All Questions
Tagged with computational-geometry ag.algebraic-geometry
50 questions
3
votes
0
answers
83
views
Practical way of computing bitangent lines of a quartic (using computers)
Are there known practical algorithms or methods to calculate the bitangent lines of a quartic defined by $f(u,v,t)=0$ in terms of the 15 coefficients? Theoretically you can set up $f(u,v,-au-bv)=(k_0u^...
- 101
2
votes
1
answer
271
views
Irreducibility of an explicit complex projective variety
Let $Y\subset \mathbb P^n_\mathbb C$ be a subvariety defined by a series of homogeneous polynomials $f_1, \ldots, f_t$. Is there an effective way to determine the irreducibility of $Y$ as an algebraic ...
- 279
2
votes
0
answers
112
views
Understanding normalization algorithms
Let $R$ be a commutative and reduced ring, finitely presented over $\mathbb Z$. Let $\overline R$ be the integral closure of $R$ in its total ring of fractions. In https://arxiv.org/abs/alg-geom/...
- 211
3
votes
2
answers
240
views
Writing a smooth plane quartic as the vanishing of $Q_0Q_2 - Q_1^2$ for quadratic $Q_0,Q_1,Q_2$
It seems well-known that any smooth plane quartic can be written as the vanishing of $Q_0Q_2 -Q_1^2$. Is there a good way to work out these quadratic factors $Q_0,Q_1,Q_2$? For example, given the ...
- 679
3
votes
1
answer
271
views
Resultants and elimination theory
Consider an ideal $I = \langle f_1,\dotsc,f_n\rangle$ in the ring $k[x_1,\dotsc,x_m]$.
Define the $i$-th elimination ideal to be $I_i = I \cap k[x_{i+1},\dotsc,x_m]$.
For any two polynomials $f$ and $...
- 1,435
-2
votes
1
answer
248
views
Are there any non-elementary functions that are computable?
Does a function $\mathit{f}:\mathbb{R}→\mathbb{R}$ being non-elementary (not expressible as a combination of finitely many elementary operations), imply that it is not computable?
The particular case ...
- 11
11
votes
2
answers
824
views
A quadratic $O(N)$ invariant equation for 4-index tensors
Consider an $O(N)$ invariant quadratic equation
$$
T_{ijkl}= T_{ijmn}T_{klmn}+ T_{ikmn}T_{jlmn}+ T_{ilmn}T_{jkmn},
$$
where $T_{ijkl}$ is a real, totally symmetric 4-tensor, and the indices run from 1 ...
- 679
0
votes
0
answers
94
views
Boolean operation on n dimensional polyhedron
A polyhedron in $R^n$ is defined by a set of half-planes: $P = \{x \in R^n \mid Ax - b \le 0\}$.
Given a set of polyhedra in $R^n$, $ P_1, P_2, \dotsc, P_k$, is there an algorithm/implementation that ...
- 21
9
votes
0
answers
290
views
Computer algebra tools for finding real dimension of an algebraic variety
I have a system of polynomial equations with the unknowns being real numbers. The set of solutions is infinite. What software can I use to compute the real dimension of the solution set?
The CAD-based ...
- 175
3
votes
0
answers
285
views
Explicit computations of finite covers of genus one curves with two points of ramification
I have an explicit genus one curve $E$ with two points $p_1$ and $p_2$ on it and am looking for an explicit degree seven cover $X\to E$ with ramification precisely over $p_i$, with a single preimage ...
- 5,186
1
vote
0
answers
58
views
Are cells of 4-polytopes a convex polyhedron by definition?
I'm going by the Wikipedia definition for a 4-polytope.
Do by definition, cells of 4-polytopes have to be a convex polyhedra?
If not, then are there polyhedra with non-convex faces?
If yes, is it the ...
- 11
1
vote
1
answer
175
views
A variation on the projective Nullstellensatz
Let $V$ be a $\mathbb{C}$-vector space, and let $f_1,\dots,f_n \in S^d(V^*)$ be homogeneous polynomials of degree $d$ for which $V(f_1,\dots, f_n)=\{0\}$.
Must there exist a positive integer $k\geq d$ ...
