# Tagged Questions

611 views

### What is the longest algebraic curve?

Consider a convex body $\Omega\subset \mathbb{R}^2$. Let $L(d)$ be the maximum over all curves $C$ of degree $d$ of the length of $C\cap\Omega$. Is $L(d)\leq d P(E)/2$, where $P(E)$ is the ...
210 views

### Polygon Problem [closed]

There are $N$ regions which are numbered from $1$ to $N$. Each region is represented by a single simple polygon on the 2D plane. Simple polygon means the boundary of the polygon does not cross itself. ...
340 views

### Do elements of the fundamental group give rise to isometries

Let $X$ be a complex algebraic variety, and let $\tilde X\to X$ be its universal cover. Suppose that there exists a Kahler-Einstein metric on $\tilde X$. Note that $\pi_1(X) \subset Aut(\tilde X)$. ...
127 views

### Difference between Kahler-Einstein and Bergman metric on a bounded symmetric domain

Let $H$ be a bounded symmetric domain. What is the difference between the Bergman metric and the Kahler-Einstein metric on $H$?
503 views

### The Icosahedron Equation

$$1728 V^5 + F^3 = E^2 \;.$$ Can anyone point me to a concise, modern derivation and explanation of the significance of the icosahedron equation, more modern and concise than Klein's description in ...
120 views

### What are interesting 3-colorings of the plane without rainbow lines?

This question is about 3-colorings of the plane in which every line is bichromatic (or monochromatic), i.e., there are no three collinear points of different colors. Such colorings trivially exist, ...
272 views

### Integer lattice points on a hypersphere

Is the following statement true? For every integer $n\ge2$ and every integer $k\ge0$ there exists a hypersphere in $\mathbb{R}^n$ (circle, sphere etc) containing exactly $k$ integer lattice ...
321 views

### Constructing a field from a spherical building

Tits proved that (sufficiently high rank) spherical buildings arise from an algebraic group and a field, so any building is some $\Delta(G, F)$. He also showed that a building isomorphism ...
447 views

### Minimizing the excursion of a sum of unit vectors

I have $n$ unit-length vectors $v_i$ in $\mathbb{R}^3$, whose sum is zero: $$v_1 + v_2 + \cdots + v_n = 0 \; .$$ Now I form the closed polygon $P$ in space by placing them head to tail. So the ...
197 views

### When is the hull of a space curve composed of developable patches?

Let $C$ be a smooth curve in $\mathbb{R}^3$ that lies entirely on its convex hull, $\cal{H}(C)$. Under what conditions on $C$ is $\cal{H}(C)$ the union of developable surface patches? I believe ...
651 views

### The Constructions of Davis and Januszkiewicz

One of the most useful tools in the study of convex polytope is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on ...
416 views

### Affine “real algebraic geometry” of hyperbolic space?

Real algebraic geometry, at least to start with, traditionally studies the zero-sets of real polynomials in a given set of variables. But treating, say, the Euclidean plane as an uncoordinatized ...
482 views

### Maximum area of intersection between annulus and circle? [closed]

Given two concentric circles $[C_1,C_2 ]$ with radii $(R_1 < R_2)$ creating an annulus; where should a third circle ($C_3$, radius $R_3$) be located such that the area of intersection between ...
588 views

What is a good reference for the following result which I believe is proved by Tchebotarev. There are exactly 5 types of Lunes that are squarable. (Hippocrates produced three and then two more were ...
413 views

### Subtlety in the definition of the Kobayashi metric

When defining the Kobayashi metric on a connected complex analytic space $X$, one makes the following auxiliary definition: A holomorphic chain from $x\in X$ to $y\in X$ is a finite sequence of ...
357 views

### Mean Three Dimensional Shape of Surfaces

If I have $n, 1 < i < n,$ surfaces composed of $f_i$ faces and $v_i$ vertices, how would I go about finding the average surface? (I'm unsure what I mean by average - intuitively it's obvious, ...
604 views

### Fitting an ellipse to an arbitrary polygon

Hello, I'd like an algorithm for fitting an ellipse to a polygon. This polygon may be convex or concave. I've read about fitting an ellipse inside or outside a polygon (maximal and minimal of size, ...
127 views

### Are spherical codes algebraic?

Jeffrey Wang in Section 4.2 writes "Since a code is the solution to a number of polynomial equalities between the shortest edges, the coordinates of each rigid point in the code are algebraic and lie ...
123 views

### Tubular neighborhood growth of zero set of polynomial of bounded degree in the torus

This question is related to my related post: Volume growth of tubular neigbhorhood of critical values of an algebraic/differentiable map The setting here is as follows: Let \$p: \mathbb{R}^{2k} \to ...