All Questions
Tagged with co.combinatorics ag.algebraic-geometry
292 questions
0
votes
1
answer
308
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
3
votes
0
answers
179
views
Polytope algebra and toric vareties
Let $\Pi$ denote the McMullen polytope algebra (over the field of rationals $\mathbb{Q}$) generated by convex polytopes with rational vertices in $\mathbb{R}^n$.
For a simple polytope $P$ let us ...
- 21.8k
11
votes
2
answers
977
views
Reference for combinatorics with view towards representation theory/algebraic geometry
I'm making this post to ask for a reference about combinatorics: I'm a PhD student in representation theory/algebraic geometry. My background is mostly in algebra and geometry (and also mostly ...
- 1,061
18
votes
5
answers
1k
views
Number of $3\times 3$ anticommuting matrices over finite fields $\mathbb{F}_p$ is (or is not?) polynomial in $p$?
There are rare algebraic varieties such that the number of points over finite fields $\mathbb{F}_p$ is given by a polynomial in $p$. One notable series of examples is the commuting variety: $[A,B]=0$ ...
- 24.9k
5
votes
1
answer
357
views
Asymptotics of degree of $\textrm{SO}_n$?
(This is an offshoot of Degree of parametrization of $\textrm{SO}_n$?)
Consider $G=\textrm{SO}_n$ as an affine subvariety of the affine space of $N$-by-$N$ matrices. There is an explicit expression ...
- 20.2k
2
votes
1
answer
215
views
The growth of the number of Fano complete intersection families
I recently calculated the number (possible multidegrees) of Fano complete intersections of dimension $n$ , because I wanted to make the remark that it grows "very rapidly" as $n \rightarrow \...
- 6,995
4
votes
0
answers
152
views
How to show the set of stable polynomials equals to the set of Lorentzian polynomials in degree 2
Give a homogenous polynomial $f\in \mathbb{R}[x_1,\dots,x_n]$ of degree $2$ in $n$ variables, we can consider $f$ as a quadratic form.
We call $L_n^2:=$ the set of quadratic forms with nonnegative ...
- 41
3
votes
1
answer
162
views
Recurrence relation for the moments of the GOE
The Harer-Zagier formula provides a three term recurrence relation for the expected value of the single-trace operator $\mathrm {Tr}(X^k)$ where $X$ is a $N\times N$ matrix from the GUE. Is there an ...
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
2
votes
1
answer
474
views
Chevalley–Warning theorem for rational field $\mathbb{Q} $
At 1st we consider some weak statement of Chevalley–Warning theorem for any finite field: If $f$ is a homogeneous polynomial of degree $d$ with $n$ independent variables over a finite field $F$. Then ...
- 923
15
votes
5
answers
1k
views
Comparing two power-series
I have been wrestling with the following problem for some time now. If possible, I would prefer a hint rather than a full solution, because I would like to "solve" it myself.
Let $f(z)$ be a ...
- 988
3
votes
0
answers
105
views
Structure of fibers of (complex) moment map of hypertoric variety
I am primarily interested in the hypertoric variety $\mathfrak M(\mathcal B_d)$ associated to the braid arrangement.
Any hypertoric variety $X$, say of complex dimension $2n$, comes equipped with an ...
- 71
15
votes
3
answers
892
views
Log-concavity of matroids: characterization of equality?
Let $M$ be a (loopless) matroid of rank $r$.
The characteristic polynomial $\chi_M(x)$ is defined by $\chi_M(x)=\sum_{F \in \mathcal{L}(M)}\mu(\hat{0},F) \cdot x^{\mathrm{rk}(F)}$, where $ \mathcal{L}(...
- 24.2k
4
votes
0
answers
87
views
Toric Bézier patches
Toric Bézier patches (as described in https://arxiv.org/abs/0706.2116) are maps from a lattice polytope $P$ to the positive part of its associated toric variety $X_P$. While they are not the inverse ...
- 1,435
5
votes
1
answer
163
views
Polynomials vanishing on prescribed layers
Given a prime $p$ and an integer $n\ge p$, what is the smallest possible degree of a polynomial $Q\in\mathbb F_p[x_1,\dotsc, x_n]$ such that $Q$ vanishes on every vector $x\in\{0,1\}^n$ of weight $w(x)...
- 23k
3
votes
1
answer
336
views
Cohomology ring of a hypersurface in toric variety
Let $X^n$ be a smooth projective toric variety corresponding to a simple polytope $P$. It is well known that the cohomology ring $H^*(X)$ can be described in terms of combinatorics of faces of $P$.
...
- 21.8k
14
votes
2
answers
571
views
Number of d-Calabi-Yau partitions
This problem arises from algebraic geometry/representation theory, see https://arxiv.org/pdf/1409.0668.pdf (chapter 2).
We call a partition $p=[p_1,...,p_n]$ with $2 \leq p_1 \leq p_2 \leq ... \leq ...
- 26.5k
4
votes
1
answer
233
views
What's known about the matroid induced by the Plücker coordinates of the representation of a matroid?