- 980
1
vote
0
answers
183
views
Using Bertini software to determine whether or not a variety is empty
I have a system of polynomials $f_1,\dots, f_n \in \mathbb{C}[x_1,\dots, x_m]$, and I would like to determine whether the set of solutions to the system $f_1(x)=\dots=f_n(x)=0$ is empty or not. Since ...
- 980
11
votes
0
answers
234
views
When is cohomology of a finitely presented dg-algebra computable?
Given a smooth affine variety $X$ defined over $\mathbb{Q}$, its singular cohomology is isomorphic to the algebraic de Rham cohomology, which is the cohomology of the complex $\Omega_X^0\to\Omega_X^1\...
- 3,772
2
votes
0
answers
113
views
Computing whether a set of polynomials cuts out a projective variety
I have a set of multivariate polynomials over $\mathbb{Q}$, and I want to compute whether they cut out a projective variety, i.e. whether the radical of the ideal $I$ that they generate is homogeneous....
- 980
-1
votes
1
answer
120
views
IntersectInP bug of Macaulay2 [closed]
I am trying to use the intersectInP command in Macaulay2, inside package ReesAlgebra. However, I tried to follow the exact code in the user-guide, but it doesn't run in my Ubuntu app (of win 10). Can ...
- 287
2
votes
0
answers
71
views
grobner basis of an ideal dependent on some parameter
Suppose $I = \langle f_1, ... , f_l \rangle$ is an ideal generated by polynomials $f \in k[x_1,\dots,x_n]$, where $k$ is a field of rational functions in some parameters $s_1,\dots,s_m$.
What are the ...
- 1,435
3
votes
0
answers
106
views
finding a good term order for grobner basis
What are the tricks to pick a "good" monomial order to find a Grobner basis for a given ideal?
By good I mean one in which the final Grobner basis has a simple expression in terms of the ...
- 1,435
5
votes
0
answers
85
views
special classes of ideals (eg. toric) that admit faster Buchberger algorithm?
I have heard that toric ideals allow one to speed up the Buchberger algorithm considerably (see Grobner bases of toric ideals, Remark 2,3). My question is two-fold:
What are the precise complexity-...
- 1,221
12
votes
1
answer
382
views
Characteristic polynomial of an $8 \times 8$ symmetric matrix with indeterminate entries related to octonionic multiplication
I consider $1,i,j,k,l,m,n,o$ the standard basis of the (complexified if you like) octonions ($\mathbb{O}$ for the octonions).
Let $a = x_1.1 +\ldots + x_8.o$, $b = x_9.1+ \ldots + x_{16}.o$ and $c = ...
- 7,300
5
votes
0
answers
350
views
Are nearby points in an algebraic curve necessarily connected?
I would like a result of the following form:
For every algebraic curve $C$ in $\mathbb{R}\mathbf{P}^{n-1}$, there exists an
explicit and easy-to-compute $\epsilon=\epsilon(C)>0$ such that ...
- 7,633
2
votes
0
answers
111
views
How to compute explicit equations for the Jacobian of a variety over a field [duplicate]
Suppose we start with a projective curve $X$ over a field $K$, given as a closed subvariety of $\mathbb P^n_K$ by some explicit list of equations. I would like to find an explicit representation of ...
- 374
5
votes
1
answer
2k
views
Intersections of quadratic planes as elliptic curves
An elliptic curve defined over a field $k$ is a smooth projective curve of genus $1$, plus a $k$-rational point. Every elliptic curve can be written in a Weierstrass form, i.e. as a plane cubic curve ...
- 3,432
1
vote
1
answer
146
views
Omitting constraints of polynomial system
Let $n_1, n_2 \geq 1$ be known integer constants.
Suppose that we have the following system of $n$ polynomial inequalities
for which we know that there exists a feasible solution $(p_1, p_2) \in (0,1)...
- 175
5
votes
0
answers
273
views
Can this set of equations be solved explicitly for algebraic curves?
In my recent work I stumbled upon a set of two equations. I'm interested in solving by eliminating auxiliary variable "$z$" and getting algebraic curve in terms of $x$ and $y$ given by the zero locus ...
