0
votes
2answers
61 views

Dimension of $\dim_{\mathbb C}\mathbb C[X,Y]/I(Y^2-X^2,Y^2+X^2)$

I'm trying to solve the question 1.36 from Fulton's algebraic curves book: Let $I=(Y^2-X^2,Y^2+X^2)\subset\mathbb C[X,Y]$. Find $V(I)$ and $\dim_{\mathbb C}\mathbb C[X,Y]/I$. Obviously ...
7
votes
0answers
76 views

What does the space of non-diagonalizable matrices look like?

Let $k$ be a field (I would be happy working entirely over $\mathbb C$). Consider the action of $G=GL_n(k)$ by conjugation on the set of $n\times n$ matrices over $k$. The collection $X$ of matrices ...
0
votes
1answer
20 views

Find a projectivity to create a graph.

I have the tetrahedron {xyzt=0} in projective space with homogeneous coordinate (x,y,z,t). I need to create a graph but the tetrahedron in affine coordinate is {xyz=0} and I can't visualize the ...
0
votes
1answer
28 views

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept.

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. f(x) = (1/5)x^4(x^2 - 3) the choice 1- 0, ...
3
votes
1answer
60 views

rank of quadrics

Consider the quadric $xw-yz$ in $\mathbf{P}^3$ (all over $\mathbf{C}$), and the Klein quadric $x_0 x_5+x_1 x_4+x_2 x_3$ in $\mathbf{P}^5$. I want to determine the rank of these quadrics. For the first ...
2
votes
2answers
38 views

The general expression of plane through the intersection of other two planes

For two planes: $$A_{1}x+B_{1}y+C_{1}z+D_{1}=0 $$ $$A_{2}x+B_{2}y+C_{2}z+D_{2}=0$$ Prove that any plane going through the intersection line of the previous planes could be expressed like where ...
1
vote
1answer
26 views

Finding signatue of a symmetric matrix.

Is it possible to find the signature of a matrix without finding the eigenvalues of the matrix? I was hoping to use the Sylvester's Law of inertia but I don't remember any algorithm to diagonalize a ...
2
votes
1answer
36 views

Question about geometry in a finite projective space

I apologize again for a dumb question! To add some context (though I think it'll largely be unnecessary): suppose $q$ is a prime, $F:= \displaystyle \mathbb{Z}/q\mathbb{Z}$ is a field. I've defined ...
2
votes
1answer
91 views

Change of coordinates (referential system) mistake? Doesn't seem to yield the proper coordinates.

Let $\varepsilon$ be an affine space with referential system $R$ characterized by $O=(1,1,1)$ as origin and $B=(c_1,c_2,c_3)$ as its basis, which is the canonical. Now, lets define a new ...
1
vote
1answer
53 views

Difference in surfaces described by equivalent quadratic forms

It is fairly straightforward to prove that a quadratic form $Q(\mathbf{x})=\mathbf{x}^{T}A\mathbf{x}$ can equivalently be written $Q(\mathbf{x})=\mathbf{x}^{T}M\mathbf{x}$ for ...
2
votes
1answer
39 views

The set of all $k$-dimensional planes which intersects $X$ is closed in $G(k,n)$

Let $X$ be irreducible algebraic set of projective n space. I am trying to show that: The set of all $k$-dimensional planes which intersects $X$ is closed in $G(k,n)$, where $G(k,n)$ is the ...
4
votes
1answer
46 views

Closed conjugacy classes in $M_n(k)$

Let $k$ be an algebraically closed field, $n$ a positive integer, and consider the action of $\mathrm{GL}_n(k)$ on $M_n(k)$ by conjugation. My professor tells me that semisimple conjugacy classes are ...
0
votes
0answers
74 views

Transform one curve into another

I have been working on something for a while now, and I can't really get my head around it. I consider two curves with data points and want to determine the most optimal transform from one to another. ...
3
votes
2answers
91 views

Manipulating identities

I'm having some trouble deriving certain identities. If $$S(z) = \prod_{i=1}^n (z-z_i)$$ then how can I write $$\frac{1}{S(z)}\frac{d^2S}{dz^2} = \sum_{i=1}^n\frac{1}{z-z_i}\sum_{j\neq ...
0
votes
1answer
30 views

Matrix Vector Multiplications

If your coordinate system is assumed to be right-handed and given in the following Orthogonal Matrix. M =[l1 m1 n1;l2 m2 n2;l3 m3 n3] suppose we multiple this ...
1
vote
1answer
51 views

Computing the intersections of arbitrary number of planes

Imagine three sets of planes (A, B, and C). The planes in each set are parallel and evenly spaced and so can be defined by a plane equation and a scalar representing the distance between planes. If ...
3
votes
1answer
34 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 ...
1
vote
2answers
65 views

Does congruence guarantee length conversion?

