Tagged Questions

3
votes
1answer
34 views

square system of polynomial equations having infinite number of solutions

Suppose we have a system of $n$ polynomial equations in $n$ unknowns over $\mathbb{C}$ and suppose that the corresponding ideal generated by these equations is not the unit ideal $(1)$. Under what ...
3
votes
1answer
63 views

Small question about a proof of Hilbert's Basis Theorem

I am currently going going through the proof of Hilbert's Basis Theorem: http://www.maths.usyd.edu.au/u/de/AGR/CommutativeAlgebra/pp806-850.pdf (it starts on slide 832) On slide 836-837 he makes the ...
7
votes
0answers
103 views

Find all maximal subrings of $\mathbb{C}[x]$

Definition: A maximal subring $S$ of $R$ is a subring such that if $S \subseteq T \subseteq R$ then $T=S$ or $T=R$. Find all maximal subrings of $\mathbb{C}[x]$. Clearly $\mathbb{C}[x^2,x^3]$ ...
5
votes
1answer
130 views

Grobner Basis and generating set

I have come across the following past exam question... Define an ideal $J:=(z^2x+y^2-2y,x^3+y^3+z^3,x^2+2z^2) \subseteq \mathbb{Q}[x,y,z]. $ Compute a generating set for $J \cap \mathbb{Q}[y]$. ...
1
vote
2answers
170 views

Zero-dimensional ideals in polynomial rings

I have the following past exam paper question, a similar sort of question seems to come up every year. And I'm completely lost with it... Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated ...
0
votes
0answers
26 views

robust computation of Groebner basis

I am trying to solve numerically polynomial systems of equations, over the reals. I am coming across the following phenomenon: let's say that i have a system of 7 equations with 7 unknowns. I am using ...
1
vote
0answers
33 views

Equality of two $k$-algebras

Let $f\in k[X_1,\ldots, X_n]$ and $1-fX_{n+1}\in k[X_1,\ldots, X_{n+1}]$. Moreover $X\subseteq k^n$ is a subset and $$I(X)=\{g\in k[X_1,\ldots, X_n]\,:\, g(x)=0\,\forall x\in X \}$$ is the ideal of ...
0
votes
0answers
77 views

How to show an ideal is zero-dimensional? [duplicate]

Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated by $\{y^2-xy-2xz,y^3+z^2+1, x^2yz-yz\}$. Show that $J$ is zero-dimensional. How do I go about showing this?
13
votes
2answers
146 views

Is a linear combination of minors irreducible?

Let $X=(X_{ij})_{1\le i,j\le n}$ be a matrix of indeterminates over $\mathbb C$. For choices $I,J\subseteq\{1,\ldots,n\}$ with $|I|=|J|=k$ denote by $X_{I\times J}$ the matrix $(X_{ij})_{i\in I,j\in ...
1
vote
1answer
31 views

A relation between homomorphisms from the polynomial ring zero on an ideal and homomorphisms from the quotient of the polynomial ring by this ideal

Let $n\geq 1$, $K$ be a field and $R\neq \{0\}$ a $K$-algebra. For Ideals $I$ and $J$ of $K[X_1\ldots,X_n]$ with $J\subseteq I$ consider $$ A(I)=Hom_{Kalg}(K[X_1,\ldots,X_n]/I,R) $$ and $$ ...
6
votes
1answer
85 views

Vandermonde identity in a ring

Let $R$ be a commutative $\mathbb{Q}$-algebra. For $r \in R$ and $n \in \mathbb{N}$ we can define the binomial coefficient $\binom{r}{n}$ as usual by $\binom{r}{0}=1$ and ...
2
votes
2answers
38 views

Example of a non-free module over some Laurent polynomial ring

This is probably a naive question. What is an example of a non-free finitely generated module $M$ over some Laurent polynomial ring $$ L_n=K[X_1,X_1^{-1},\ldots,X_n,X_n^{-1}] $$ where $K$ is a field. ...
3
votes
2answers
138 views

