All Questions
12 questions
10
votes
2
answers
1k
views
Do relatively prime polynomials $f$ and $g$ in $k[x,y]$ generate an ideal of finite codimension?
Let $k$ be a field and $f,g \in k[x,y]$ be two non-constant polynomials in two variables. Is it true that the following conditions are equivalent:
$f$ and $g$ are relatively prime in $k[x,y]$, in the ...
7
votes
0
answers
369
views
Intersections of ideals in polynomial rings with countably many variables
Fix a field $k$ and let $R = k[x_1,x_2,\ldots]$. Say that an ideal $I \subset R$ is generated in finite degree if there exists a generating set $S$ for $I$ (possibly infinite) and an integer $n$ such ...
- 71
5
votes
1
answer
1k
views
Ideal generated by two univariate, coprime, integer polynomials
Let $f(x)$, $g(x)$ be two univariate, coprime, integer polynomials and let $I=\big(f(x),g(x)\big)$ the ideal of $\mathbb{Z}[x]$ generated by $f, g$. Let $I \cap \mathbb{Z}$, that is, the elements of $\...
- 7,746
4
votes
0
answers
238
views
When $\langle u,v,w \rangle$ is a maximal ideal in $\mathbb{C}[x,y]$?
Let $u,v,w \in \mathbb{C}[x,y]$ and let $\langle u,v,w \rangle$ be the ideal generated by $u,v,w$.
It is known that for two elements the following result holds:
$\langle u,v \rangle$ is a maximal ...
- 2,837
2
votes
1
answer
279
views
Intersecting a ideal generated in degree $\leq a$ with one generated in degree $\leq b$ in a polynomial ring
Question 1. Let $R$ be a polynomial ring over a field $k$. Assume that $R$ is graded in the usual way (i.e., each variable has degree $1$). Let $I$ and $J$ be two ideals of $R$ such that $I$ is ...
- 33.8k
1
vote
1
answer
134
views
If $(f,g)$ and $(f,h)$ are maximal ideals, then $ag+bh=P(f)$ for some $a,b \in k, P(t) \in k[t]$?
Let $k$ be an algebraically closed field of characteristic zero, for example $k=\mathbb{C}$.
Let $f,g,h \in k[x,y]$, $g \neq h$, satisfy the following two conditions:
(1) $(f,g)$ is a maximal ideal of ...
- 2,837
1
vote
1
answer
153
views
Ideals: If $\langle f_1,f_2 \rangle = \langle g_1,g_2 \rangle$, then $\langle f_1-\lambda,f_2-\mu \rangle = \langle g_1-\delta,g_2-\epsilon \rangle$?
The following question appears in MSE without answers.
Let $f_1,f_2,g_1,g_2 \in \mathbb{C}[x,y]-\mathbb{C}$.
Assume that $\langle f_1,f_2 \rangle = \langle g_1,g_2 \rangle \subsetneq \mathbb{C}[x,y]$,
...
- 2,837
1
vote
1
answer
108
views
Finding a characteristic for which the zero-locus of an ideal is not empty
I have a set of polynomials $f_1, \dots, f_m \in \mathbb{Z}[x_1, \dots, x_n]$ and I am interested in finding if these polynomials have a common root inside either $\mathbb{C}[x_1, \dots, x_n]$ or $\...
- 113
1
vote
0
answers
205
views
Finding if an ideal is the radical of another one
Let's suppose we have, in the ring $\mathbb{Z} [x,y,z,w,v]$, the following polynomials:
$f=xw-yz$,
$g=x^2z-y^3$,
$h=yw^2-z^3$,
$k=xz^2-y^2w$.
The question is to prove that $I=(f,g,h,k)$ is the radical ...
0
votes
1
answer
162
views
$\mathbb{C}(u(x,y),v(x,y),f(x)+g(y))=\mathbb{C}(x,y)$ implies $\mathbb{C}(u(x,y),v(x,y))=\mathbb{C}(x,y)$?
The following question is a direct continuation of this question:
Let $u,v \in \mathbb{C}[x,y]$.
Assume that for every $f \in \mathbb{C}[x]$ and every $g \in \mathbb{C}[y]$ (excluding the cases where $...
- 2,837
0
votes
1
answer
137
views
$k(F_i)_{i=1}^{n}=k(G_j)_{j=1}^{m}$ iff there exist $a_i,b_j \in k$ such that $\langle F_i-a_i \rangle_{i=1}^{n} = \langle G_j-b_j \rangle_{j=1}^{m}$
Let $k$ be an algebraically closed field of characteristic zero, for example $k=\mathbb{C}$ and let $F_1,\ldots,F_n,G_1,\ldots,G_m \in \mathbb{C}[x,y]$, $n,m \in \mathbb{N}-\{0\}$.
Claim:
$\mathbb{C}(...
- 2,837
0
votes
0
answers
105
views
An ideal invariant under an automorphism
The following question appears here; hopefully, it is appropriate for MO.
Let $k$ be a field of characteristic zero, and let $\beta: k[x,y] \to k[x,y]$ be the following involution $\beta: (x,y) \...
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