Questions tagged [geometric-invariant]
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22 questions
6 votes
2 answers
674 views
Mismatching Euler characteristic of the Torus
Why is it that when I try to compute the Euler characteristic for the Torus using a drawing like the following , then the number that I get is not the number that the Torus should have? Which is $0$? ...
- 667
0 votes
1 answer
91 views
Proof for "any $p \in \operatorname{Sym^3(\mathbb C^2)}$ can be written as a product of linear factors?"
Here is the paper I am trying to understand: I need a proof that "any $p \in \operatorname{Sym^3(\mathbb C^2)}$ can be written as a product of linear factors $(a_1 x + b_1 y)(a_2 x + b_2 y)(a_3 x ...
- 69
1 vote
2 answers
75 views
Why are the linear factors of $g.p$ are given by $(a_i, b_i)g^{-1}$?
I am reading a part of the paper below that computes the semistable locus in case of $\operatorname{Sym^3}(\mathbb{C}^2).$ Here is the part of the paper I do not understand: Specifically, I do not ...
- 69
1 vote
1 answer
112 views
Role of finite generation of the ring of invariants in the existence of a categorical quotient
From the Geometric Invariant Theory book [Mumford - Fogarty - Kirwan], we have the following theorem ([MFK,Theorem 1.1) Let $X$ be an affine scheme over a field $k$, let $G$ be a reductive algebraic ...
- 3,429
6 votes
1 answer
233 views
Clairaut differential equations and elliptic discriminants
I was solving this math.SE question, which was asking to solve the Clairaut differential equation $y= xy' - (y')^3$. Just to have nicer signs, I then looked at the equivalent equation $$ y= xy' + (y')^...
- 1,337
1 vote
1 answer
180 views
Polygon / Any shape invariant for comparison or fiting
For my personal curiosity, I was wondering which would be simplest algorithmic way to compare two shapes to say whether they are the same or not. After some researches, I found out that there are many ...
- 123
2 votes
2 answers
245 views
Insight on difference between Euler characteristics of 2 manifolds: $\chi(U)-\chi(V)$?
For the Euler characteristic, we have the inclusion-exclusion principle: $$\chi(U\cup V) = \chi(U)+\chi(V)-\chi(U \cap V),$$ and also the connected sum property: $$ \chi(U\#V) = \chi(U)+\chi(V)-\chi(...
- 189
-1 votes
1 answer
320 views
Do two exponential spirals intersect?
I have lists of complex points: orbit of complex point z under quadratic function f(z) = z*z I know that lists are: z, z^2, z^4, z^8, ... (r,t), (r^2, 2*t), ......
- 1,786
-2 votes
2 answers
249 views
The sum of squares of distances from the vertexes of regular polygon to the any line that passes the center of it. [closed]
To prove that it is geometric invariant I need to find some others. I was thinking about proving it by the Pythagorean theorem, using the fact that in all cases the distance from the vertex to the ...
9 votes
2 answers
2k views
How can I get better at solving problems using the Invariance Principle?
I have some questions regarding the Invariance Principle commonly used in contest math. It is well known that even though invariants can make problems easier to solve, finding invariants can be really,...
0 votes
0 answers
4k views
How to find Invariant Lines and Lines of Invariant Points, without utilising Eigenvectors?
this is my first post so I do apologise regarding any formatting issues! I have a question regarding invariant lines and lines of invariant points; from what I can gather, an invariant line is one of ...
6 votes
1 answer
843 views
Is dot product the only rotation invariant function?
I am looking for rotation invariant scalar functions $f(x,y): x,y \in R^3$ that are not some scalar function over the dot product (or norm), i.e. $ f \neq g(x\cdot y, \Vert x \Vert, \Vert y \Vert ) $ ...
- 61
2 votes
1 answer
1k views
Invariance of the second moment of area of a regular polygon
Consider a $n$-sided regular (convex) polygon and its circumscribed circle of radius $r$, centered in $(0,0)$. Fixing $(r,0)$ as the coordinate of the first vertex, the $n$ vertices of the polygon ...
- 3,564
2 votes
2 answers
56 views
Dimension of the group of all motions in $\mathbb{R}^n$ which leaves a fixed r-plane invariant.
As the title, I would like to ask the dimension of the group of all motions in $\mathbb{R}^n$ which leaves a fixed r-plane $L^0_r$ invariant. Here is my observation, but I don't know if it is useful ...
- 31
1 vote
1 answer
90 views
What is known about rational points on the ideal of relations / syzygy ideal?
What is known about rational points on the ideal of relations / syzygy ideal? Let $G$ be a finite group, with $|G|=n$. Then $G$ acts on $\mathbb{Q}[x_1,\cdots,x_n]$ through the regular representation (...
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