Points and maximal ideals in polynomial rings

Let $k$ be a field, then I want to prove the following statement: for every $P=(b_1,\ldots,b_n)\in K^n$, the ideal $\mathfrak{m}_P=(x_1-b_1,\ldots,x_n-b_n)$ is maximal in the polynomial ring ...
0
votes
0answers
71 views

An Algorithm to Find the Generators of the Radical of a Monomial Ideal

Working over $R=\mathbb{C}[x_1,...,x_n]$, I'm given a ring homomorphism with $i\in{1,...,n}$ and $t\in \mathbb{C}$. $\phi_{i,t}(x_j)=x_j$ for $j\neq i$ to themselves. From this I've proven that an ...
3
votes
3answers
84 views

Ring of invariants of Klein Four group

Assume $F$ is a field and assume $f\in F[x_1,\ldots,x_4]$ is a polynomial that is invariant under the Klein Four group $V_4$. How can I show that this polynomial can then be rewritten as a polynomial ...
0
votes
0answers
39 views

How to find all possible polynomials set generate the same variety?

Some concepts I am not clear a. Does statement $\langle f_1, f_2\rangle$ = $(\sum u_if_i, i=1,2)$ means $f_1 = u_1f_1 + u_2f_2$ and $f_2 = u_3f_1 + u_4f_2$ ? b. Obviously $u_1 = 1$ and $u_2 = 0$ ...
1
vote
0answers
46 views

Minimal syzygies for polynomial ideals

Let $I$ be an ideal of $S=k[x_1,\dots,x_n]$. I am asked to find a minimal free graded resolution of $I$, by means of syzygy matrices. I suppose there has to be an algoritmic approach to it, provided ...
-3
votes
4answers
92 views

Radical of an ideal I

Let $I$ be the ideal of $\mathbb{C}[x,y]$ generated by $x^8$, $x^2y^3$, $x^7 - y^5$, $y^{42}$. Find a simple expression for the radical $\sqrt{I} = \{ f \in \mathbb{C}[x,y] : f^n \in I\;\text{for ...
4
votes
4answers
127 views

Irreducible element of the ring.

Element $X_1 X_2 \cdots X_n - 1$ is irreducible in $K[X_{1},\ldots,X_{n}]$ for $n\ge 1$, where $K$ is a field. For $n=2,3$ it is easy to see that the element is irreducible but for higher value of $n$ ...
1
vote
1answer
40 views

Nice closed form for polynomial defined as an antiderivative

Let $n$ be an integer $\geq 1$, and let $f_n(t)=(t(1-t))^n$. Let $F_n(t)$ denote the antiderivative of $f_n(t)$ satisfying $F_n(0)=0$. Of course, using Newton’s binomial formula we have an expansion ...
3
votes
1answer
91 views

Is there an irreducible polynomial vanishing on two components? (In the Zariski sense)

The polynomial $$f(x,y) = (x^2 − 1)^2 + (y^2 − 1)^2$$ is an example of an irreducible polynomial in $\mathbf{R}[x,y]$ which is irreducible but whose zero set has multiple components in the Zariski ...
0
votes
1answer
55 views

On the alphabetical order of monomial

I found this definition of alphabetical order for monomials in $k[x_1,\ldots,x_n]$. We say that $x_1^{a_1}\cdots x_n^{a_n}>x_1^{b_1}\cdots x_n^{b_n}$ if for the least $i$ such that $a_i\neq b_i$ we ...
4
votes
0answers
77 views

Some elementary facts

What is the simplest and the most conceptual proof of some basic facts on algebraic geometry? 1) Hilbert's Nullstellensatz 2) Regular functions on projective variety - only constants 3) elemination ...
2
votes
2answers
159 views

Maximal ideals in multivariate polynomial rings

Maximal ideals in univariate polynomial rings $R[X]$ have a nice characterization in that they all are of the form $(E)$, for some irreducible $E\in R[X]$. This allows for a systematic way to ...
5
votes
2answers
125 views

In an ideal, pairwise non-coprime implies globally non-coprime?

Let $R$ be a polynomial ring $R=k[X_1,X_2, \ldots ,X_n]$. Let $I$ be an ideal of $R$ such that any two elements of $I$ have a non-constant gcd. Does it follow that there is a non-constant $D$ dividing ...
-1
votes
2answers
245 views

$R[x]$ has a subring isomorphic to $R$.

