A computer algebra system (CAS) is a program which is able to carry out various symbolic manipulations with mathematical expressions. Some well-known computer-algebra systems: Mathematica, Maple, Wolfram Alpha, GAP. For questions about Mathematica please see the ...

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How does one solve equations over finite fields in SAGE?

Sage has the method solve (or function, I'm not sure what's the correct terminology) that finds solutions to 'symbolic expressions'. In particular, if one wants to find solutions for a given set of ...
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Elementary “bugs” in computer algebra systems?

There's a discussion of bugs in CAS's here, but these are technical errors of interest mainly to the professional mathematician. I am more interested in simple errors which might arise in the use of ...
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Why are these CAS calculated integrals equal

I let Wolfram evaluate this integral: $$\int\frac{-2\sin(x)\cos(x)}{2\cos^2(x)+\cos(x)-1}dx$$ 1st result from WolframAlpha online was $$\frac43 \log(\cos(\frac{x}2))+\frac{1}{3\log(1-2 \cos(x))}+C$$ ...
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Graded piece of ring in Macaulay2

I apologize for the basic question, but I couldn't figure how to do this. I'm looking at the homogenous ring $R = QQ[w,x,y,z]$, I take $I = ideal(w*z-x*y)$ and $S=R/I$. Then I construct some ideal $J$ ...
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54 views

Solving an equation in charcateristic 2 in sage OR finding 3-torsion points of an elliptic curve over field with char 2

Problem: show that an elliptic curve over a field of char 2 has nontrivial 3-torsion points Method: I used SAGE to unwind the duplication formula for an elliptic curve given in short Weierstrass form ...
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72 views

Silverman exercise 3.1 proving that two polynomials are relatively prime iff the discriminant is non-zero

Silverman, p. 104: Show that the polynomials $$f=x^4−b_4x^2−2b_6x−b8 \qquad \text{and}\qquad g=4x^3+b_2x^2+2b_4x+b_6$$ appearing in the duplication formula (III.2.3d) are relatively prime if ...
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65 views

Computing univariate resultant via modified Euclidean algorithm

In an answer to the question Resultant of Two Univariate Polynomials, a PDF of course slides was linked which describes a modification of Euclid's algorithm for computing univariate polynomial ...
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61 views

How to get 2 percentages to a 100%

First of im new to this site and I've never been the sharpest at math, im a web developer by trade. My question is math related but ill just give you a quick background about my website so that you ...
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51 views

What are elementary field extensions?

While reading about symbolic integration I encountered some concepts of Differential Algebra. I do not know much of D.A and Fields in general also I have encountered as an extension of Rings. I ...
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48 views

How to calculate resultant of two polynomials without knowing the roots.

So in Rothstein - Trager's Method of evaluating logarithmic part they need resultant of two polynomial as shown in the image. My question is that how do they calculate the resultant without knowing ...
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111 views

Knuth-Bendix completion algorithm: word problem

Can someone explain me how to set up an algorithm to find the 12 normal forms of the group $A_4$ by making use of the Knuth-Bendix completion algorithm? So we have that $RRR=1, SSS=1$ and $RSRS=1$. ...
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51 views

Help in understanding risch algorithm

This is an excerpt from the paper that explains risch algorithm for symbolic integration. I do not understand how do we get to the residue directly and also how does it help us in the integration. ...
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34 views

Printing long lines in MAGMA without line break

In MAGMA, how do I print a long list of items to a file without automatic line breaks? It seems to me that there is an internal length limit and anything longer than it will result in a automatic ...
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118 views

Galois group command for Magma online calculator?

I need to test if a family of 7th deg and 13 deg equations are solvable. I'm new to Magma, so my apologies, but what would I type in, http://magma.maths.usyd.edu.au/calc/ to determine the Galois ...
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Computing the decomposition of a representation of $S_n$

I have an explicitly defined representation of the symmetric group that I would like to decompose into irreducibles. How to do this most easily? The best approach I have so far is as follows: Find a ...
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31 views

Canonical Separation of variables

Do the functions of the form $\psi(x)\phi(y)$ span $L^2(\mathbf{R}^6)$? Insert proper grammar here.
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How to integrate $\left(1+\ln(x)\right)\sqrt{1+(x\ln(x))^2}$ with Risch algorithm?

How would you integrate $\left(1 + \ln\left(x\right)\right)\, \sqrt{1 + \left(x\ln\left(x\right)\right)^{2}\,}$ using the Risch algorithm? I want to know this because Mathematica is using the Risch ...
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Coercing a vector into a module in MAGMA.

Given a $1\times n$ row vector $v$ over a ring $R$, and an $n$-dimensional $R$-module $M$, is there any way to coerce the vector $v$ into $M$ (the obvious $M!v$ does not work).
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In maple how do you evaluate combinations?

In Maple how do you have it evaluate combinatorics such as $\binom{5}{2}$ and give you the answer 10? (What is the name of what I want to do anyways, is it evaluate?) Thanks. Seems so easy now that I ...
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205 views

CAS for algebraic geometry, which one?

I use Maple to compute Groebner bases and find it very efficient/fast for my current needs. However, several introductory textbooks on algebraic geometry refer to Singular, which I never used before. ...
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33 views

Can Maple solve recurrence relations when n is only certain numbers?

