0
votes
0answers
15 views

Characterization of ideals generated by homogeneous polynomials in terms of $f^{(d)}$ in Gathmann's notes.

On pg. 37 of Gathmann's Algebraic Geometry notes, the following is mentioned: For every $f\in k[x_0,x_1,\dots,x_n]$ be an ideal. The following are equivalent: I can be generated by ...
0
votes
1answer
43 views

The ring of fractions $K(x)$ is the field generated by $K$ and $x$.

I would like to show that the ring of fractions $K(x)$ of $K[x]$ in an extension $L$, where $K\subset L$ fields, is the field generated by $K$ and $x$ (let's call it by $\tilde{K(x)}$). I know just ...
0
votes
2answers
19 views

Rational functions are decomposed in polynomial products

I'm trying to understand why this is true: Since $K(x)$ is a field, $K(x)$ is an UFD, then $K(x)$ can be written uniquely as products of irreducible elements of $K(x)$. I didn't understand why ...
1
vote
2answers
116 views

roots of a polynomial inside a circle

I am asked to show that for $n$ larger or equal to $2,$ the roots of $1 + z + z^{n}$ lie inside the circle $\|z\| = 1 + \frac{1}{n-1}$ Attempt1: Induction for the case $n = 2,$ the roots of $1 + z + ...
6
votes
1answer
91 views

Is $\mathbb{C}$ algebraically closed (in a strong sense)?

Let $p,q$ be polynomials in $\mathbb{C}[x,y]$ such that the ideal $(p,q)$ is a proper ideal of $\mathbb{C}[x,y]$. Does there exist complex numbers $z,w$ such that $$p(z,w)=0,\,\,\,\,\,q(z,w)=0\ ?$$ ...
3
votes
3answers
91 views

Are surjective polynomial maps injective?

An injective polynomial map $p:\mathbb{C}^n\mapsto\mathbb{C}^n$ is surjective (Ax-Grothendieck theorem). What is known about the reverse implication (surjective implies injective)? Why does the ...
1
vote
1answer
36 views

Show that there is no non-constant morphism from $\mathbb{A}^1\rightarrow E = Z(Y^2-X^3+X)$

I think this is supposed to be over an arbitrary field $K$, but if it's only true when $K$ is, say, algebraically closed, then feel free to assume whatever conditions are required to make the ...
2
votes
3answers
69 views

Proving the area of an equilateral triangle

How do you prove that How do you prove that for any equilateral triangle with side length s, area is $\frac{s^2 √3}{4}$ ? I tried using an equilateral triangle in a square, but I keep coming up with a ...
1
vote
0answers
64 views

Criterion to decide the invertibility of polynomial maps

Consider a polynomial map $f:\mathbb{R}^{n-1}\to V\subset\mathbb{R}^n$ where $V$ is $n-1$-dimensional variety in $\mathbb{R}^n$. Are there any conditions on $f$ to determine whether it defines ...
1
vote
0answers
35 views

Birational Variety

Given a polynomial map $f:\mathbb{R}^2\to V\subset \mathbb{R}^3 $ defined as follows: $$ (z_1,z_2)\mapsto (2z_1-z_2, 2z_1^2-z_2^2, 2z_1^3-z_2^3) $$ This map defines a Variety ($V$) of dimension $2$ in ...
1
vote
1answer
85 views

Invertibility of a Polynomial map.

Given following polynomial map $f:\mathbb{R}^2\to V\subset \mathbb{R}^3 $: $$ (z_1,z_2)\mapsto (2z_1-z_2, 2z_1^2-z_2^2, 2z_1^3-z_2^3) $$ Is this map a bijection? If so, how?
1
vote
0answers
90 views

Too many independent cubic polynomials in an ideal $I\subset \mathbb C[x,y,z]$

Let us consider the ideal $I=(x^2-x,y,xz)\subset \mathbb C[x,y,z]$. I want to prove that $I$ contains (exactly) $5$ linearly independent polynomials of degree $3$. In three variables, we have ...
8
votes
2answers
518 views

How does Hilbert's Nullstellensatz generalize the “fundamental theorem of algebra”?