Say $R$ is a commutative ring. Does there exists a subring of $R[x]$ that is isomorphic to $R$? My approach would be to define the subring of $R[x]$ that generates $R$. Any thoughts?
2
votes
1answer
91 views

Making the fundamental theorem of Galois theory explicit

I encountered the present question when investigating that other recent question of mine. Let $x_1,x_2, \ldots, x_8$ be indeterminates. Let $s_1,s_2, \ldots s_n$ denote the elementary symmetric ...
2
votes
1answer
71 views

Describe invariant polynomials under action of commutative group of order eight.

I believe the question below should be fairly standard in invariant theory ; I hope someone more familiar with it than me can explain a bit more or point to a reference. Let $F$ be polynomial field ...
1
vote
2answers
101 views

A question regarding the quotient map in a polynomial ring over a field

Let $F$ be a field, $F[X]$ the polynomial ring in one variable and $I$ an ideal of $F[X]$. Then does the quotient map $\pi:F[T]\longrightarrow F[T]/I$ map prime ideals to prime ideals?
2
votes
1answer
123 views

Multivariable Gauss's Lemma

Gauss's Lemma for polynomials claims that a non-constant polynomial in $\mathbb{Z}[X]$ is irreducible in $\mathbb{Z}[X]$ if and only if it is both irreducible in $\mathbb{Q}[X]$ and primitive in ...
5
votes
1answer
117 views

polynomials over a local Artinian (or finite) ring

In this question " Zero-divisors and units in $\mathbb Z_4[x]$ " it looks like it has been shown that the set of zero divisors of $\mathbb{Z}_4[x]$ coincides with its nilpotent elements. Since the ...
3
votes
1answer
111 views

Commutative Algebra - Polynomial Rings

Let $Z$ be the ring of integers, $p$ a prime and $F_p = Z/pZ$ the field with $p$ elements. Let $x$ be an indeterminate. Set $R_1 = F_p[x]/(x^2-2)$, $R_2 = F_p[x]/(x^2-3)$. Determine whether the rings ...
5
votes
2answers
156 views

$F, G \in k[X_1, \dots , X_n]$ homogeneous of degrees $r$ and $r+1$ $\implies$ $F+G$ is irreducible

I have a question about Exercise 2-34 from William Fulton's Algebraic Curves book. The exercise is as follows. Suppose that $F, G \in k[X_1, \dots , X_n]$ are forms (i.e. homogeneous ...
3
votes
0answers
102 views

Extension of the theorem of Jacobson

Let $A$ be a ring. Let $E$ be the set of polynomials $\{X^n-X \in \mathbb{Z}[X]|n \in \mathbb{N}^*-\{1\}\}$. By the theorem of Jacobson, we know that if for each $a\in A$ there is an element of $E$ ...
1
vote
1answer
75 views

Subring of polynomials

Let $k$ be a field and $A=k[X^3,X^5] \subseteq k[X]$. Prove that: a. $A$ is a Noetherian domain. b. $A$ is not integrally closed. c. $dim(A)=?$ (the Krull dimension). I suppose that the first ...
2
votes
2answers
262 views

Jacobson radical of $R[X]$, where $R$ is domain

Let $R$ be a commutative domain. Prove that the Jacobson radical of $R[X]$, i.e. the intersection of all maximal ideals, is the zero ideal. Thank you.
8
votes
4answers
179 views

Show $\mathbb{Q}[x,y]/\langle x,y \rangle$ is Not Projective as a $\mathbb{Q}[x,y]$-Module.

