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
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4
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Computational algebra: where?
I'm on my last semester of a math B.Sc. and about to start studying for a math M.Sc in the same institute.
It now seems like a good time to start thinking of a PhD.
I'm interested in both algebra and ...
7
votes
1
answer
849
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Computer power in plane geometry
I often hear that modern computer programs "may prove any theorem in elementary Euclidean geometry". Of course, as stated it is false - say, they can not prove theorems about $n$-gons for ...
4
votes
0
answers
111
views
Recommendations for distributed calculations of Groebner Bases
There are many computer algebra systems available which can compute a Groebner basis, including:
Mathematica
Singular
Macaulay2
Magma
Maple
CoCoA
However (please correct me if I've missed something) ...
0
votes
2
answers
399
views
Fastest way to solve non-negative linear diophantine equations
Let $A$ be a matrix in $M_{n \times m}(\mathbb{Z}_{\ge 0})$ without zero column. Let $V$ be a vector in $\mathbb{Z}_{> 0}^m$.
Question: What is the fastest way to find all the solutions $X \in \...
5
votes
1
answer
213
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Classification of multiplicative lattices
Question 1:Is there a classification of finite lattices which admit a multiplication making them into a finite multiplicative lattices? (see https://encyclopediaofmath.org/wiki/Multiplicative_lattice ...
3
votes
4
answers
820
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Compute the two-fold partial integral, where the three-fold full integral is known
I have the following trivariate ($\rho_{11}, \rho_{22}, \mu$) function
\begin{equation}
4 \mu ^{3 \beta +1} \rho_{11}^{3 \beta +1} \left(-\rho_{11}-\rho_{22}+1\right){}^{3 \beta +1}
\rho_{22}^{3 \...
6
votes
0
answers
223
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Proving the spectrum of the Young-Jucys-Murphy elements by formal computation in the degenerate affine Hecke algebra
This is really a followup to Why are Jucys-Murphy elements' eigenvalues whole numbers? , specifically to Igor Makhlin's beautiful answer. I'm trying to make it even more beautiful by getting rid ...
1
vote
0
answers
208
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Computer algebra programs that can solve polynomial systems on algebraically closed fields besides MAGMA
I was wondering which computer algebra programs out there can solve polynomial systems on the algebraic closure of $\mathbb{Q}$ analytically and efficiently.
So far, I only found MAGMA with its ...
1
vote
0
answers
42
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Is there a more efficient computer algebra system to solve the system of nonlinear equations in N-R method or other numerical methods?
Consider the system of infinite series
\begin{align}
&F=x+\frac{y^{3^2}}{3}+\frac{x^{3^5}}{3^2}+\frac{y^{3^7}}{3^3}+\frac{x^{3^{10}}}{3^4}+\frac{y^{3^{12}}}{3^5}+\cdots=0
\\
&G=y+\frac{x^{3^3}}...
1
vote
1
answer
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Are algebraic power series in positive characteristics D-finite?
We know that in characteristic $0$, all algebraic series are differentiably finite.
Is this true in positive characteristic? I look at the proof, indeed we need to the
characteristic to be $0$ for the ...
2
votes
3
answers
518
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Useful software for variable elimination
I have three non-homogeneous trivariate polynomials in $\mathbb Z[x,y,z]$ and I want to eliminate the variables $y$ and $z$ to get a polynomial in $x$. The monomials of the polynomials are $\{1,x^4,x^...
1
vote
0
answers
273
views
Algebraic independence criterion
Is there any criterion for checking algebraic independence of a set of polynomials in $n$ variables in terms of the leading monomials with respect to some monomial order ? The Jacobian criterion is ...
1
vote
1
answer
161
views
Find $x$ that solves $x\left(e^{\frac{a}{x}}-1\right)-y=0$
When trying to solve the equation in the title with WA, it produced the following as the solution:
now, if you divide the numerator and denominator by $y$ and set $z:=-\frac{a}{y}$ the solution ...
1
vote
0
answers
41
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Computer verification for hyperbolic trigonometry
I am currently writing a paper that requires some lengthy computations using basic hyperbolic trigonometry. So, several hyperbolic figures appear, and we apply the law of sines and so on in order to ...
50
votes
5
answers
14k
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The unification of Mathematics via Topos Theory
In her paper The unification of Mathematics via Topos Theory, Olivia Caramello says "one can generate a huge number of new results in any mathematical field without any creative effort". Is ...
13
votes
3
answers
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Is computer algebra or symbolic computation an active area of research?
I'm interested in doing PhD in computer algebra or symbolic computation, and was wondering if this is an active area of research? Would this area of research also help me in the transition to ...
3
votes
0
answers
106
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Finite global dimension via the Cartan determinant
Let $A=T(KQ)$ be the trivial extension algebra of a path algebra of Dynkin type $KQ$.
