Questions tagged [computer-algebra]
Using computer-aid approach to solve algebraic problems. Questions with this tag should typically include at least one other tag indicating what sort of algebraic problem is involved, such as ac.commutative-algebra or rt.representation-theory or ag.algebraic-geometry.
368
questions
1
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1
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101
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Factorial series $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$ and hypergeometric functions
For positive integer $D$, define $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$.
For $D \le 6$, sage finds closed form in terms of hypergeometric functions
at algrebraic arguments and fails to find closed ...
2
votes
0
answers
110
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How to find a single-variable polynomial in a zero-dimensional ideal?
Given finitely many multivariate polynomials with algebraic coefficients that generate a zero-dimensional ideal, is there an easy way to find a nonzero single-variable polynomial in this ideal?
If we ...
0
votes
1
answer
39
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relations between non-negativity of multivariate polynomials and SOS over gradient ideal
We know there is a necessary condition for the non-negativity of multivariate polynomials in the paper "Sum of Squares Decompositions of Polynomials over their Gradient Ideals with Rational ...
1
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0
answers
28
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Finite right-triple convex sets in planes
Let $\mathcal{S}$ be a set of points in $\mathbb{R}^2$. We say that $\mathcal{S}$ is right-angle convex, if for any two distinct points $P,Q\in \mathcal{S}$ there always exists another point $R\in \...
-2
votes
0
answers
132
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Elimination over $\mathbb F_p[x,y]$
Let $p$ be a prime. Consider the two independent modular equations:
$$a_1x^2+b_1y^2+c_1xy\equiv d_1\bmod p$$
$$a_2x^2+b_2y^2+c_2xy\equiv d_2\bmod p$$
Is it possible to extract the common roots $(x,y)\...
65
votes
2
answers
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Does there exist a complete implementation of the Risch algorithm?
Is there a generally available (commercial or not) complete implementation of the Risch algorithm for determining whether an elementary function has an elementary antiderivative?
The Wikipedia article ...
0
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0
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71
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Potential typo in "Complete Systems of Two Addition Laws for Elliptic Curves" by Bosma and Lenstra
Here is a link to the article: https://www.sciencedirect.com/science/article/pii/S0022314X85710888?ref=cra_js_challenge&fr=RR-1.
Pages 237-238 give polynomial expressions $X_3^{(2)}, Y_3^{(2)}, ...
2
votes
1
answer
103
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Efficiency of Groebner basis for constraints of the form $(a_i x_i+b_i)(a_j x_j+b_j)$
This is based on numerical experiments in sage.
Let $K$ be a ring and define the ideal where each polynomial
is of the form $(a_i x_i+b_i)(a_j x_j+b_j)$ for constant $a_i,b_i,a_j,b_j$.
Q1 Is it true ...
1
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1
answer
199
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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 ...
7
votes
1
answer
454
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Computing homology groups with GAP
I’m studying the homology groups of arithmetic groups such as $SL(5,\mathbb{Z})$. I saw in the answer to this post that we can use GAP to compute some of the homology groups for $SL(3,\mathbb{Z})$. Is ...
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votes
1
answer
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Application of Resultant in Computer Algebra [closed]
Can you guys give me some application of resultant in Computer Algebra, it will be amazing if you guys can give me some paper or book to read more. Thanks so much
12
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1
answer
412
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Tarski-Seidenberg for strict inequalities and bounded quantification
This theorem says that quantifiers over real variables can be eliminated from classical first order formulae built from equations and inequalities between polynomials with rational coefficients, ie in ...
149
votes
38
answers
38k
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Computer algebra errors
In the course of doing mathematics, I make extensive use of computer-based calculations. There's one CAS that I use mostly, even though I occasionally come across out-and-out wrong answers.
After ...
5
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0
answers
129
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Macaulay2 seems to have divergent behavior on rings with differently ordered variables
I noticed the following strange behavior which I cannot explain.
I wanted to compute the integral closure of the following ring,
$$ A = \mathbb{F}_5[x,t]/(t^2 (1 - x^4) - x^5) $$
Call the integral ...
7
votes
1
answer
254
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How to construct such a real algebraic curve
Suppose $L$ is a real line in $\mathbb{RP}^2$, my question is: for a given postive integer $n$, is it possible to find a real projective algebraic curve $C$ of degree $n$ with maximum connected ...
1
vote
3
answers
501
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Finding maximum value of degree-3 homogeneous polynomials when variables sum to 1
I would like to be able to find maximum values of degree-3 homogeneous polynomials, when the variables are non-negative real numbers that sum to 1. For example,
For example, the maximum value of $xy^...