Disclaimer: Though I have been re-reading my notes, and have scanned the relevant texts, my commutative algebra is quite rusty, so I may be overlooking something basic. I want to show $\mathbb{Q} ...
7
votes
2answers
133 views

Relation between projective modules over $R$ and $R[T]$

Let $R$ be a commutative ring and $R[U]$ the polynomial ring in one variable. What is the relation between projective modules over $R$ and projective modules over $R[U]$? Is every projective module ...
1
vote
1answer
94 views

Simplifying quotient or localisation of a polynomial ring

Let $R$ be a commutative unital ring and $g\in R[X]$ a polynomial with the property that $g(0)$ is a unit in $R$ and $g(1)=1$. Is there any possible way to understand either $$R[X]/g$$ or ...
1
vote
1answer
87 views

localization of rings and polynomial functions

Let $f$ and $g$ be two polynomials (polynomial functions in $n$ variables); if in some localization of the ring $k[X_1,\ldots, X_n]$ exists the class $\frac{f}{g}$, it defines in a unique way the ...
0
votes
0answers
61 views

Showing that an alternating polynomial is the product of some symmetric polynomial and the Vandermonde polynomial

For simplicity, consider polynomials of two variables. Let $f(x,y)$ be an arbitrary alternating polynomial. I want to show that $f(x, y)$ is the product of some symmetric polynomial and the ...
1
vote
1answer
67 views

Is the ideal $(X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ prime?

Consider the ideal $(f = X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ in the polynomial ring $k[X_0,\ldots, X_n]$. Is this a prime ideal? If so, what is its height? I'm stuck trying to show that $f$ is ...
3
votes
1answer
132 views

Generating set for sum of two ideals

Suppose there are two ideals $I,J \in \mathbb{C}[x_1,\dots,x_k]$ and two sets of generating polynomials $\langle f_1, \dots, f_s\rangle$, $\langle g_1, \dots, g_t\rangle$. Now I want to describe $I + ...
2
votes
0answers
83 views

What does it mean for a subset of $\mathbb{C}[x_1,\dots,x_n]$ to be algebraically independent?

What does it mean for a subset of $\mathbb{C}[x_1,\dots,x_n]$ to be algebraically independent? Particularly I'd like to know the formulation thereof which concerns the kernel of a surjective ring ...
4
votes
2answers
160 views

Showing polynomials in $k[x_1, \ldots , x_n]$ are irreducible

It is often the case when I wish to show a particular polynomial in $k[x_1, \ldots ,x_n]$ is irreducible. Assuming that the polynomial is sufficiently friendly (i.e. one I would encounter as part of a ...
6
votes
0answers
98 views

On the order of $\mathbb{Z}[X]/(f,g)$ and the resultant $R(f,g)$.

I suspect that $\#\mathbb{Z}[X]/(f,g)=|R(f,g)|$ holds for any two non-constant polynomials $f,g\in\mathbb{Z}[X]$, where $R(f,g)$ is the resultant of $f$ and $g$. I am however unable to prove it. I'd ...
1
vote
2answers
175 views

What is Hilbert polynomial of this projective variety?

Suppose you have a map $\varphi\colon\mathbb{C}^m\times\mathbb{C}^n\to\mathrm{Mat}_{m,n}(\mathbb{C})$ defined by sending $(\mathbf{u},\mathbf{v})\mapsto\mathbf{u}\cdot\mathbf{v}^T=(u_i,v_j)$. So ...
5
votes
2answers
217 views

(Minimal?) Polynomials using the Nullstellensatz

I'm struggling with an exercise that was asked in class: Let $\alpha = \sqrt[3]{3} + \sqrt{7}\sqrt[4]{2}.$ Show that there is a polynomial $p$ in the ideal $I=\left<a^3 - 3, b^2 - 7, c^4-2, ...
7
votes
2answers
150 views

Why is the (-1)-th coefficient of $f^n f'$ equal to 0, without dividing by $n+1$?

Let $R$ be a commutative ring, and $n$ be a nonnegative integer. Let $f\in R\left[t,t^{-1}\right]$ be a Laurent polynomial in one variable $t$ over $R$ (this means a formal $R$-linear combination of ...
3
votes
1answer
116 views

Minimal generating sets for homogeneous polynomial ideal in two variables

This question is (somehow) related to System of generator of a homogenous ideal Let $A$ be the ring $A={\mathbb R}[X,Y]$, and let $m \geq 1$. Let $$ {\cal S}_m=\lbrace X^m, X^{m-1}Y,X^{m-2}Y^2, ...

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