# Tagged Questions

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### Is there a hyperplane avoiding two independent sets?

Let $V$ be a vector space over a field with $5$ elements, $A,B \subseteq V$ independent subsets. Must there be a subspace of $V$ of codimension 1 disjoint from $A \cup B$?
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### global sections of structure sheaf on the toric Calabi-Yau

Let P be a lattice polytope and lying in $N \times {1} \subset N \times \mathbb{R}$. Let $\sigma$ be the cone over this polytope and $X_\sigma$ be the corresponding toric variety, which is an ...
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### Connection between the number of vertices and the number of lattice points of the integer hull of a polytope?

Is there a connections between the number of vertices and the number of lattice points of $P_I$, the integer hull of a polytope $P$? Which is usually more difficult to determine? Or if I have a bound ...
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### Smith Normal Form of powers of a matrix

What invariants of a matrix determine the Smith Normal Form (SNF) of all the powers of a matrix? The question makes sense over any PID $R$. If we let $M = M_n(R)$ and $G=Gl_n(R)$, then SNF is a ...
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### A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface?

In a 1986 paper, Harer and Zagier proved the recursion: $$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$ where e(g,n) is the number of ways of grouping sides $S_1...S_{2n}$ of a 2n-gon ...
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### Reference request: Lascoux's formulas for Chern classes of tensor products and symmetric powers

Let $E$ and $F$ be vector bundles on a smooth projective variety, say. A. Lascoux ("Classes de Chern d'un produit tensoriel", C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 8, A385–A387) gave ...
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### Combinatorical surface is restricted to a closed face an injection

Hello :) I'm third year student of mathematics. In my own intrest i'm studying topology in combinatorical sense. Herefore i found also an lecture note in knot theory from Roberts. I want to understand ...
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Some work I am doing is connected with a sequence 1, 3, 40, 1225, 67956, $\dots$ which agrees with http://oeis.org/A012250 for all eight terms. The only useful information in OEIS on this sequence is ...
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### Family of hypersurfaces in (C^*)^2 corresponding to tropical family

Edit: I realize the mathematics below is lacking a precise phrasing. I hope that the intuitiion behind the question is clear enough that a reader will understand the question and provide guidance. The ...
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### Systems of simultaneous real quadratic equations

Starting from a problem in spectral graph theory, I got dragged into a problem in combinatorial matrix theory about constructing $n\times n$ real orthogonal matrices with a specified pattern of ...
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### Tensor rank of anti-symmetric tensor

Let $V$ be a vector space of dimension $n$. Let us consider $V^{\otimes n}=V\otimes V \ldots \otimes V$. This vector space contains one dimentional vector space $\wedge^n V$. My question is does it ...
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### elements in the weyl group

Let W be the Weyl a group of a semisimple simply connected group over C. Let I={1,...,r} the set of simple roots. For $w\in W$, I denote by supp(w) the subset of I corresponding to the simple ...
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### Chern Classes of Exterior Products of a vector bundle.

This is mostly a question in combinatorics. Is there a clean way in terms of determinantal identities to write down $c(\wedge^k V)$ i.e. the individual summands in terms of the individual summands of ...
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### Finding the matroids with a specified set of non-bases

I'm a grad student in algebraic geometry, and I've encountered a problem which requires me to produce an algorithm involving matroids. Since this isn't my area of expertise, I'm hoping someone knows ...
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### Smooth bases of matroids

Motivated by algebraic geometry, I've come up with a purely combinatorial definition within the theory of matroids. The question is: is this concept known? If you like matroids but not algebraic ...
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### 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 ...
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### Shimura-Taniyama-Weil VS Grothendieck's dessins

When listening to the beautiful lectures by Gilles Schaeffer at the SLC68, the following (perhaps crazy) question occurred to me: did anyone attempt (succeed?) to combinatorially prove modularity of ...
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### On the number of lines of given points

Hi all, I have a question Concerning Beck's theorem. I have read it from http://en.wikipedia.org/wiki/Beck%27s_theorem and I have two questions : I suppose Beck's theorem doesn't hold when instead ...
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### Detecting tilings by toric geometry

This is probably a silly question, but I figured that if there is a good answer, this would be a good place to ask. Ever since I got my hands on the book "Toric Varieties" by Cox, Little and Schenck, ...
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### Incidences of Lines / Circles in the Plane

During linear algebra class, I was explaining that given 2 equations + 2 unknowns where we expect there to be a unique solution but sometimes there can be 0 solutions or a line's worth. At some ...
We define the affine Grassmannian to be the quotient $Gr = GL_n(\mathbb{C}((t)))/GL_n(\mathbb{C}[[t]])$ where $\mathbb{C}((t))$ is the field of formal Laurent series and $\mathbb{C}[[t]]$ is the ring ...