All Questions
Tagged with co.combinatorics ag.algebraic-geometry
292 questions
1
vote
0
answers
164
views
Combinatorial question related to Hankel-type matrices
Let $\mathbb{N}$ be the set of non-negative integers. Let $n\geq 2, d$ be positive integers. I would like a lower bound on the largest integer $r$ for which the following property holds:
For any ...
- 980
1
vote
0
answers
158
views
Is there a better/newer list of Kazhdan-Lusztig polynomials?
I am essentially just re-asking this question, as it's now over a decade old, and I'm hoping that more extensive lists exist. I've started looking at the papers cited in the previous question, and ...
1
vote
0
answers
68
views
Leibniz formula for Fulton's divided difference operators for quantum Schubert polynomials
Schubert polynomials are polynomials in the ring $\mathbb{Z}[x]$ where $x$ is an infinite set of variables indexed by the positive integers and they can be expressed in terms of "standard ...
- 2,168
1
vote
0
answers
84
views
Number of polyhedral covers of a triangulation of $S^2$
For a given triangulation (combinatorial Type I. or Type II.) of a $2$-sphere, what is the number of unique polygonal covers with $n$ polygons where ($n$ goes from $2$ to $N$)?
Under polygonal cover, ...
- 183
1
vote
0
answers
162
views
Difficulty understanding a step in the proof of multiset version of Cauchy-Davenport Theorem
In a paper "G. Kós, L. Rónyai, Alon’s Nullstellensatz for multisets, Combinatorica, 32(5) (2012) 589-605", the authors prove a multiset version of the Cauchy-Davenport Theorem (please see ...
- 167
1
vote
0
answers
118
views
Schur polynomial with integer values
There is a way to characterize for which $x_1,...,x_d$ a Schur polynomial, that can be defined as
$$s_\lambda(x_1,...,x_d)=\sum_{T\in SSYT(\lambda)}x_1^{t_1}...x_d^{t_d}, $$
with the sum running over ...
1
vote
0
answers
173
views
The geometry of a commutative ring and the topology of its ideal complex
Suppose $R$ is a commutative Noetherian ring. Let $\mathcal{P}(R)$ be the poset of ideals of $R$ ordered by inclusion, and let $\Delta(R)$ be the order complex of $\mathcal{P}(R)$. $\Delta(R)$ is a ...
- 19
1
vote
0
answers
231
views
Cohomology of realization space of matroid
Do we know any thing about cohomology of realization space of matroid (the space of all set of vectors in $\mathbb{C}^k$ which captures the independence structure of matroid $M$), more simple, for ...
- 91
1
vote
0
answers
75
views
Symmetric matrices of hyperbolic and elliptic type with certain kind of trace zero
I have been working on a problem related to determinantal varieties in symmetric matrices. I am stuck at the following point and would like to get some reference/help for the following question.
Let $\...
- 179
1
vote
0
answers
205
views
on bilinear form on free abelian group
Let $P$ be a free abelian group of rank $n-2$ with an integral symmetric bilinear form $\left<,\right>$. A sequence of elements $A_1,\ldots,A_n$ in $P$ is called an abstract toric system iff it ...
- 1,982
1
vote
0
answers
90
views
Partition of sets of monomials
Given a degree $d>0$ and $x_1,\dots,x_n$, consider the set $S$ of monomials
\begin{equation*}
x_1^{i_1} \cdots x_n^{i_n}
\end{equation*}
with $0\leq i_j\leq d$ for $1\leq j\leq n$ (the exponents ...
- 53
1
vote
0
answers
207
views
Singular Locus of a Schubert variety
I am trying to compute the singular locus of the schubert variety $X_w$ in $G_{2,7}$ where $w=(4,7) \in I_{2,7}$. Following the notation in the book "The Grassmannian Variety: Geometric and ...
- 43
1
vote
0
answers
104
views
connectedness of semi algebraic sets
We know the inequalities $x_ix_j >\theta_{ij}$ or $x_ix_j<\theta_{ij}$ for some $\theta_{ij}$>0, some $i,j\in\{1,\cdots,n\}$, $i\neq j$ defines the easiest semi algebraic set in $R^n_{\geq 0}$, ...