Suppose that a linear transformation $M:R^2 \rightarrow R^2$ maps a triangle $ABC$ to a congruent triangle $A'B'C'$ ($\{A, B, O\}, \{B, C, O\},\{C, A, O\}$ are not colinear, and $A,B,C\neq O$) Is it ...
0
votes
3answers
35 views

Proof of normalvector on a plane

I found that, for the plane with linear equation: Ax + By + Cz = 0, that the vector a with coordinates: (A, B, C), is a normal vector on that plane. Where does that come from? And can someone provide ...
0
votes
2answers
45 views

Distorted Unitary matrices

Let $U$ be an unitary and $D$ be a diagonal matrix. We know that for all vectors $v$ on the sphere $Uv$ is on the sphere and, $$\langle Uv,Uv\rangle=\langle v,v\rangle.$$ What are the vectors $v$ on ...
0
votes
0answers
44 views

Vectors on a Sphere

Let $S$ be a sphere centered at origin in $\Bbb R^{2n}$ of radius $\sqrt{2n}$. Let $D$ be a diagonal matrix. Let $U$ be unitary matrix. Let $r\in\Bbb Z_+$ be a fixed integer. $(1)$ For a vector $v$ ...
5
votes
2answers
274 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, ...
1
vote
0answers
43 views

Is Grassmann-Plucker relation implied by 3-term Grassmann-Plucker relation?

It's a problem from book "Oriented Matroid", problem 3.16. More exactly: For $n$ elements $i_1,\dots,i_n$ and an anti-symmetric function $\det$ on${\{i_1,\dots,i_n\}}^r$. we have: ...
0
votes
0answers
42 views

Relations between minors of a matrix.

motivation: I'm looking at the Segre embedding, given by (for this example) $\mathbb{P}(U) \times \mathbb{P}(V) \rightarrow \mathbb{P}(U\otimes V)$, $([u],[v]) \mapsto [u\otimes v]$. This is an ...
2
votes
3answers
166 views

Find radius of a circle which is tangent to three known lines

I need to find the equation for a circle which is tangent to the following three lines: y=0 x=0 y=-x+0.338334 For the last tangent line equation, I know that it is tangent at the point (0.169167, ...
8
votes
1answer
137 views

Linear combination of matrices

Let $A, B, C, D$ be four linearly independent symmetric 3 x 3-matrices over $\mathbb K$. Show that some linear combination of these matrices is a matrix of rank 1. I know it is supposed to be a ...
1
vote
0answers
63 views

Invariants of the Determinant Form

Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial $$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n} $$ After the linear change of ...
2
votes
1answer
126 views

Projective equivalence

Definition Two projective plane curves $F$ and $G$ are projectively equivalent if there is a $\varphi_A\in PGL_2(k)$ such that $F(x,y,z)=G(a_{00}x+a_{01}y+a_{02}z,a_{10}x+a_{11}y+a_{12}z, ...
2
votes
0answers
29 views

Why is the definition of intersection multiplicity independent of choice of homogeneous coordinates?

I am using the following definition of intersection multiplicity of two algebraic curves $\mathcal C, \mathcal D$ of degrees $m,d$ without common components in $\Bbb P^2(\Bbb K)$: Let $P \in \mathcal ...
1
vote
1answer
34 views

Why $d_xG_{|T_xX}=0\implies d_xG=\lambda_1d_xF_1+\cdots+\lambda_md_xF_m$?

I'm trying to understand this proof of this theorem of a book I'm reading in basic algebraic geometry: Theorem Let $X$ be a closed affine subset and $x\in X$. The restriction of $d_x$ to $I_X(x)$ ...
-1
votes
1answer
67 views

find a common plane which contains two points (NP hard?)

In this problem the coordinates of 4 points are given. $p_{0}=(x_{0},y_{0},z_{0})$, $p_{1}=(x_{1},y_{1},z_{1})$, $p_{2}=(x_{2},y_{2},z_{2})$ and $p_{3}=(x_{3},y_{3},z_{3})$ I need to find the ...
2
votes
2answers
907 views

Decomposing an Affine transformation

An affine transformation is composed of rotations, translations, scaling and shearing. In 2D, such a transformation can be represented using an augmented matrix by $$ \begin{bmatrix} \vec{y} \\ 1 ...
3
votes
3answers
157 views

What is the dimension of this Grassmannian?

Why is $2\times 3$ the dimension of $Gr_2(\mathbb{R}^5)$? and can one use the dimensions of Lie groups to derive this dimension? Note: $Gr_2(\mathbb{R}^5)$ denotes the Grassmannian of all ...
0
votes
1answer
51 views

What is this dimension?

