, does the choice of max ideal matter?
I'm looking at Serre's paper How to use Finite Fields for Problems Concerning Infinite Fields. Specifically I'm trying to use the techniques in the proof of Theorem 1.2 to write out the details of the ...
- 1,149
15
votes
2
answers
1k
views
Can you use Chevalley‒Warning to prove existence of a solution?
Recall the Chevalley‒Warning theorem:
Theorem. Let $f_1, \ldots, f_r \in \mathbb F_q[x_1,\ldots,x_n]$ be polynomials of degrees $d_1, \ldots, d_r$. If
$$d_1 + \ldots + d_r < n,$$
then the ...
- 27.9k
5
votes
1
answer
517
views
Schwartz-Zippel lemma for an algebraic variety
Let $X $ be a smooth affine subvariety of $(\overline{\mathbb{F}_q})^n$ defined by a prime ideal $I$. Let $f$ $\in \mathbb{F}_q[x_1,\ldots,x_n]$ be a polynomial such that $f \notin I$.
Let $r_1, \...
- 2,578
4
votes
2
answers
537
views
Irreducible algebraic sets via irreducible polynomials
There are many results about irreducible polynomials over finite fields:
we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, ...
- 2,578
28
votes
1
answer
1k
views
Algebraic dependency over $\mathbb{F}_{2}$
Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$
such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall i\in[n]:f_{i}(a)=a_{i}$....
- 583
19
votes
3
answers
2k
views
Classification of rings satisfying $a^4=a$
We have the famous classification of rings satisfying $a^2=a$ (for each element $a$) in terms of Stone spaces, via $X \mapsto C(X,\mathbb{F}_2)$. Similarly, rings satisfying $a^3=a$ are classified by ...
- 62.1k