All Questions
33
questions
2
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0
answers
62
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Two polynomials have finitely many common roots [duplicate]
Let $\bar k$ be an algebraically closed field. Consider $f,g\in\bar k[x,y]$ such that $f$ and $g$ are co-prime. We want to show that the set of common roots of $f$ and $g$ is finite.
Here is my ...
0
votes
1
answer
41
views
Showing a curve is not a variety using Bezout's theorem.
I have a question about a step in the following proof:
To show that $y-e^x +1 =0$ cannot be written as the solution to a system of polynomial equations $F_1=F_2=...=F_n=0$, first note that any such ...
2
votes
0
answers
54
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Hilbert's Nullstellensatz and irreducible polynomials [duplicate]
In p.35 of Miranda's book Algebraic Curves and Riemann Surfaces, the following theorem is stated and is called as Hilbert's Nullstellensatz.
Theorem. Let $f\in \Bbb C[x_1,\dots,x_n]$ is an irreducible ...
- 1,802
2
votes
2
answers
323
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If I erase one of the variables in an irreducible homogeneous polynomial, it's still irreducible
If $f$ is an irreducible homogeneous polynomial in $k[x_0,...,x_n]$ then $f$ is supposed to cut out an irreducible hypersurface in $\mathbb{P}_k^n$. So if I look in an affine chart, I should see an ...
1
vote
0
answers
56
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Points in "general position" with respect to polynomials
I have a question about points in general position. In my mind I am imagining that I am working over the field $\mathbb{C}$ and in dimension $d=3$, but I am stating the problem with slightly more ...
- 3,830
1
vote
1
answer
57
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Checking irreducibility of polynomials in two variables
There are a few exercises in Hartshorne about checking singularity of an affine curve. For example, $Y$ defined by $x^2 = x^4 + y^4$ over a field $k$ (with ${\mathrm{char}}k \neq 2$). This is easy.
...
- 808
3
votes
1
answer
427
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Intersection of the zero sets of two multivariate polynomials
Suppose now I have two multivariate polynomials, say $p_1: \mathbb{R}^n\to\mathbb{R}$ and $p_2: \mathbb{R}^n\to\mathbb{R}$, $n\ge 2$. Suppose they are both irreducible, and $A$, $B$ are the zero sets ...
- 89
0
votes
1
answer
59
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Irreducible bivariate complex polynomial whose zero-locus contains two given points
Let $\alpha,\beta\in\mathbf{C}^2$ be distinct pairs of complex numbers. Is there an irreducible polynomial $f\in\mathbf{C}[x,y]$ vanishing at $\alpha$ and $\beta$?
- 473
1
vote
2
answers
49
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Is $p\in k[z]$ irreducible iff $q(x,y)=y^{b\cdot\deg(p)}p(\frac{x^{a}}{y^{b}})\in k[x,y]$ is irreducible, where $a,b$ are coprime integers
Let $a,b\in\mathbb{N}$ be coprime, $p\in k[z]$ a polynomial of degree $g$. Then the bivariate polynomial $q\in k[x,y]$, given by
\begin{equation*}
q(x,y)=y^{b\cdot g}p\left(\frac{x^{a}}{y^{b}}\...
- 1,532
3
votes
0
answers
237
views
Special irreducible polynomials in $k[x,y]$
Let $k$ be a field of characteristic zero, $n \in \mathbb{N}$.
Definitions:
(1) $0 \neq f \in k[x_1,\ldots,x_n]$ is always irreducible, if for every $\lambda \in k$, $f+\lambda$ is irreducible in $...
- 6,705
3
votes
0
answers
207
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Is $x_0^4+x_1^4+x_2^4+x_3^4-ax_0x_1x_2x_3$ irreducible for every $a \in \mathbb{C}$? [duplicate]
I'm trying to solve the problem Sec. 2 - 1.16 in Shafarevich's book Basic Algebraic Geometry, vol. 1, second ed. My attempt is to use the previous exercise Sec. 2 - 1.12. So I only need to show that ...
- 1,239
0
votes
0
answers
101
views
For a line $L$ and an algebraic curve $C$ of an irreducible polynomial, prove $C \cap L$ contains at most d points unless C = L.
Artin Algebra Chapter 11
This has been answered here. My questions are about the solution of Brian Bi:
By stronger, does he mean that $C \ne L$ and $f$ is irreducible $\implies l \nmid f?$ If so, ...
user198044
2
votes
1
answer
3k
views
What's the degree of a multivariate polynomial in Artin Algebra?
According to Degree of a polynomial, it is highest degree among the monomials. Where is this in Artin Algebra?
In Chapter 11.9, an exercise gives a degree to an irreducible complex polynomial of two ...
user198044
1
vote
1
answer
450
views
In proving an irreducible curve has only finitely many singular points, is $f_x \not \equiv 0$?
Artin Algebra Chapter 11
This has been answered on the site, but I want to ask about the solution given by Brian Bi:
The constant polynomials are not considered irreducible, so f is not constant. ...
user198044
0
votes
2
answers
57
views
Example of polynomial in two variables
Can you please give me an example of a polynomial $F \in K[X,Y]$ such that $V(F)$ is finite?
I found in Fulton the following proposition:
If F is an irreducible polynomial in $K[X,Y]$ such that $V(...
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