Let $M$ be a linear matroid with ground set $E$ and independent subsets $\mathcal I$, represented by $\rho: E \rightarrow V$.
This induces a map
$$
\hat\rho: \mathcal I \rightarrow \mathbf P(\Lambda V)...
- 213
4
votes
0
answers
144
views
Generalized Catalan generating series
Let
$$
\mathscr{B}_k(z) = \sum_{n\geq 0}{kn+1\choose n}\frac{1}{kn+1}z^n\,,
$$
then it is well known that
$$
\tag{1}\label{1}
\text{log}\mathscr{B}_k(z)= \sum_{n\geq 1}{kn\choose n}\frac{1}{kn}z^n\,.
$...
- 988
1
vote
1
answer
338
views
Polynomial form/Fourier transform of rational function over finite affine space
I am certainly going to make a mess of any serious algebraic terminology, so bear with me as I present my problem arising from a probability problem.
Consider the space of sequences of $n$ zero-one ...
- 526
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
10
votes
1
answer
299
views
Map from Bruhat stratification to Catalan stratification for the space of totally nonnegative upper-triangular matrices
$\DeclareMathOperator\SL{SL}$This question came up in a class "Total Positivity and Cluster Algebras" being taught by Chris Fraser.
Let $N^+$ denote the space of uni-upper-triangular ...
- 24.2k
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
7
votes
1
answer
481
views
Geometric foundation of the Grothendieck polynomials
Grothendieck polynomials were firstly defined in
Alain Lascoux and Marcel-Paul Sch¨utzenberger. Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une vari´et´e de drapeaux....
- 719
7
votes
1
answer
489
views
Combinatorial meaning of Kazhdan-Lusztig-Stanley polynomial
This question is motivated by
Why do combinatorial abstractions of geometric objects behave so well?
The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials
Kazhdan-Lusztig-Stanley polynomials ...
- 5,230
8
votes
0
answers
313
views
Cohomology of the complement of the resonance hyperplane arrangement
Here was a question about resonance arrangement. It is defined as follows.
Let $x_i$ be the standard coordinates on $\mathbb{C}^n$. For each nonempty $I\subseteq\{1,\dots,n\}$, define the hyperplane $...
- 743
5
votes
0
answers
270
views
Connected relative Gromov Witten invariants
I am currently interested to compute relative Gromov Witten invariants(GW) over $\mathbb{P}^1$.
In the paper
https://arxiv.org/pdf/math/0204305.pdf
eq 3.1 gives the count of relative disconnected GW ...
- 685
9
votes
0
answers
361
views
Bernoulli-like polynomials
Let $\psi_0 (x,t)=\frac{te^{xt}}{1-e^{-t}}$. Then
$$\psi_0(0,t)=\frac{t}{1-e^{-t}};$$
$$\psi_0(x,t)=1+\sum_{n=1}^\infty \frac{t^n}{n!} B_n(x)$$
where $B_n$ is a monic polynomial of degree $n.$
Now ...
- 401
5
votes
1
answer
278
views
Set-theoretic generation by circuit polynomials
Let $P$ be a prime ideal in $S=\mathbb{C}[x_1,\ldots , x_n],$ and write $[n] = \{ 1, \ldots , n \}.$ The algebraic matroid of $P$ can be defined according to circuit axioms as follows: $C\subset [n]$ ...
- 396
3
votes
0
answers
157
views
F-vectors of simplicial complex and f-vectors of non-faces of simplicial complex
Is there any result which gives us a relation between f-vector of simplicial complex and f-vector of nonfaces of a simplicial complex?
Thank you
- 51
15
votes
5
answers
2k
views
Striking existence theorems with mild conditions, and simple to state: more recent examples?
I would like to write an article about powerful existence theorems that assert, under mild and simple conditions, that some basic pattern or regularity exist. See some examples below. By mild ...
8
votes
1
answer
422
views
Conjecture by Ekedahl on Weyl groups and Abelian varieties
A conjecture was made on p.14 in "Cycle Classes of the E-O Stratification on the Moduli of Abelian Varieties" by Torsten Ekedahl (late, excellent contributor to MO) and Gerard Van Der Geer concerning ...
- 10.5k
4
votes
0
answers
206
views
A conjectural inequality of the constant terms of functions
Could someone help me with the following question? This is equivalent to my previous question
A conjecture about the barycenter of a polytope
Let $D$ be a differential operator defined as follows,
\...
- 1,121
4
votes
0
answers
130
views
program to compute hurwitz numbers
Is there a computer program available to compute Hurwitz numbers easily? In fact I only care about counting covers $C\to\mathbb{P}^1$ branched over $0,1,\infty$, and am even willing to restrict to the ...
- 735
2
votes
0
answers
112
views
Getzler's stable graphs for modular operads
In The semi-classical approximation for modular operads, Getzler displays a table at the bottom of page two enumerating certain stable graphs. (This is related to the MO-Q "Stable graphs: Feynman ...
- 10.5k
8
votes
1
answer
381
views
A "polar dual" for projective varieties?