The indecomposable module of $A$ correspond to the roots of $Q$ (and not just the positive roots as for $KQ$).
Let ...
0
votes
0
answers
86
views
Computational tool for checking the existence of non-trivial rational zero of a cubic form
Suppose we consider a arbitrary cubic homogeneous form $f$ in in four or five variables over the rational field. Is there any computational tool or algorithm to check whether this cubic homogeneous ...
3
votes
1
answer
460
views
Computing conjugacy between two elements of $\mathrm{SL}_2(\mathbb{Z})$
The conjugacy classes of $\mathrm{SL}_2(\mathbb{Z})$ are well characterized (see, e.g., this question). Assuming two matrices $A, B \in \mathrm{SL}_2(\mathbb{Z})$ are conjugate, is there a way to ...
3
votes
0
answers
65
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Finding generators and relations for special commutative algebras with a computer
Let $K[x_1,...,x_n]$ be the polynomial ring in $n$ variables and $a_1,...,a_m$ elements in the quotient field $K(x_1,...,x_n)$.
Let $A:=K[a_1,...,a_m]$ the ring generated by the $a_i$ in $K(x_1,...,...
5
votes
0
answers
76
views
Conjugacy classes in normalized unit group of a group ring
Let $V(FA_4)$ be the normalized unit group of the group ring $FA_4$, where $F$ is the field containing 4 elements and $A_4$ is the alternating group on 4 symbols. How can I find conjugacy classes of ...
3
votes
0
answers
176
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Enumeration of stable graphs of genus $g$
Let $G=(V,E)$ be a connected undirected finite graph, let us call $G$ stable if each vertex has degree at least $3$.
Is there a computer algorithm to efficiently enumerate (repetition allowed) all ...
11
votes
3
answers
657
views
An open triangle problem in plane geometry
Some years ago, I asked some 'famous' people in an advanced Plane Geometry forum about the following:
Let $ABC$ be arbitrary triangle, how can one construct a point $P$ in the plane such that $P$ is ...
1
vote
0
answers
129
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Can PARI compute class numbers without factoring the discriminant?
When calculating properties of algebraic number fields, one of the hardest steps is factorizing the discriminant of a defining polynomial. This is necessary in the Pohst-Zassenhaus algorithm for ...
1
vote
0
answers
62
views
Finding multivariate binomials with a common zero [closed]
I have a problem for which I have to find binomials over a multivariate polynomial Ring which all have a common zero.
Let $\mathbb{F}[x_1,\dots,x_n]$ be some multivariate polynomial ring over some ...
2
votes
0
answers
243
views
Integer points on genus 1 curves using CAS
How can I practically find integer points on genus 1 curves with small coefficients using computer algebra systems (CAS), like Mathematica, Maple, SageMath, Magma, etc.?
As a specific example, do ...
8
votes
3
answers
2k
views
Is there a MAGMA function to calculate the absolutely irreducible components of an algebraic curve defined over the rationals?
Given a curve defined over the rationals, is it computationaly possible to find all its absolutely irreducible components?
Is there an implementation of this in the MAGMA program?
20
votes
5
answers
3k
views
How can you find an integer coefficient polynomial knowing its values only at a few points (but requiring the coefficients be small)?
Example: How can you guess a polynomial $p$ if you know that $p(2) = 11$? It is simple: just write 11 in binary format: 1011 and it gives the coefficients: $p(x) = x^3+x+1$. Well, of course, this ...
7
votes
0
answers
119
views
Softwares to determine semi-simple types of Lie algebras generated over $\mathbb{R}$ or $\mathbb{C}$ by a set of matrices
I wish to determine the type of a Lie algebra generated over $\mathbb{R}$ or $\mathbb{C}$ by a set of square matrices with irrational elements. For example,
\begin{align}
n^+ =
\begin{pmatrix}
...
4
votes
1
answer
152
views
Link invariants from modular categories (strictification and computation)
By the theory of Reshetikhin and Turaev, a modular tensor category $C$ gives rise to a link invariant. While $C$ is strict as a monoidal category (e.g. $\mathbb{Fib}$), calculating the link can be ...
2
votes
0
answers
146
views
How to decide if an algebraic number is a root of a given polynomial?
Let $p$ be a polynomial with rational coefficients and $\alpha = \sqrt[n]{q}e^{i2k\pi/m}$ for some natural numbers $n,m,k$ and a rational number $q > 0$. Is there an effective algorithm for ...
2
votes
1
answer
520
views
Efficiently computing Gröbner basis to prove no solution to polynomial constraints
In a similar vein to these now quite old questions on advice for calculating a Gröbner basis:
Fast computation of a Groebner basis. What is possible?
What is the state of art in Groebner bases
I am ...
1
vote
1
answer
198
views
Primary decomposition of huge ideals using M2/Singular
I used to ask similar questions in other communities, but so far never received any feedback.
