All Questions
Tagged with computer-science ag.algebraic-geometry
15 questions
2
votes
0
answers
122
views
What do we know about efficiently finding a solution to a system of multivariate polynomials over finite fields?
Consider the following (NP-complete) problem:
Given a system of polynomials $f_1, f_2, \ldots, f_m \in \mathbb{F}_q[x_1, x_2, \ldots, x_n]$ of total degree at most $d$, find an $\mathbb{F}_q$-rational ...
- 121
4
votes
1
answer
291
views
Discrepancy in the calculation of $2$-Selmer group by Magma and LMFDB
The result of LMFDB claims (https://www.lmfdb.org/EllipticCurve/Q/1640/c/1 )
that (2-part of) Tate-Shafarevich group $\mathrm{Sha}(E/\Bbb{Q})$ of elliptic curve $y^2=x^3-8747x-314874$ has order $16$. ...
- 1,531
1
vote
0
answers
62
views
Polynomial sized arithmetic map from circle to ellipse preserving integral points
Let $n$ be a square free integer and a product of $O(m/\log m)$ number of primes $1\bmod 4$ where $m$ is $\log_2n$.
Take the circle around origin of radius $n^2$. It has ${\exp}(m/\log m)$ number of ...
- 13.9k
2
votes
1
answer
278
views
Is good reduction decidable?
Let $X$ be a smooth projective geometrically connected variety over $\mathbb{Q}$. It is said to have good reduction at a prime $p$ is there is a smooth projective $\mathcal{X}\to \mathrm{Spec}\:\...
user158636
5
votes
1
answer
462
views
Polynomial size embeddings of toric varieties from polytopes?
Background: Let $P$ be a integral polytope, and $X_P$ the toric variety associated to the normal fan.
$X_P$ is always projective, because the collection of characters corresponding to the points $\...
- 1,462
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
5
votes
1
answer
453
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an algebraic variety for a boolean circuit
There is a polynomial reduction from a $3-CNF$ $SAT$ problem to some system of polynomial equations over $\mathbb{F}_2$.
I mean there is polynomial reduction $F$ such that for every boolean ...
- 2,576
6
votes
0
answers
217
views
Nonclassical polynomials, circles, and groups
Tao and Ziegler have introduced a generalization of polynomials over a prime field called nonclassical polynomials, useful for studying the Gowers norm.
A nonclassical polynomial of degree $d$ is a ...
- 539
1
vote
0
answers
89
views
Counting models in first order logics without existencial quantifiers
My question is about the posibility of to construct a parameter space of models in a first order theory, finitely presented, with out existencial quantifiers (parameter space in the sense of ...
- 527
5
votes
2
answers
901
views
Given a formal power series ,decide whether there exists a polynomial the series satisfies and if it exists,how to write it down?
Given a formal power series $$y(x)=\sum_{i=0}^{\infty} a_i x^i$$ Is there an algorithm that decides whether there exists a polynomial$$ P(x,y)=p_n(x)y^n+p_{n-1}(x)y^{n-1}+\cdots+p_0(x)=0,p_j(x)\in F[x]...
- 3,094
38
votes
0
answers
1k
views
Computer calculations in A_infinity categories?
Is there a good computer program for doing calculations in A-infinity categories?
Explicit calculations in A-infinity categories are an important, useful, yet very tedious task. One has to keep track ...
- 1,439
4
votes
2
answers
2k
views
viewing the second fundamental form as a tensor
Dear all,
Thank you for your time reading this post. I am a student in computer science so this viewpoint of the second fundamental form may be interesting to you.
I would like to understand the ...
- 565
1
vote
1
answer
455
views
Algorithm for generating a size k error-correcting code on n bits
I want to generate a code on n bits for k different inputs that I want to classify. The main requirement of this code is the error-correcting criteria: that the minimum pairwise distance between any ...
- 2,383
50
votes
4
answers
4k
views
What algorithm in algebraic geometry should I work on implementing?
This summer my wife and one of my friends (who are both programmers and undergraduate math majors, but have not learned any algebraic geometry) want to learn some algebraic geometry from me, and I ...
3
votes
1
answer
2k
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Conditions that allow unique solutions for Linear Diophantine equations
(This posting became very long, so I should note that there are two alternative but nearly equivalent formulations of the same question being given. The first one asks for the optimal strategy for ...