Given a projective variety $X$ (over $\mathbb{C}$, say) with an affine paving $X=\sqcup_i C_i$, one can construct a poset $P_X$ on the set of cells $\{C_i\}$ by saying $C_i \leq C_j$ whenever $C_i \...
- 2,753
2
votes
0
answers
94
views
Anything similar to cone product formula (for convex polytopes)?
The convex polytope flag vector ring $\mathcal{R}$ satisfies the cone product formula
$$
C(U) C(V) = C(J(U, V)) + DUV
$$
where
$$
J(U, V) = U C(V) + C(U) V - e_1 UV
$$
is the join formula.
Note: ...
- 459
1
vote
1
answer
241
views
Integral zeros of a multivariate polynomial
Consider the multivariate polynomial
$$f(x_1,\ldots,x_m)=mk\sum_{i=1}^mx_i^2-mk(k-1)\sum_{i=1}^mx_i-\left(\sum_{i=1}^mx_i\right)^2,$$
for integers $m,k\ge2$. We are looking for integral zeros of $f$ ...
- 33
6
votes
1
answer
539
views
Proofs of Euler's characteristic formula for n-Dim polytopes
Twenty proofs of Euler's formula V - E + F - 1 = 1, which applies to convex polyhedrons, i.e., 3-dimensional polytopes, are presented at the Geometry Junkyard.
I'm interested in proofs of the more ...
- 10.5k
2
votes
0
answers
150
views
Projection of conormal bundle of Schubert variety under Springer resolution
Let $G=\mathrm{GL}_n(\mathbb{C})$ and $X_{\omega}=\overline{B_-wB/B}\subset G/B$ be a Schubert variety. Denote by $C(X_\omega)$ the conormal variety inside $T^*(G/B)$ ,
$\mu:T^*(G/B)\to \mathcal{N}$ ...
- 849
2
votes
1
answer
186
views
Non-commutative projective lines
There have been many approaches to the notion of projective line:
combinatorial approaches (e.g. as certain permutation groups, such as $\mathrm{PGL}_2(k)$ in its natural action on $\mathbb{P}^1(k)$, ...
- 4,547
16
votes
2
answers
539
views
Surprising appearances of Painlevé transcendents
What are some of your favorite examples of enumerative problems whose answer ended up being (related to) a solution to one of the Painlevé equations?
I have seen examples from enumeration of classes ...
1
vote
1
answer
102
views
Software for Hilbert series of quotients of exterior algebras
Is there some software which computes Hilbert series of quotients of exterior algebras? In commutative case, Maple can compute Hilbert series. Thank you very much.
- 6,201
7
votes
0
answers
276
views
Cyclic shift acting on finite Grassmannian
The (twisted) cyclic shift $(v_1,v_2,\ldots,v_n) \mapsto (v_2,v_3,\ldots,v_n,(-1)^{k-1}v_1)$ acting on the Grassmannian $\mathrm{Gr}(\mathbb{C};k,n)$ of $k$-planes in $\mathbb{C}^n$ is an important ...
- 24.2k
5
votes
1
answer
453
views
Polynomial defined recursively by a resultant
Cross posting from MSE.
Definition:
For any natural number $n\ge 3$, define the polynomial $P_{n}\left(x_1,x_2,...,x_{n-1},x_{n} \right)$, with indeterminates $x_{i}$, where $i\in\{1,2,...,n-1,n\}$, ...
- 149
7
votes
1
answer
148
views
Combinatorial curves in combinatorial projective planes
Suppose $\mathcal{P}$ is an axiomatic projective plane (a nondegenerate point-line incidence geometry in which there is a unique line on any two different points, and a unique intersection point for ...
- 4,547
3
votes
0
answers
243
views
Interlacing sequences by polynomials?
Given $t=2^\ell$ where $\ell\in\mathbb N_{>0}$ and $M\in\mathbb Z$ and two sets of integers $\{a_1,\dots,a_t\}$ and $\{b_1,\dots,b_t\}$ with $0<a_1\leq \dots\leq a_t<M$ and $0<b_1\leq \...
- 13.9k
4
votes
1
answer
196
views
Polynomial time decodable binary linear codes achieving $GV$ bound?
Are there explicit or random construction of linear codes that achieve the $GV$ bound with polynomial time decodable property with alphabet size $q=2$?
Tsfasman, Manin, Vladut beat the bound at ...
- 13.9k
3
votes
1
answer
138
views
Intersection of quadratic equations with planted solutions?
Suppose we have two quadratic equations in $\mathbb R[x_1,\dots, x_n]$. What is the expected dimension of their intersection?
In general what can we say about intersection of $k$ quadratics? How many ...
- 13.9k
9
votes
0
answers
2k
views
Exactly Counting the Number of Lattice Points in an $n$-Dimensional Sphere
Let $S_n(R)$ denote the number of lattice points in an $n$-dimensional "sphere" with radius $R$. For clarification, I am interested in lattice points found both strictly inside the sphere, and on its ...
- 311