Given four Hermitian $n\times n$ matrices $A_1,A_2,B_1,B_2$ together with the constraints $[A_i,B_j]=0$, I ...
11
votes
5
answers
6k
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Finding minimal or canonical expressions for Boolean truth tables
This is not an urgent question, but something I've been curious about for quite some time.
Consider a Boolean function in n inputs: the truth table for this function has 2n rows.
There are uses of ...
3
votes
2
answers
235
views
Finding the "Q-span" of vectors in Q(q)
Apologies if this question is quite basic.
Consider the $\mathbb{Q}(q)$-vector space $V = \mathbb{Q}(q)^n$ with standard ordered basis $\{e_1,\ldots,e_n\}$.
Suppose someone hands you some vectors $v_1,...
1
vote
0
answers
76
views
Fitting point on a Quadric curve [closed]
I am working on research project. I am currently using CloudCompare for my project, which calculates the Gaussian curvature and mean curvature to extract the geometric features of the points. I have a ...
1
vote
1
answer
159
views
Sufficient syntactic conditions for zero-dimensionality of polynomial systems
Consider a system $S$ of polynomial equations, $p_1=0,...,p_m=0$, for $p_i\in K[x_1,...,x_n]$, for a field $K$: the system $S$ is zero-dimensional if it has finitely many solutions. It is well-known ...
65
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 ...
0
votes
0
answers
222
views
Algorithm to compute automorphism group of a finite group
Is there an algorithm to compute automorphism group of a finite group?
GAP has a function to do this, but while perusing their GitHub repo, I could not find an implementation. I'm struggling to find ...
3
votes
1
answer
237
views
Number of rings with additive group $(\mathbb{Z}_{16})^2$. A341547(16) in OEIS
I would like to know if somewhere the number of non-isomorphic rings with additive group $(\mathbb{Z}_{16})^2$ is mentioned. If not, is someone able to calculate it?
And (easier) the commutative case? ...
32
votes
4
answers
6k
views
How does Mathematica do symbolic integration?
I suppose there was at least once in our lifetime the point where we resorted to mathematica for help with an integral.-Unless you chose not to have the pleasure of using the continuum in your ...
7
votes
0
answers
246
views
Is it easy to certify whether a given set of solutions to a polynomial system is complete?
Given a system of complex polynomial equations, we seek the solution set. If we have more equations than variables, then we might expect a finite solution set. One may obtain the solution set by ...
2
votes
0
answers
43
views
Transforming a symmetric matrix into pentadiagonal form
Given a symmetric matrix $A$, which has complex values in the diagonal, but whose all other entries are real, I am interested in finding an orthonormal transformation $Q$ such that $Q^tAQ$ is a ...
1
vote
1
answer
275
views
Find basis for the set of torsion points E[m]
In paper "On the Cost of Computing Isogenies Between Supersingular Elliptic Curves" (source) reads
Let ${P, Q}$ be a basis for $E[2^{e/2}]$. Let $R_0 = [2^{e/2}−1]P , R_1 = [2^{e/2}−1]Q, ...
12
votes
1
answer
967
views
Where to publish a long classification?
Suppose that the classification of some mathematical (say algebraic) notions requires (say) 70 pages. Let clarify that (say) 90% of the pages are used to write the result itself, whereas only 10% are ...
4
votes
1
answer
127
views
String compression algorithms for simplifying an expression by introducing variables
I have a very long algebraic expression computed with Maple, and when I inspect it visually, it is clear that it consists of a set of terms that appear over and over again. For purposes of human ...
2
votes
0
answers
68
views
Checking for norm-Euclidean with a computer
Let $K$ be a finite Galois extension of $\mathbb{Q}$ and $O$ the ring of integers in $K$.
Question: Is there an algorithm to test whether $O$ is norm-euclidean?
In case the question has a positive ...
1
vote
1
answer
89
views
Algorithms for Polynomials Over a Real Algebraic Number Field, a reference
I need to find "Algorithms for Polynomials Over a Real Algebraic Number Field
Ph.D. thesis, University of Wisconsin, Madison (1974) by Rubald". However I cannot find it online nor in my ...
4
votes
0
answers
75
views
Finding all nice ideals for quiver algebras
Let $Q$ be a finite, connected and acyclic quiver which is simply-laced.
Let $k$ be a field and $kQ$ the path algebra of $Q$ over $k$.
Recall that an ideal $I$ of $kQ$ is called admissible if it is ...
3
votes
0
answers
147
views
For which $n$ is this ring an euclidean domain?
Let $f_n(x)=x^n-\sum\limits_{i=0}^{n-1}{x^i}$ and $A_n$ the number field corresponding to $f_n$.
Question: Is $A_n$ for all $n$ an euclidean domain? Is there a good choice for an euclidean function?
...