5
votes
2
answers
327
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Reliability of ILP approach to number-theoretic optimization
This question is inspired by the recent answer, where @RobPratt proposed to use integer linear programming (ILP) for solving a number-theoretic optimization problem. I will consider a very similar ...
1
vote
0
answers
70
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Hall-Littlewood polynomials with sage
I need to make some computations with Sage involving Hall-Littlewood polynomials. I couldn't find any satisfying information in the Sage manuals/tutorials that I found on the internet. What I found is ...
2
votes
1
answer
249
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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 ...
3
votes
0
answers
87
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Isomorphism and counting for tree quivers
Let $Q$ be a quiver which is a connected tree and let $A=KQ/I$ be a quiver algebra with $I$ an admissible ideal, meaning that $I$ is generated by paths of length $\geq 2$. Let $n$ be the number of ...
3
votes
1
answer
284
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The associated graded algebra of a finite dimensional algebra
$\DeclareMathOperator\rad{rad}$Let $A$ be a finite dimensional algebra (we can assume that $A \cong KQ/I$, for a quiver $Q$ and an admissible ideal $I$ if that helps).
Denote by $A_G$ the associated ...
7
votes
0
answers
97
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Optimizing computations with nilpotents in a group algebra
Of course, I have a very concrete problem at hand, which has been vexing me for about a year now. But let me start with a question that has a better chance of having been answered.
Let $G$ be a ...
4
votes
1
answer
328
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GCD in $\mathbb{F}_3[T]$ with powers of linear polynomials
This is a continuation of my previous question on $\gcd$s of polynomials of type $f^n - f$.
Let us call $n > 1$ simple at a prime $p$ when $p-1 \mid n-1$ but $p^k - 1 \not\mid n-1$ for all $k > ...
6
votes
2
answers
131
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Is every parametric function family, expressed as an elemetary function, a solution to an ODE with elementary functions?
When given an ODE of the form $F(x, y, y', \ldots, y^{(n)}) = 0$, where $F$ is an elementary function, chances are that it has no solution of the form $y = G(x, c_1, \ldots, c_n)$, where $G$ is also ...
8
votes
3
answers
2k
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Rational exponential expressions
Consider the following extension of polynomials. The rational exponential expressions (REXes) are given by:
The leaves 1 and $x$ for $x$ drawn from a class of variables; and
Closed under the binary ...
2
votes
0
answers
157
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Help with Macaulay2 computation of invariant ring
Consider the algebraic group $G:=\operatorname{SL}_{2}\times\operatorname{SL}_{2}$ acting on $V:=\operatorname{Mat}_{2\times 2}\oplus\operatorname{Mat}_{2\times 2}$ via the action $(A,B)\,\cdot\,(X,Y)=...
4
votes
2
answers
296
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Algorithm for computing rational points if the rank of Jacobian is 0
Is there a general algorithm that can compute in finite time all rational points on any curve of genus $g\geq 2$ whose Jacobian has rank $0$?
If not, for what special cases such algorithm is known? ...
9
votes
0
answers
285
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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 ...
4
votes
0
answers
94
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Koszul algebras among finite dimensional commutative algebras
Given a local commutative artinian algebra $A$ of the form $K[x_1,...,x_n]/I$ with quadratic ideal $I$ and $K$ a field.
Question 1: Is there a computer algebra system that can check whether such an ...
2
votes
0
answers
75
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Gröbner implicitization with relationships between the variables
I have the following parametric equations, where cost$=\cos t$, cos2t$=\cos 2t$, and $A^2+B^2=1$:
...
3
votes
2
answers
187
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What can be said about the cube-free part of $x^3 -3xy^2 +y^3$?
For $x$ and $y$ in $\mathbf{Z}$, not both zero, let $cfp(x,y)$ be the cube-free part of $x^3 -3xy^2 + y^3$ (normalized to be $> 0$). One sees:
(#) $cfp(x,y)$ is either a product of primes $p$, with ...
8
votes
1
answer
1k
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Software for computing multi-graded Hilbert series
The ring of invariants $S^T$ of $k[a,b,c,d]$ under the algebraic torus action $T = k^{*}$ with weights $(1,1,-1,-1)$
is $S = k[ac,ad,bc,bd]$ which has multigraded Hilbert series
$$
\frac{1 - abcd}{(1-...
5
votes
0
answers
97
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Size of minimal generating set of ideal over Laurent polynomial ring
Recently in attacking a problem in algebraic topology relating to the construction of stably-free non-free modules over integral group rings I’ve noticed that it is often fairly easy to reduce to ...
6
votes
2
answers
340
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Checking for a normal p-complement with a computer
Let $G$ be a finite group.
