# All Questions

152,187
questions

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### What are the primitive notions and axioms in model theory?

I know every theory has its primitive notions and axioms. Now I am reading Basic Model Theory, and there is no term or sentence referred as to a primitive notion or an axiom. But, I think I know that ...

0
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46
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### Elimination over $\mathbb F_p[x,y]$

Let $p$ be a prime. Consider the two independent modular equations:
$$a_1x^2+b_1y^2+c_1xy\equiv d_1\bmod\mathbb F_p$$
$$a_2x^2+b_2y^2+c_2xy\equiv d_2\bmod\mathbb F_p$$
Is it possible to extract the ...

3
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0
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44
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### Is the Whitehead bracket $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$ injective?

Let $X$ and $Y$ be finite CW-complexes and $p,q\geq 2$. The Whitehead bracket induces a homomorphism $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$, $\alpha\otimes \beta\mapsto [\alpha,\beta]$....

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51
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### Hilbert scheme of points on an arithmetic surface

Let $X$ be a smooth surface over a field $k$. Fogarty proved that the Hilbert scheme of points $\mathrm{Hilb}^n(X)$ is regular. My primary question is: if $X$ is a smooth (quasi-projective) surface ...

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34
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### What is the proper name for this "tersest path" problem in Infinite Craft?

The web game Infinite Craft gives you a starting set of elements $V_0\subset V$ and a mapping $E$ of type $V\times V\rightarrow V$. In fact, $F$ is commutative: $E(v_a,v_b) = E(v_b,v_a)$. So another ...

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22
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### One parameter subgroups of reductive algebraic groups

If I have a reductive algebraic group $G$ defined over a non-archimedean local field $F$. We can define a one-parameter subgroup to be a group homomorphism from $G_{m}$ to $G$. I was wondering, if I ...

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35
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### Is it possible for the dihedral angles of a polyhedron to all grow simultaneously?

(Originally on MSE.)
Suppose $P$ and $Q$ are combinatorially equivalent non-self-intersecting polyhedra in $\mathbb{R}^3$, with $f$ a map from edges of $P$ to edges of $Q$ under said combinatorial ...

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0
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48
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### If Kolmogorov continuity criterion gives the optimal Hölder regularity then does the process have all moments?

Although very useful in the Gaussian (or other infinite moment) setting, Kolmogorov continuity criterion is non optimal in the finite moment setting. For example, let $X(t)=Zt$ where $Z$ is a random ...

2
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37
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### Some questions about induced subgraphs of the directed hypercube graph

Let $Q^n$ be the hypercube graph in $n$ dimensions. Hao Huang famously showed that any induced subgraph on more than $2^{n-1}$ must have maximum degree $ \geq \sqrt{n}$. It is also known that this ...

2
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19
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### Norm estimate for parabolic SPDE solution

When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|...

1
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0
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101
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### Tame-Wild dichotomy; why cant tame algebras be wild?

I would like to understand the Tame-Wild dichotomy, and in particular why an algebra cannot be tame and (semi-)wild at the same time. I've looked in the papers by Drozd and Crawley-Boevey[D80, CB88].
...

2
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21
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### Smoothness of the Fréchet Function

Let $M$ be a compact Riemannian manifold and $d$ be the induced distance function. Suppose $\mu$ is a probability measure on $M$ with continuous density. The Fréchet function is defined as
$$
F(x) = \...

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0
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43
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### Affine Springer fibers for symmetric spaces

Springer fibers are defined to be the varieties of "isotropic" full flags which are fixed by a certain element in the symmetric space. In a similar manner, affine Springer fibers can be ...

5
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1
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206
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### Trying to understand the topology of the Weil group for the local Langlands conjecture

I am trying to study the representation of the Weil group from the book "The Local Langlands Conjecture for $GL(2)$". I have some problem with the topology of this group.
Let $F$ be a non ...

3
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73
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### Intersection complex of genus-zero curves?

I would like to have a very explicit description of $\bar M_{0, n}$, especially its boundary divisors and how they intersect. All I can do in my construction is add divisors and blow up at strata, ...

3
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0
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62
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### Equivalence between vector bundles with integrable connections to isocrystals

Let $k$ be a perfect field, $W(k)$ its Witt ring, and $K$ the fraction field of $W(k)$. Let $X_k$ be a smooth proper curve over $k$, and let $X_K$ be the schematic generic fibre of a smooth proper ...

4
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74
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### Hecke algebra $\mathcal{H}(K_1\backslash \mathrm{GL}_n(\mathbb{F})/K_1)$

$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers, and let $\frak{m}$ be its maximal ideal Let $\GL_n(\mathcal{O})$ be the group ...