What is Hilbert's Nullstellensatz in the sense of the generalization of "fundamental theorem of algebra"? I've seen that in some texts it was referred to as the generalization of the fundamental ...
5
votes
1answer
55 views

Topological closure of ideal in $A[[T]]$ - Proposition 1.3.7 in Liu

In Proposition 1.3.7 of Liu's book, one proves that if a ring $A$ is noetherian then so is $A[[T]]$. We take an ideal $I$ of $A[[T]]$ and prove that there exist $F_1,\ldots,F_m\in I$ such that for all ...
2
votes
1answer
64 views

Projecting an affine hypersurface away from a point in its projective closure is never a finite map?

Let $X\subset \mathbb{A}_k^r$ be an irreducible hypersurface defined by a polynomial $g$, where $k$ is an algebraically closed field. Embed $\mathbb{A}^r\hookrightarrow\mathbb{P}^r$ in the usual way. ...
0
votes
1answer
54 views

If f is 0 at enough points, it is the 0 polynomial?

Let $f \in \mathbb{C}[x_1,...,x_n]$, and let d be the largest $x_i$-degree of f for $0 \leq i \leq n$. Prove that f is the zero polynomial, if $f(a_1,...,a_n)=0$ for all points $(a_1,...,a_n) \in ...
3
votes
1answer
37 views

Generic points as coefficients of polynomial kernels?

I am reading the paper Dual-to-Kernel Learning with Ideals. Here is part of it: The definition/motivation of genericity in Wikipedia are A generic point of the topological space $X$ is a point ...
5
votes
1answer
103 views

Irreducibility of $x^2+y^2+z^2-xyz-2$

Are there some general criteria for deciding the irreducibility of polynomials? For example the one in the title?
4
votes
0answers
52 views

The ring of homogeneous polynomials

I think I found an error in my textbook, but I am not completely sure. The book is Hulek, Elementary algebraic geometry, pag. 73. There is a theorem showing that $U_i$ and the affine space ...
2
votes
1answer
32 views

Proof of Projective Nullstellensatz

In the proof of the Nullstellensatz for projective varieties, I can't understand the following remark (which comes from Hulek, Elementary algebraic geometry, pag. 72) " ... if $f=\sum f_i$ is a ...
2
votes
1answer
82 views

Showing that if $f,g \in k[x,y]$ are irreducible and not associates then $(f,g) \cap k[x] \ne 0$

There is a part of example 10.25.3 at http://stacks.math.columbia.edu/tag/00EX that I'm having trouble understanding. Here, $k$ is a field and $f,g \in k[x,y]$ are irreducible and are not associates. ...
3
votes
1answer
128 views

Number of common zeros of two quadratic polynomials in ${\Bbb C}[t,x]$

The following theorem is in Artin's Algebra(2nd edition): Theorem 11.9.10 Two nonzero polynomials $f(t,x)$ and $g(t,x)$ in two variables have only finitely many common zeros in ${\Bbb C}^2$, ...
5
votes
2answers
284 views

Is evaluation homomorphism surjective?

Let $A^n$ be an affine space over $\mathbb{C}$ and let $\mathbb{C}[X_1,\cdots,X_n]$ be the polynomial ring of $n$ variables. Then $A^n\to (\mathbb{C}[X_1,\cdots,X_n])^*$ by evaluation homomorphism, ...
6
votes
1answer
110 views

Homogeneous polynomials between vector spaces

Consider $\mathbb{C}$ vector space $V$ = span$(e_1,\cdots,e_n)$. Consider the following algebra embedding $$ \mathbb{C}[X_1,\cdots,X_n]\hookrightarrow F(V,\mathbb{C})$$ where $f\mapsto(\sum_i a_i ...
4
votes
1answer
54 views

Preimage of a point by a power map in quaternions

Suppose we have a point $x_0\in{\bf H}$ (where by $\bf H$ I denote the ring of quaternions). What I'm curious about is what can the set of solutions of $x^2=x_0$ look like? From what I've checked, ...
2
votes
0answers
49 views

