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Results tagged with polynomials
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user 23758
Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.
3
votes
Is a finite order automorphism of k[x,y] necessarily linear?
As explained in the article of McKay and Wang, the result of Jung says that $\operatorname{Aut}_k(k[x,y])$ is an amalgamated product of the affine automorphisms and Jonquières ones, that preserve the …
- 7,722
14
votes
Accepted
Is a wild automorphism of $k[x_1,\ldots,x_n]$, $n \geq 3$, necessarily of infinite order?
The answer is no.
Let $K$ be a field of characteristic zero and let us take the Nagata automorphism of $\mathbb{A}^3_K$ given by
$$N\colon (x,y,z)\mapsto (x+(x^2-yz)z,y+2(x^2-yz)x+(x^2-yz)^2z,z)$$
…
- 7,722
7
votes
0
answers
576
views
Factors of the polynomial $X^n-a$
I am interested in the polynomial $X^n-a$ in $\mathbb{Q}[X]$, for some $a\in \mathbb{Q}^*$, and would like to know the irreducible factors of it. Is there something in the literature which gives a low …
- 7,722
4
votes
Accepted
Two bivariate polynomials (or rational functions) that generate $\mathbb{C}(x,y)$
The two polynomials that you are giving provide a morphism $\tau\colon\mathbb{C}^2\to \mathbb{C}^2$, given by $(x,y)\mapsto (f(x,y),g(x,y))$. … In practical, you can extend your morphism to a birational map from $\mathbb{P}^2$ to $\mathbb{P}^2$ by homogenising, and use Bézout Theorem: you find the common roots of the two homogenised polynomials …
- 7,722
2
votes
Accepted
Special elements of the Cremona group
The monoid that you are looking for is the set of birational endomorphisms of the affine plane. It is of course closed under compositions and the invertible elements are the automorphisms. You would l …
- 7,722
6
votes
Accepted
If $(f,g)$ and $(f,h)$ are maximal ideals, then $ag+bh=P(f)$ for some $a,b \in k, P(t) \in k...
The answer is no. Let
$$f=x, \quad g=y+xy^2, \quad h=y+1+xy^3.$$
Then
$$
(f,g)=(x,y)\quad\text{and}\quad (f,h)=(x,y+1)
$$
are maximal ideals of $k[x,y]$, but for all $a,b\in k$ not both zero, you find …
- 7,722
11
votes
4
answers
1k
views
Explicit large finite fields in characteristic $2$
This might be too much to ask for all degrees, so an infinite sequence of irreducible polynomials would be good.
EDIT: Some sequences have already be given. …
- 7,722
3
votes
Accepted
A variation on Abhyankar–Moh–Suzuki theorem
The answer is no. You can simply choose $$f=t(t^2-t+1)$$
$$g=t(t^2+1)$$
These satisfy all your conditions, but $\mathbb{C}[f,g]\subsetneq \mathbb{C}[t]$.
Let us check this:
$(1)-(2)$: $f=tf_1$ and $g= …
- 7,722