Question 1: What are the fastest available programs to test whether $G$ has a normal $p$-complement (see https://en.wikipedia.org/wiki/Normal_p-complement for a definition)?...
26
votes
5
answers
6k
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Minimal polynomial of cos(π/n)
I know that $\cos(\pi/n)$ is a root of the Chebyshev polynomial $(T_n + 1)$, in fact it is the largest root of that polynomial, but often that polynomial factors. For example, if $n = 2 k$ then $\cos(\...
129
votes
74
answers
20k
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Most helpful math resources on the web
What are really helpful math resources out there on the web?
Please don't only post a link but a short description of what it does and why it is helpful.
Please only one resource per answer and let ...
4
votes
0
answers
188
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Is it possible to compute Lie bialgebra structures with SageMath?
Is it possible to use SageMath (or some Linux open source program) to compute the bialgebra structures on a given finite dimensional Lie algebra? I wonder if such program can compute all the ...
9
votes
2
answers
663
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Finite subgroups of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$
Question 1:Is there a reference that lists all possible finite subgroups and their orders of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$ for $n=4$ or even higher $n$ over the real numbers?
I can only find ...
24
votes
5
answers
11k
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Fastest algorithm to compute the sum of primes?
Can anyone help me with references to the current fastest algorithms for counting the exact sum of primes less than some number n? I'm specifically curious about the best case running times, of ...
10
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2
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Degree 8 multilinear operations on Jordan algebras
I am interested in the dimension, or, even better, in the $S_8$-module structure of the space of degree 8 multilinear operations on Jordan algebras.
Recall that a Jordan algebra is a commutative but ...
3
votes
1
answer
272
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Siegel modular forms in Mathematica
Is there a convenient way to work with Siegel modular forms in Mathematica? I am interested in doing analytic computations using the $\chi_{10}(\Omega)$ Siegel modular form, where $\Omega$ is the $2\...
7
votes
2
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569
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Deriving consequences of identities
Suppose we are given a variety in the universal algebra sense.
For concreteness, suppose that we have two binary operations $+,\cdot$, three unary operations $-,\ast,'$, and two zeroary operations $0,...
5
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0
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78
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Compute the principal polarization on $J_0(N)$ in terms of modular symbols
If we consider the modular curve $X = X_0(N)$ as a curve over $\mathbb C$ then one can describe the jacobian $J(X)$ as $H^0(X,\Omega^1_X)^\vee/H_1(X,\mathbb Z)$ as one can do for any curve $X$. ...
1
vote
2
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250
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Method to solve system of exponential sums of the form $a^x+b^x=c$ given more equations than variables [closed]
Cross post with mse
For example, let's say I have the following equations.
\begin{gather*}
a^{x-1}+b^{x-1}=337 \\
a^{x}+b^{x}=1267 \\
a^{x+1}+b^{x+1}=4825 \\
a^{x+2}+b^{x+2}=18751.
\end{gather*}
What ...
3
votes
0
answers
97
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Checking the generic rank of a matrix
Suppose that $A,B\in M_{p,q}(\mathbb{Z})$ are two rectangular integer matrices of the same size. Suppose that one has a conjecture stating that the rank of the matrix $A+tB$ for Zariski generic values ...
0
votes
1
answer
142
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Grobner basis of a submodule of a free module over polynomial ring
Let $A=\mathbb{Q} [x_1,\dots,x_n]$ be the ring of polynomials with rational coefficients. Let $T$ be an $m\times n$ matrix with entries from $A$. Consider it as a morphism of $A$-modules $T\colon A^n\...
7
votes
1
answer
349
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About the complexity of some operation involving integers
There are two integers: $A, B$. Given the below four allowed operations (and only them):
$A+1$, $A-1$, $\sqrt{A}$, $A^2$
Also, it is only allowed to take the square root of $A$ when this square root ...
0
votes
1
answer
115
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Software to compute generators of a module over polynomial ring
Let $A=\mathbb{R}[x_1,\dots,x_n]$ be the algebra of real polynomials in $n$ variables. Fix polynomials $p_1,\dots,p_k\in A$.
Consider the subset
$$M:=\{(q_1,\dots,q_k)\in A^k|\, p_1q_1+\dots+p_kq_k=0\}...
4
votes
0
answers
127
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7D simple Lie algebras over $\mathbb{F}_3$
Up to isomorphism, what are all the seven-dimensional simple Lie algebras over the field with three elements?
17
votes
4
answers
2k
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An experiment on random matrices
A bit unsure if my use/mention of proprietary software might be inappropriate or even frowned upon here. If this is the case, or if this kind of experimental question is not welcome, please let me ...