- 243
7
votes
1
answer
362
views
Does generic projection into $\mathbb{R}^3$ preserve real-algebraic-curve-ness?
I'm interested in the topological properties of certain real algebraic curves in high-dimensional spaces. I want to visualize these curves (say, like this), and so I'm pursuing dimensionality ...
- 7,633
2
votes
0
answers
185
views
How can I find a basis of $H^0(Y, -K_Y)$ for the del Pezzo surface $Y$ of degree 5?
Consider the surface $X \subset \mathbb{P}^2_{x, y, z}\times\mathbb{P}^1_{t, s}$:
$$
s^3y^2 + t^3yz = (t+s)x^2 + tsxz + t(t^2 +s^2)z^2
$$
over a finite field $k = \mathbb{F}_{2^d}$, $\mathrm{gcd}(d, 6)...
- 1,833
5
votes
1
answer
169
views
Connectedness of semialgebraic sets via CAD
I do not know whether there is a standard or some traditional ways to decide whether a semialgebraic set is connected or not.
One way I know is the cylindrical algebraic decomposition (CAD) algorithm. ...
- 61
1
vote
0
answers
104
views
connectedness of semi algebraic sets
We know the inequalities $x_ix_j >\theta_{ij}$ or $x_ix_j<\theta_{ij}$ for some $\theta_{ij}$>0, some $i,j\in\{1,\cdots,n\}$, $i\neq j$ defines the easiest semi algebraic set in $R^n_{\geq 0}$, ...
- 53
4
votes
1
answer
1k
views
How to compute the tangent space of a quotient by a finite group
Let $I\subseteq R:=\mathbb C[x_0,\ldots,x_n]$ be a homogeneous ideal defining a subscheme $X\subseteq\Bbb P^n$. As in my previous question, the permutation group $\mathfrak S_{n+1}$ acts on $R$ by ...
- 4,902
5
votes
0
answers
193
views
Determining N d-points yielding equal sums of Euclidean distances from M s-points
Given M source points (s-points), determine N, the number of destination points (d-points), and their locations (coordinates), such that the sum of the N Euclidean distances from each source point to ...
- 51
1
vote
1
answer
242
views
Finding generators of symmetric cones
I have a bunch of vectors $\mathbf v_i$ in $\mathbb R^n$. I would like to consider the cone $C$ spanned by these vectors, together with all the other vectors that can be obtained by permuting the ...
user47305
3
votes
0
answers
234
views
Uniqueness of Riemann Constant Vector Solution
Let $X$ be a compact, genus $g$ Riemann surface (given as the desingularization and compactification of a plane algebraic curve), $J(X)$ its Jacobian, and $A : X \to J(X)$ the Abel map
$$A(P) = \left(...
8
votes
0
answers
229
views
Nearest point to a real algebraic set
Suppose I have a compact bounded real algebraic (eventually: or analytic or semialgebraic or semianalytic set) $V \subset \mathbb R^3$ and a point $x\in\mathbb R^3 \setminus V$. How much do we know ...
- 2,275
1
vote
1
answer
558
views
Is there a Gröbner basis analogue that exists for vector spaces?
Suppose I have a coordinate system $t_1,\ldots t_N$ with a lexicographical ordering. Let LT denote choosing the lowest term of a polynomial with respect to this ordering. e.g. LT$(t_1 + t_2)=t_2$.
...
- 131
12
votes
1
answer
2k
views
Ways to show a system of polynomial equations has no solution
I came across the following system of polynomial equations on $X_1,\dots,X_{m-2}$:
$$
\begin{cases}
2X_{2s}+\sum\limits_{t=1}^{2s-1}(-1)^tX_tX_{2s-t}=0,\quad s=1,\dots,\frac{m}{2}-1,\\
X_sX_{m-s}+(-1)^...
- 767
2
votes
1
answer
387
views
Hilbert function of points in $\mathrm{P}^2$
Let $\Gamma$ be a collection of $d$ points in $\mathrm{P}^2$, and $I$ the graded ideal of $\Gamma$.If
$$
\mathrm{gin_{rlex}}(I)=(x_1^k,x_1^{k-1}x_2^{\lambda_{k-1}},...,x_1x_2^{\lambda_1},x_2^{\...