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34
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### Normality in a tower of cyclic extensions of global fields, as in Artin-Tate

Let $L_0$ be a global field without real places, that is, a global function field or a totally imaginary number field,
and let $V_f(L_0)$ denote the set of finite (that is, non-archimedean) places of $...

1
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0
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39
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### Formula and smallest solution for the A260711

Let $a(n)$ be A260711 without initial $0$ (i.e., numbers of the form $x^2 - y^2$ with $x > y$ where $x$ and $y$ are odd, $x + y$ is a power of $2$).
The sequence begins with
$$
8, 16, 32, 48, 64, ...

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1
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66
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### Closed unbounded sets and partitions

Let $\kappa$ be a regular, uncountable cardinal. Let $S\subseteq \kappa$ be a closed and unbounded set. Suppose that we partition $S$ into $<\kappa$ pieces. Does one of those pieces contain a ...

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73
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### Discrete spectrum of $A \otimes 1+ 1 \otimes B$

Let $A, B$ be unbounded self-adjoint operators on Hilbert spaces $\mathcal{H_1}, \mathcal{H_2}$, with both non-empty discrete spectra. Let us say, for instance, $\inf \, \sigma(A) = \lambda_1^A$ and $...

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96
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### Vanishing of first group cohomology

Let $\Gamma$ be a finitely presented group, acting by isometries on a normed vector space $(V,\|\cdot\|)$. The first group cohomology $H^{1}(\Gamma,V)$ is the space of crossed homomorphisms modulo ...

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51
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### Gradient estimates of linear elliptic PDE

Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain. Assume that $u(x)$ is the classical solution solving
$$a_{ij}(x)\partial_{ij}u(x)+b_i(x)\partial_iu(x)+c(x)u(x)=f(x)$$
$$u(x)\Big|_{\...

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0
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112
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### Effective bound for odd numbers expressed as sums of three primes

I am interested in the representation of odd numbers greater than five as sums of three primes, inspired by Harald Helfgott's seminal proof of the ternary Goldbach conjecture and the nuanced findings ...

4
votes

1
answer

132
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### Fppf or étale extension of group algebraic spaces

Let $S$ be a scheme and let
$$0 \to A \to B \to C \to 0$$
be an exact sequence of abelian sheaves on $(\mathrm{Sch}/S)_\text{fppf}$. Assume that $A$ and $C$ are representable by flat algebraic spaces. ...

-6
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26
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### Similar Rectangles [closed]

Garden a with a length of 6ft and a width of 8feet is similar to garden b with a length of 15 feet. it takes you 45 minutes to plant garden a. how long does it take you to plant garden b?enter image ...

4
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68
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### Growth of spheres in FINITE nilpotent groups - Gaussian approximation (central limit theorem)?

Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is ...

2
votes

1
answer

57
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### Let $G$ be a graph of genus $g$. Is the number of (non necessarily disjoint) 5-clique subgraphs at most $f(g)$ for some function $f$?

For a graph of genus $g$, it holds that it cannot have too many disjoint 5-cliques, as each clique requires a new handle. It feels that given a graph of genus $g$, it cannot have an unbounded number ...

2
votes

0
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50
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### Cohomological dimension of functors from fields to vector spaces

Let $K$ be an algebraically closed field. Denote by $\mathcal F_d$ the category of extensions $K\to F$ of transcendent degree $d$.(Objects are pairs $F,j$ consisting of a field $F$ and the embedding $...

4
votes

1
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146
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### Subset of the reals with zero inner measure and "full" outer measure in $\mathsf{ZF}+\mathsf{DC}$

Working in $\mathsf{ZF}+\mathsf{DC}$ (that is, we are allowed to use Dependent Choice but not full choice), suppose that there exists a non-measurable subset of the unit interval $[0,1]$ (just non-...

3
votes

1
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119
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### Solution of SDE at finite time, continuity of pdf

I'm looking at the Langevin dynamics described by the following SDE
$$d X_t = - \nabla U(X_t) \, d t + \sqrt {2 \Sigma} \, d B_t,$$
where $X_t \in \mathbb R^d$, $\nabla U(\cdot)$ has some regularity ...

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72
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### A Combinatorial Geometric Problem [closed]

It's a math (geometry, combinatorial geometry) problem.
Ten cowboys are on the ground (which is totally plane), when the clock strikes at 12 o'clock，they shoot at the one who is nearest to them. How ...

5
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0
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68
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### Spherical functions in the space of functions on real Grassmannians

Let $G=O(n)$ be the orthogonal group. Let $K=S(O(k)\times O(n-k))$ be the subgroup of $O(n)$.
Then the pair $(G,K)$ is symmetric, and the homomogeneous space $G/K$ is the Grassmannian of $k$-...

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0
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31
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### Product of two circles and holomorphic functions

Assume that $f$ and $g$ are holomorphic functions in the unit disk having boundary values on the unit circle $T$ almost everywhere. Assume further that $$\int_0^{2\pi}\int_0^{2\pi}|f(e^{it})+g(e^{is})|...