What is the real dimension of the cone of $2$ by $2$ Hermitain matrices with at lease one eigenvalue that is $0$?
37
votes
4answers
1k views

How to identify surfaces of revolution

Given a surface $f(x,y,z)=0$, how would you determine whether or not it's a surface of revolution, and find the axis of rotation? The special case where $f$ is a polynomial is also of interest. A ...
0
votes
1answer
70 views

what are the various fields in which circle is treated as infinite sided regular polygon?

What are the various fields in which circle is treated as infinite sided regular polygon? What I actually mean is , "can u suggest me some applications where circle is treated as infinite sided ...
2
votes
1answer
107 views

How to extract the indeterminates from a set of polynomial?

I am a biologist and I am facing a huge problem. I would like to extract the indeterminates of a set of polynomials, for example, I have: $f_{1} = \{\\x_{3}^{2} + x_{1}*x_{2} + x_{1} + x_{1}*x_{3},\\ ...
0
votes
1answer
351 views

How to find whether the line is inside the polygon or outside.

I have a polygon How can i prove whether the black color line lies outside the polygon or inside the polygon . Given the coordinates of the black line and all the vertices of the polygon.
2
votes
1answer
264 views

Rotation matrix convention; successive rotations in intermediate coordinate systems or not

I am getting very confused about the different conventions used for rotation matrices. Thing is I want to accomplish (3) successive rotations each time in the newly defined coordinate system. I use ...
1
vote
1answer
61 views

How do i define a plane orthogonal to a given one?

I have a plane $P_1$ given by the equation: $$ ax+by+cz+d=0$$ I want to find an orthogonal plane to that. I know that their normal vectors should be orthogonal so the normal vector $v_1$ of $P_1$ is ...
4
votes
0answers
90 views

What does “generic” mean in this context, and is it related to generic points in algebraic geometry?

In "The characteristic polynomial and determinant are not ad hoc constructions" by Skip Garibaldi, available at http://arxiv.org/abs/math/0203276 the characteristic polynomial is defined as the ...
0
votes
0answers
30 views

What can I say about these two points… [duplicate]

Two points on the graph of $y=kx^p$ are labeled $A$ and $C$. Point $A$ has coordinates $(a,b)$, where $0<a<1$ and point $C$ has coordinates $(c,d)$, where $1<c$. If we are told that that ...
1
vote
1answer
36 views

How to properly sort a set of axis-aligned boxes so they are drawn correctly under this projection?

Given a set S of axis-aligned, non-overlapping boxes {x,y,z,w,h,l}, where x,y,z are their center-positions and w,h,l their width, height and lengths, and given the following orthographic projection: ...
2
votes
2answers
68 views

Quadratic Bezier curves representation as implicit quadratic equation

A quadratic bezier curve from points P1=(x1, y1) to P3=(x3, y3) with control point ...
1
vote
1answer
50 views

Proof convex polyhedron with line does not contain a corner if closed

The excercise I am struggling with is the following: Given a convex closed polyhedron that contains a line, the question is, whether this polyhedron can also contain a corner. My idea was to make a ...
1
vote
0answers
42 views

Edge in a convex polytope

I want to show that a convex polytope $A$ that is an intersection of half-spaces contains an edge if $ A=\{x \in \mathbb{R}^n|Ax=0 \wedge x \ge 0\}$, where x greater equal 0 means, that all components ...
1
vote
0answers
87 views

Classification of Cones

I am attempting to classify (convex rational) cones in $\Bbb{R}^2$. We say here that $\sigma\subset\Bbb{R}^2$ is a cone if there exist $u,v\in\Bbb{Z}^2$ which $\Bbb{R}$-span the whole plane ...
0
votes
1answer
131 views

Points from an affine subspace with equal distance from given points

Given vector space $\mathbb{R}^3$ with dot product defined as $x \cdot y = 2x_1y_1 + 3x_2y_2 + x_3y_3$ where $x = (x_1,x_2,x_3),y = (y_1,y_2,y_3)$ and given an affine subspace $W: x - y - z - 2 = 0$ . ...
2
votes
1answer
69 views

Polynomial Equations for the Rank of a Power of a Matrix

If I have some $n \times n$ matrix $X$ (in my case, I happen to know that X is nilpotent and in Jordan normal form), how can I write the condition that $\text{rank}(X^r)= k$ as a polynomial equation ...
4
votes
2answers
211 views

Arrangements of affine hyperplanes

Fix $n>0$ and $X\subseteq\mathbb{R}^n$. Call a function $f:X\longrightarrow \mathbb{R}$ linear if it is of the form $$ f(\bar{x})=a_1x_1+\ldots+a_nx_n+b $$ for some $a_i,b\in\mathbb{R}$. Now ...