- 73
6
votes
1
answer
301
views
Software computation with arithmetic schemes
For rings such as $\mathbb{Z}[x,y]$ is there software to compute any of:
1.) The integral closure of $\mathbb{Z}[x,y]/(f)$. de Jong has a very general algorithm that works in this context (http://...
- 3,555
66
votes
3
answers
4k
views
Reasons to prefer one large prime over another to approximate characteristic zero
Background:
In running algebraic geometry computations using software such as Macaulay2, it is often easier and faster to work over $\mathbb F_p = \mathbb Z / p\mathbb Z$ for a large prime $p$, rather ...
- 7,328
1
vote
0
answers
517
views
Solving 3D equation system (inverse-projecting a triangle)
Please, how is the equation system below named exactly (to search further literature)?
Does it have an analytical solution? If it doesn't, then what could be the fastest numerical method for it (...
- 119
11
votes
3
answers
960
views
Algorithms in Invariant Theory
Let $V$ be a polynomial representation of the general linear group $\Gamma:=\DeclareMathOperator{\Gl}{Gl}\Gl_n(\newcommand{\C}{\mathbb C}\C)$.
In chapter 4.6 of his book "Algorithms in Invariant ...
- 262
10
votes
3
answers
756
views
Degree of generators of irreducible components
Let $V$ be a Zariski-closed subset of $\mathbb{A}^n_k$, where $k$ is an algebraically closed field. Assume that $V$ may be defined by polynomials of degree at most $d$ (or to put it otherwise $V$ is ...
- 4,132
1
vote
1
answer
573
views
'Reference' request: Program to work with cyclic quotient singularities.
I'm looking for program code to deal with cyclic quotient singularities on normal surfaces. In particular, at the moment I need that given a singularity $p$ like $p=\frac{1}{n}(1,a)$ the code computes:...
- 2,215
10
votes
1
answer
2k
views
Can taking the projective closure of an affine variety increase the degrees of its ideal generators?
Say we have some equations $f_1(x)=0, \ldots f_k(x)=0$ defining a variety $X$ in ${\mathbb C}^n$ (not necessarily a minimal number of generators, and not necessarily of minimal degree), and suppose we ...
- 11.3k
1
vote
0
answers
204
views
Which rational subfields are corresponding to the two dimensional subspaces of holomorphic differentials
I implemented the algorithm that Felipe Voloch's suggested in his reply to the question:
Subfields of a function field
the algorithm is here:
Subfields of a function field
I considered the ...
- 601
14
votes
2
answers
1k
views
Subfields of a function field
Is there an algorithm for generating (some or all) subfields of a certain genus of a given function field (even a random one,I mean for example generating a random elliptic subfield of a certain given ...
- 601
4
votes
2
answers
3k
views
Numerical solution for a system of multivariate polynomial equations
Hi all,
I have a system of 6th-order polynomial equations in 4 variables $q_1, q_2, q_3, q_4$ (i.e. polynomials with all the terms such as $q_1^6, q_2^6, q_2^4 q_3^2$):
$P_k(q_1, q_2, q_3, q_4) = 0$ ...
- 113
5
votes
1
answer
2k
views
Intersections of irreducible components
Let $V$ be an algebraic variety (not irreducible) over $\mathbb{C}$, defined by an ideal $I = \{f_1,f_2,\dots, f_n\}$. $V$ is not necessarily pure dimensional. Suppose $V = R_1\cup R_2\cup\dots\cup ...
- 1,510
2
votes
2
answers
354
views
formulate edge length problem as convex optimization problem
I want to us convex optimization to describe a problem in computational geometry.
Let $E = (E_1, E_2,\ldots , E_m) $ be a sequence of line segments in the plane, where $E_1$ and $E_m$ may be points ...
2
votes
0
answers
227
views
Finding equations for projective bundles associated to vector bundles over explicitly given varieties
Suppose I have a projective variety V over a field k. It is given explicitly in terms of homogeneous equations. Moreover, say I have an explicitly given vector bundle E (in terms of a module ...
- 21