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0
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145
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### Degree 6 Galois extension over $\mathbb{Q} $

Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...

3
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0
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102
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### A question about Gromov's proof of a "more effective version of the main theorem"

In the paper "Groups of polynomial growth and expanding maps" Gromov proves the following "effective version of the main theorem"
For any positive integers $d$ and $k$, there ...

1
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0
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16
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### Riemannian gradient flow over a convex set

Given a function $A: \mathbb{R}^n \rightarrow \mathbb{R}^{n \times n}$ that is Lipschitz continuous, and $A(x)$ is positive definite for all $x \in \text{int} C$ and $A(x) = 0$ for all $x \in \text{bd}...

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0
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82
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### A cell complex constructed from singular knots

Let $\mathcal K_n$ be the set of all $n$-singular knots up to isotopy,i.e. an immersion of $S^1$ into $\mathbb R^3$ with $n$ transverse double points that is an embedding when restricted to the ...

1
vote

1
answer

117
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### How is interpolation used in the proof of Lemma 4.1 in Tao's article Endpoint Strichartz Estimates?

In the proof of Lemma 4.1, pp. 962–963 in "Endpoint Strichartz Estimates" by Tao and Keel (1997) (see MR1646048 or Zbl 0922.35028), the authors first proved the statements hold for some ...

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0
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60
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### Are the alternating pentultimate and the $2$-alternating pentultimate isomorphic?

Consider a physical puzzle which is in the shape of a dodecahedron where the pieces that move are the face centers and you can slice it through a plane parallel to two opposite faces going through the ...

6
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175
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### Do acyclic amenable groups exist?

Is there an example of a nontrivial discrete amenable group with vanishing integral homology?
To put the question in contrapositive. Given arbitrary acyclic group $Q$, is there some reason for the ...

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0
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61
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### Lower bounding a partition-related sum

We say the $\mathbb{N}$-valued, non-increasing, eventually zero sequence $\lambda=(\lambda_1\geq\lambda_2,\cdots)$ is a partition of $N$ if $|\lambda|:=\sum_{k\geq 1}\lambda_k=N$, and denote $m_k(\...

3
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51
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### Subspaces of $\mathbb{F}_2^N$ containing many pairs of far apart vectors

Let $S$ be a subset of vectors in $\mathbb{F}_2^{3n}$ having Hamming weight $n$. Suppose that $S$ contains $m$ pairs of vectors having disjoint supports (that is, they are at Hamming distance $2n$ ...

2
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1
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160
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### Construction of the Mayer-Vietoris spectral sequence

Given subspaces $\{U_i\}_{i \in I}$ of a topological space $X$ with $X = \bigcup_i U_i$ satisfying some conditions, there is a Mayer-Vietoris spectral sequence converging to the homology of $X$. Here ...

0
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1
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131
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### Does weak convergence in $L^2$ imply convergence a.e. of a subsequence?

The title pretty much explains it all. Let $u_n\in L^2(\mathbb{R}^n)$ be a sequence converging weakly in $L^2$ to some $u\in L^2(\mathbb{R}^n)$, that is $\int u_k v \to \int u v$ for all $v\in L^2(\...

0
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0
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67
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### Number of solutions $x$ of equation $a_1 b_1^x + \dotsb + a_n b_n^x=0$ over a finite field

Let $F$ be a finite field and let $a_1, b_1, \dotsc, a_n, b_n \in F$ be field elements. I am interested in the number of solutions $0\leq x \leq |F|-1$ such that
\begin{equation}\label{e:1}
a_1 b_1^x +...

5
votes

2
answers

286
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### A continuous version of Carathéodory's convex hull theorem

A well-known theorem of Caratheodory states that any point in the convex hull of a set $X\subset R^n$ lies in the convex hull of at most $n+1$ points of $X$. I am wondering about a version of this ...

1
vote

1
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86
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### Defect between modulus and conductor of ray class field

I have following question about a remark in J. Neukirch's
Algebraic Number Theory around page 397.
The context: We consider ideal theoretic formulation of global class field theory of a number field $...

2
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0
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104
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### A relative cycle class map

Suppose I have a smooth projective morphism $p: X \to S$ between varieties, and a relative cycle $Z \subset X \to S$ which is assumed to be as nice as can be (rquidimensional with fibers of dimension $...

3
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0
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50
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### Fast checking that a system of polynomial equations is satisfiable over $\mathbb{F}_2$

I have a (fairly large) system of polynomial equations, of the form
$$
c_1d_1=0,\ c_1d_2+c_2d_1=0,\ldots
$$
(In case it is relevant, all the polynomials are homogeneous of degree 2, except for exactly ...