- 53
1
vote
0
answers
54
views
Lattice-isotopic essentialization of arrangements
I'm working on a problem related to
$\textbf{Randell's isotopy theorem}$ for complex hyperplane arrangements. I have a question which seems quite obvious. However, I haven't found a rigorous proof ...
- 741
1
vote
0
answers
207
views
Polynomial existence over finite field
Denote $\mathcal{F_n}$ as collection of multiaffine polynomials $f\in\Bbb F_2[x_1,\dots,x_n]$.
Denote total degree of $f\in\mathcal{F_n}$ as $deg(f)$ (note $deg(f)\leq n$).
Denote $e_i=(0,\dots,0,\...
- 13.9k
1
vote
0
answers
96
views
Finding stable ideals of $\mathbb{F}_3[[X,S]]$ by group action
Let $k > 1$ be a positive integer and define the action $\sigma_k$ on $\mathbb{F}_3[[X,S]]$ by:
$\sigma_k: X \mapsto X + S + X^k$
$\sigma_k: S \mapsto S + S^3$.
Conjecture: There exists a ...
- 87
1
vote
0
answers
56
views
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 ...
- 11
1
vote
0
answers
513
views
Maximal disjoint hyperplanes
Assume a set of $n^{r}$ points $X_{r} = \{ x_{1}, \cdots, x_{n^{r}} \}$ is given occupying a codimension $t^{r}$ subspace in $\mathbb{R}^{n^{r}}$. Let $M_{r}$ be the set of $t^{r}$-tuples of these ...
- 800
1
vote
1
answer
273
views
Coefficients of the monomials appearing in a Schubert polynomial
It is known that the coefficients of the monomials appearing in a Schubert polynomial are always positive. My question is: Is it always true that at least one such coefficient must be $1$? If that is ...
user111492
0
votes
3
answers
622
views
When does the rigidity matrix of a graph have full row rank?
Intuitive description: In the 2D plane, there are $m$ bars connected by $n$ joints. The length of each bar is fixed. These joints and bars can be viewed as a graph (see the figures below). Denote $s_i$...
- 61
0
votes
1
answer
225
views
Is the Euler characteristic of a certain nonlinear variety related to that of a certain linear variety?
(This is a generalization of a question I posted a week ago.)
I'm looking at a variety sitting inside the algebraic torus $(\mathbb{C}\setminus 0)^n$ generated by the ideal $I = (*x_1^{\alpha_1} + \...
0
votes
1
answer
390
views
Moment maps and flat degenerations of toric varieties
We have a flat family of projective varieties with a torus $T$ action, over $\mathbb{A}^1$.
How do the moment map images of the fibers change when we pass from the generic fiber to the special fiber ...
- 1,719
0
votes
1
answer
307
views
About Alexander method in mapping class group
The Alexander Method in Farb and Maraglit's "A Primer on Mapping Class Groups"
For a surface $S$ with marked points, we say that a collection $\left\{\gamma_{i}\right\}$ of curves and arcs ...
- 119
0
votes
1
answer
91
views
Construct next polynomial from predecessor and resulting GCD
I have a sequence of polynomials built from an interpolation derived in a combinatorial problem. For each integer value of a parameter $n$ there is a different polynomial.
After trying to find a way ...
- 47
0
votes
1
answer
130
views
Complexifications of Real Flags are dense in complex Flag Varieties
Let $E$ be a complex vector space endowed with a Hermitian metric, it can be seen as the complexification of a real vector space $E_R$, with its metric coming from an inner product on $E_R$. Let $X$ ...
- 291
0
votes
2
answers
254
views
A specific polynomial triplet question
Notation
$P_k[n]=\{$multilinear polynomials in $\Bbb R[x_1,x_2,\dots,x_{n-1},x_n]$ of total degree exactly $k\}$.
$k=1$ is just linear polynomials.
QUESTION
Is there a triplet $(p,f,g)\in (P_{k}[4]...
- 13.9k
0
votes
1
answer
79
views
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 ...
0
votes
1
answer
367
views
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 ...
