All Questions
36 questions
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: $C\subset [n]$ ...
- 1,046
2
votes
1
answer
262
views
Randomly fixing elements and transcendence degree
Given $f_1,\ldots,f_n \in \mathbb{F}_q[x_1,\ldots,x_m]$ such that $\deg(f_i) \leq d < q$. Suppose we have for some $1 \leq j \leq m$
$$ \operatorname{trdeg}_{\mathbb{F}(x_1,\ldots,x_j)}\{f_1,\ldots,...
- 341
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
2
votes
0
answers
148
views
Nilpotent polynomial matrices over $F_q$ - polynomial count variety ? ( Nilpotent cone for Hitchin-Gaudin like integrable system)
Context: Number of nilpotent $n\times n $ matrices over $F_q$ is $q^{n(n-1)}$ classical result due to Ph.Hall, M.Gerstenhaber (see very nice exposition by T.Leinster at n-cat-cafe/arxiv) which have ...
- 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?
- 869
5
votes
0
answers
107
views
Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$
Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
- 10.5k
3
votes
0
answers
107
views
Non-tree models of Lagrange inversion polynomials
The specific Lagrange inversion / series reversion polynomials (LIPs) I'm addressing are illustrated in OEIS A134685 with a general linear term and in Lang's pdf for A176740 with the coefficient of ...
- 10.5k
3
votes
0
answers
116
views
Intersection numbers of moduli spaces and noncrossing partitions
The coefficients of the monomials $u_1^{e_1}u_2^{e_2} \ldots u_n^{e_n}$ of the partition polynomials (ParPs) $[M=M1]$ on pg. 831 of The Handbook of Mathematical Functions by Abramowitz and Stegun are ...
- 10.5k
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
11
votes
2
answers
1k
views
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 ...
- 2,896
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
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
19
votes
3
answers
2k
views
Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the ...
- 10.5k
6
votes
1
answer
260
views
Vanishing linear combinations of minors
Let $V$ be the set of $k$ by $n$ matrices ($k<n$) with entries in $\mathbb{C}$, and let $\mathbb{C}[V]$ denote the set of polynomial functions on $V$. For any subset $I \subseteq [n] = \{1,2,\dotsc,...
- 7,300
3
votes
1
answer
451
views
Commutative algebra for the Conway games
I was reading the book On Number And Games and I have some question. In this book Conway constructed the set of "games" with a addition and a multiplication. I understand that the surreal numbers are ...
- 6,453
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
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
8
votes
2
answers
577
views
Pfaffian representation of the Fermat quintic
It is known (see for instance Beauville - Determinantal hypersurfaces) that a generic homogeneous polynomial in $5$ variables of degree $5$ with complex coefficients can be written as the Pfaffian of ...
- 7,300
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.
- 7,875
18
votes
1
answer
1k
views
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 ...
3
votes
0
answers
175
views
Geometric interpretation of homological quantities in Artinian local Gorenstein algebras
By corollary 3.5. of http://www.ams.org/journals/tran/2012-364-09/S0002-9947-2012-05430-4/S0002-9947-2012-05430-4.pdf the classification of local artinian Gorenstein algebras (all algebras here are ...
CommunityBot
- 1
1
vote
1
answer
274
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 ...
- 15.8k
8
votes
1
answer
604
views
Number of zeros of the derivatives of a polynomial
What is the maximum total number of zeroes a univariate polynomial $f\in\mathbb{C}[z]$ of degree $d$, together with all of its derivatives, can have at $k$ given points of $\mathbb{C}$?
I am ...
- 22.5k
2
votes
0
answers
255
views
Ideals with the same Hilbert series
Consider a polynomial ring $\mathbb C[x_1,\ldots,x_n]$ that is $\mathbb Z_{\ge 0}$-graded by degree. Let $I$ and $J$ be two homogeneous ideals therein with the same Hilbert series, i.e. with their ...
- 3,513
10
votes
2
answers
820
views
Simple question about polynomials
Starting from a problem in combinatorics, I ended up with a very simple problem about polynomials, which, unfortunately, I am not able to solve.
Say we work over $\mathbb C$. Fix $d>1$.
Is it ...
- 5,958
4
votes
0
answers
95
views
Topological hyperfields
I am trying to generalize the notion of reorientation class of an oriented matroid to the context of matroids over hyperfields (compare Baker and Bowler, 2016). I have already got some results in this ...
- 741
3
votes
1
answer
221
views
Alternating multisymmetric functions
I am looking for a reference on certain modules of invariants. I think that the question is quite natural so that I believe there should be some results already, but I am not able to find anything.
...
- 3,271
5
votes
1
answer
208
views
Zariski openness of Newton non-degenerate polynomials
Suppose you are given a convex polyhedron $\Delta$ in $\mathbb{R}^n$ (i.e. a convex hull of finitely many points in $\mathbb{Z}^n$) and consider a finite dimensional vector space $V$ over $\mathbb{C}$ ...
- 156k
5
votes
1
answer
528
views
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 affine,...
CommunityBot
- 1
11
votes
0
answers
629
views
Inversion, Koszul duality, combinatorics and geometry
According to this MO answer Koszul duality is related to operations on generating series;
1) multiplicative inversion for quadratic algebras,
2) compositional inversion for quadratic operads,
3) ...
CommunityBot
- 1
3
votes
2
answers
362
views
Kahler differentials on cluster varieties
On affine toric varieties there is a classical theorem of Danilov which gives some combinatorial ways to describe the global sections of an appropriate sheaf of Kahler differentials as a vector space. ...
- 156k
12
votes
1
answer
949
views
Discrete version of Nullstellensatz?
Hi. I was reading the paper "On the foundations of combinatorial theory (VI): The idea of a generating function" by Doubilet, Rota and Stanley, and there is a relation treated which is very ...
- 902
13
votes
3
answers
1k
views
Reference for combinatorics of cell decomposition of the Hilbert scheme of points in the plane
It is known from either Morse theory or Bialynicki-Birula decomposition that the fixed points of a ${\mathbb{C}}^*$ action on a smooth algebraic variety over $\mathbb{C}$ determine a cell ...
- 156k
11
votes
2
answers
1k
views
Geometric motivation for the Stanley-Reisner correspondence
The Stanley-Reisner ring of an abstract simplicial complex $\Delta$ on the vertex set $\{1,...,n\}$ is the $k$-algebra
$$
k[X_1,...,X_n]/I_\Delta
$$
where $I_\Delta$ is the ideal generated by the $X_{...
- 1,500
12
votes
0
answers
529
views
A commutative monoid associated with a finite abelian group
Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as
$$
v_{m,...
- 181
6
votes
3
answers
1k
views
Is there a software package that does Schubert Calculus computations?
Is there a good software package for doing computations in the cohomology ring of Grassmannians? Things like, I can write down a polynomial in, in fact, special Schubert classes, but it's one where ...
- 4,394