Polynomial functions and polynomial maps

What follows comes from Hulek, Elementary Algebraic Geometry, first chapter. Definition Let $V$ be an affine variety in $\mathbb{A}^n_{k}$. A polynomial function on $V$ is a map $f:V\longrightarrow ...
0
votes
1answer
113 views

Zeros of multivariate polynomials

Consider the ring of polynomials $k[x,y]$ where $k$ is an infinite field. (1) If $f$ and $g$ are two non-constant polynomials with no common irreducible factors then $V(f,g)$ is finite. (2) If $V$ ...
1
vote
2answers
264 views

Homogeneous polynomial in $k[X,Y,Z]$ can factor into linear polynomials?

My question is quite simple. Let $k$ be a closed algebraic field and $f\in k[X,Y]$. We know that $f$ can factor into linear polynomials. I would like to know if there is some generalization of ...
1
vote
1answer
85 views

Can this quick way of showing that $K[X,Y]/(Y-X^2)\cong K[X]$ be turned into a valid argument?

I've been trying to show that $$ K[X,Y]/(Y-X^2)\cong K[X] $$ where $K$ is a field, $K[X]$ and $K[X,Y]$ are the obvious polynomial rings over the indeterminates $X$ and $Y$ and $(Y-X^2)$ is the ...
12
votes
3answers
381 views

Polynomial map is surjective if it is injective

A friend of mine told me the following fact: If $k$ is any algebraically closed field, then a polynomial map $f\colon k^n\to k^n$ of affine space $k^n$ is surjective if it is injective. The ...
1
vote
1answer
33 views

Triple of powers is subvariety?

Consider the set $B=\{(t^2,t^3,t^4)\mid t\in \mathbb{C}\}$. Is it a subvariety of $\mathbb{C}^3$? That is, is it the set of common zeros of some (finite number of) polynomials? I'm thinking about ...
3
votes
1answer
79 views

Example of a curve of genus $4$

I'd like to put my hands on some polynomial defining a curve of genus $4$, living in the plane or in the 3D space. Do you know about any? Is there any procedure to build one? The best would be one ...
0
votes
0answers
79 views

$f$ is irreducible iff $V(f)$ is irreducible

I would like to know if the following statement is true: $f$ is irreducible iff $V(f)$ is irreducible. My tools I'm trying to use to prove this are Study's Lemma and basic algebra. If $f$ is ...
3
votes
1answer
48 views

If singular set is finite then the ideal is radical

Let $F\in K[X,Y]$ and if the zero set $V(F,\frac{\partial F} {\partial x},\frac{\partial F} {\partial y})$ is finite then $\sqrt {(F)} = (F)$. I don't see the relation between $\frac{\partial F} ...
2
votes
0answers
24 views

Show that the (Veronese-like) surface is given by zero locus of the following polynomials

Consider the following set in $\mathbb R^6$: $$ S= \bigl \{(x_1^2,x_2^2,x_3^2,x_1x_2,x_2x_3,x_3x_1) \mid (x_1,x_2,x_3) \in \mathbb R^3, \; x_1^2+x_2^2+x_3^2 = 1 \bigr\}. $$ If we denote by ...
0
votes
0answers
75 views

This might be interesting. The ring $R[x_1, \dots]$ (Non-specific)

Let $R$ be a commutative ring and define $\mathcal{R} = R \oplus \bigoplus_{i=1}^{\infty} x_i R[x_1, \dots, x_i]$. Then $\mathcal{R}$ is an $R$-algebra of polynomials in any finite number of ...
5
votes
1answer
151 views

What ideal is this?