- 1
0
votes
0
answers
54
views
Functional equations with coupled arguments and additive sructure
Let $G$ be a locally compact abelian group and let $f: G \to \mathbb{R}^+$ be a continuous function satisfying the functional equation
$$f(x + \phi(y)) + f(y + \phi(x)) = 1 + f(x+y)$$
for all $x, y \...
0
votes
0
answers
53
views
A question on bounding the size of the polynomial
Suppose we are given the following n polynomials in $\bar{\mathbb{F}}_2[x_1,...,x_n]$:
$f_1 = x_1 + x_n^2$
$f_2 = x_2 + x_1^2$
$\cdot$
$\cdot$
$f_{n-3} = x_{n-3} + x_{n-4}^2$
$f_{n-2} = x_{n-2} + x_{n-...
- 341
0
votes
0
answers
61
views
Combinatorial counting question related to count (anti)commuting N-tuples of matrices (more generally $(X_1,...X_n): F(X_i,X_j)=0$ - only one F)
Consider some finite set $S$ (can be matrices over $F_p$), consider some symmetric relation $F(s1,s2)$ which values are True or False (for example - matrices (anti)commutate or not).
Question 1: can ...
- 24.9k
0
votes
0
answers
57
views
Reference for packing property and König property
Can someone please suggest reference material to study about the packing property and König property of ideals and some examples?
- 21
0
votes
0
answers
91
views
How to compute the multiplicity of a strongly convex, rational, polyhedral cone $ \sigma $?
In David Cox, John Little and Hal Schenck's book Toric Varieties page 302, Chapter 6, Section 4, Proposition 6.4.4, the authors state the following proposition. If $ \Sigma $ is a simplicial fan of ...
- 912
0
votes
0
answers
349
views
Relation between $3$-term Plücker relations and more than $3$-term Plücker relations
$\DeclareMathOperator\Gr{Gr}$Let $\Gr(k,n)$ be the Grassmannian variety of $k$-planes in an $n$-dimensional vector space. The coordinate algebra $\mathbb{C}[\Gr(k,n)]$ is generated by Plücker ...
- 6,201
0
votes
0
answers
67
views
Neccessary and sufficient condition for trivial rational solution of rational homogeneous cubic polynomials
If we consider a cubic homogeneous polynomial in $ 5 $ variables , $ ax_{1}^{3} + bx_{2}^{3} + cx_{3}^{3} + dx_{4}^{3} + ex_{5}^{3} + \sum_{i < j<k =1}^{5} f_{ijk} x_{i}x_{j}x_{k} $ where a,b,c,...
- 923
0
votes
0
answers
42
views
When is the set of faces of a convex polytope algebraically independent?
This is related to another question of mine
Let $V=\Bbb R^n$. Morelli defined the (commutative unital) ring $L(V)$ to be the additive group generated by the indicator functions of convex polytopes in ...
- 3,079
0
votes
0
answers
193
views
What breaks down in the theory of affine hyperplane arrangments?
It appears to me that there is a substantial amount of combinatorial algebra and geometry supporting the theory of central hyperplane arrangements (See Topics in Hyperplane Arrangements, Aguiar and ...
- 101
0
votes
0
answers
74
views
Upper bound on number of cells created by varieties of co-dimension 1
Say I have polynomials $p_1,p_2,\dots,p_m$ in $\mathbb{R}^n$ (ie. over $n$ variables), each of degree $d$. Is there an upper bound on the number of "regions" created by the surfaces $p_i = 0$? Let's ...
- 143
0
votes
0
answers
230
views
Toric morphism fiber and kernel dimensions
Given a morphism between two smooth toric varieties $f: X \rightarrow Y$, is the dimension of the kernel of $\mathrm{d}f$ at any point $p \in X$ equal to the dimension of the fiber at $f(p) \in Y$?
...
- 1,719
0
votes
0
answers
2k
views
Ignore this question [closed]
This question is a hacky way to create some tags for you to use. Move along.
-1
votes
1
answer
304
views
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 ...
- 22.8k
-1
votes
1
answer
139
views
What is the period of this expression? [closed]
I already know that this expression is periodic, but what I don't know is what is the period. In other words what values of $n$ will give me zeros for this:
$$ \Sigma_{i=1}^n\sin(6\Sigma_{x=1}^i x^2-...
- 1