All Questions
Tagged with co.combinatorics ag.algebraic-geometry
115 questions with no upvoted or accepted answers
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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
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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 ...
- 11
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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
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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
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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 \...
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53
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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
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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
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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
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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
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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
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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
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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
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193
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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
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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
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230
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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 $f(p) \in Y$?
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