Let $k$ be a field and $R = k[X]$ all polys over $k$ in $X$. Choose $p \in R$ and define $I_p = \{ f \in R : f\circ p(X) \in I \}$, where $I$ is some ideal in $R$. Then $I_p$ is an additive ...
1
vote
0answers
76 views

Questions on polynomial ring in several variables

Let $K$ be an infinite field. Prove that different polynomials in $R=K[X_1,X_2,...,X_n]$ don't lead to the same function $K^n \to K, x \mapsto f(x)$. (solved) Find $I \subset R$ and different ...
2
votes
1answer
54 views

Is this a homogeneous polynomial

If $f(ta,tb) = f(a,b)$ $\forall t \neq 0$, then can we conclude that $f$ is a homogeneous polynomial of degree 1?
2
votes
1answer
78 views

If $X$ is a cone, show that $I(X)$ is homogeneous.

The exercise is 1.3(3) from HP Kraft, "Appendix A: Basics from Algebraic Geometry." If a closed subset $X\subseteq \mathbb C^n$ is a cone, show that $I(X)$ is generated by homogeneous functions. ...
5
votes
2answers
109 views

What's the most basic yet interesting Algebraic Geometry result regarding this polynomial?

Let $f(x,y,z) = x^a + y^b - z^c$, where $a,b,c \gt 0$. What is the most basic yet interesting result about this polynomial from Algebraic Geometry?
2
votes
1answer
49 views

Any finite set in $k^n$ is an algebraic set.

I'm trying to show that given a field $k$, and a finite set of points $\{a^i: i = 1\dots n\} \subset k^n$ is an algebraic set or equivalently is the set of common zeros of some set of polynomials $S ...
0
votes
0answers
37 views

How do we calculate the Euler numbers of this

Suppose we are given two cubics X(a) and Y(a) in $CP^2$; $X(a)={ (4-a^3) xyz-a^3(x^3+y^3+z^3) =0 }$ $Y(a)={ a(x^3+y^3+z^3)-(2+a^3)xyz =0 }$ where a is a parameter in C satisfying $a^3 \not=1$ and ...
0
votes
0answers
69 views

Dimension of local ring as vector space over $\mathbb C$

I want to know what the dimension of each of the local ring $\mathbb C[x,y]_p/(y^2-x^7,y^5-x^3)$ is, where $p\neq (0,0)$ over $\mathbb C$-vector space. I know the dimension of it in the origin point, ...
0
votes
1answer
76 views

Inconsistent system of simultaneous equations

Let $F$ be an algebraically closed field, and $f_1,\ldots,f_n$ polynomials in $k$ variables over $F$. The system of simultaneous equations $$\mathcal{F}: ...
2
votes
2answers
41 views

Geometrical interpretation of $P(x) + Q(y) = 0 $ when P,Q are polynomials of degree 2?

Special cases are circles ( $ (x-x_0)^2 + (y-y_0)^2 = R $ ) and ellipses. Is there a geometric interpretation in the general case $ ( ax^2 + bx + c ) + ( dy^2 + ey + f ) = 0$?
2
votes
1answer
56 views

Vanishing of a multivariable polynomial on a lattice

Let be $p(x_1,...,x_n) \in K[x_1,...,x_n]$ be a polynomial of degree $d$. Suppose there is a $n$-dimensional hyperbox $B = I \times \stackrel{n}{...} \times I = I^n$. Divide $I$ to $d$ segements by ...
3
votes
1answer
103 views

homogenization of irreducible polynomial

This is the last detail in an exercise that I'm working on in hartshorne and I can't seem to figure it out. If $f$ is an irreducible polynomial in $k[x_{0},\cdots,x_{n}]$ (where $x_{i}$ does not ...
3
votes
1answer
57 views

Can the method of resolvents be used to give a proof of Bezout's Theorem?

Can the method of resolvents be used to give a proof of Bezout's Theorem? It seems to me like it should but I am unable to finish the proof. Here is what I have so far. Take two homogeneous ...
8
votes
1answer
289 views

Is an ideal generated by multilinear polynomials of different degrees always radical?

Definition. A polynomial $f\in\Bbbk[x_0,\ldots,x_n]$ is called multilinear if $\deg_{x_i}(f)=1$ for each $0\le i \le n$. In other words, $f$ is linear in each variable. If $f$ is homogeneous of ...