If you have an equation $a(n)=a_{n-1}+2a_{n-2}$ only defined when n is powers of 2, how would convey that to Maple that n has a restriction? What if you wanted only $n \ge 2$?
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How to define a recurrence relation in Maple?

I'm new with Maple and want to define a recurrence relation. I want to 1) have Maple solve it to the explicit formula 2)have Maple output a few evaluations of it for various values of ...
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Cornacchia's Algorithm

Can any one give the representation for a prime number 9444732965601851473921 as $x^2+15y^2$. I tried cubic cornacchia algorithm in nzmath implemented through python, but in vain. I also tried ...
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66 views

Resultant of Two Univariate Polynomials

I am trying to implement an algorithm for computing Res(f(x),g(x),x) where f(x) and g(x) uni variate polynomials with integer coefficients. Could any one list the various algorithms for computing ...
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Is there a known algorithm for approximating all the real and imaginary zeros of any well behaved equation of a single variable?

Does there currently exist a general algorithm (or set of algorithms used together) that will approximate all the zeros of any well behaved non-differential equation of a single variable which has a ...
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Revealing output slowly in Magma

When working in Magma, is there a command to gradually reveal the output. For example, if you have a finite group $G$ and you want to calculate SubgroupLattice(G), then the output may be thousands of ...
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Using Wolfram Alpha to solve a system of linear equations

How do I input the below system of equations in Wolfram Alpha in order to solve for the unknowns? I'm wondering if there's some kind of code that can be written in order to make wolfram alpha ...
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65 views

R code for triple integration

I am trying to write R code for triple integration. Here, I am just using a simple function as an example: f(x,y,z) = x^2*y+z. Here's what I have : fvec = function(x,y,z) sapply(x, function(t,y,z) ...
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Evaluating polynomials at elements of algebras in Magma

I've defined a $2$ dimensional algebra, $A$ in Magma as follows: R:=FieldOfFractions(PolynomialRing(GF(2),10)); A:=Algebra< R,2|[1,0,0,1,0,1,1,1]>; The sequence of $1$'s and $0$'s just ...
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Picking out Vector Coefficients in Magma

Suppose $K/k$ is a finite field extension, with fixed basis $e_1,\ldots,e_d$, and $f\in K[x_1,\ldots,x_n]$. Using our basis, $f$ has a unique expression as a sum of polynomials in ...
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Producing a set of matrices in MAGMA

As part of my work I need to define the set of all $n\times n$ matrices with entries in $\{0,1,\ldots,n\}$ (considered as a subset of $\mathbb{Z}$), such that each row and column of the matrix sums to ...
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109 views

How to extract the indeterminates from a set of polynomial?

I am a biologist and I am facing a huge problem. I would like to extract the indeterminates of a set of polynomials, for example, I have: $f_{1} = \{\\x_{3}^{2} + x_{1}*x_{2} + x_{1} + x_{1}*x_{3},\\ ...
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29 views

software with a routine for the vanishing ideal of a finite set of points

I am looking for an algebraic software package that provides a routine that computes the vanishing ideal of a finite set of points. So far i am working with Macaulay2 but i have not been able to find ...
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1answer
119 views

Using Janet Basis to solve a nonlinear polynomial system

I am trying to solve a nonlinear polynomial equation system using Janet basis, when they have finite many solutions. For example the solution of the system: $$xy^2-y^3-3x^2=0,x^2+y^2+xy=0.$$ There ...
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366 views

Solving large, sparse system of linear equations

I have a system of linear equations as follows: $$(A+I)x=B$$ where $I$ is the $n\times n$ identity matrix, $A$ is a $n\times n$ matrix such that the first and last rows are blank, and, for every ...
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Maple command for algebraic combinaiton of polynomial in $\mathbb{Z}_p[x]$

Let $F(x) , f_1(x),\ldots, f_r(x)$ be polynomials with coefficients in $\mathbb{Z}_p$ ($p$ is odd prime) Is there a Maple command to determine if$$ F(x) = a_1(x)f_1(x)+\dots+ a_r(x)f_r(x) \pmod{p} ...
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104 views

Vectorfields on Surfaces with MuPAD

i am working with MuPAD. I can make vectorfields in 3D and also Surfaces in 3D via plot::VectorField3D and plot::Surface. But now i want to draw a vectorfield ON a surface. If $X$ is a vectorfield ...
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137 views

Is there a computer programm or CAS (maybe GAP?) that can calculate with projective (indecomposable) A-modules (A is a finite dimensional k-algebra)?

I have the following question(s): I have an "Algebra-With-One" $R$ as a subalgebra of a full matrix algebra in GAP. Furthermore, I have 5 primitive orthogonal idempotents $e_1,...,e_5$, which sum up ...
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Extracting Information from Lists in GAP

If I have a list $L$ in GAP, and a certain list of properties $a,b,c$, how can I count the the number of elements in my list that have all three properties? I've searched the manual (chapter on ...
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188 views

Computing Conjugacy Classes of Subgroups in GAP

GAP has the command ConjugacyClassesSubgroups which gives a list of the conjugacy classes of a finite group $G$. Is there a way I can specify further what types of subgroups GAP reports? For ...
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Test for equivalence of algebraic expressions

We are looking for the most efficient (most recent, or best) techniques to check if two algebraic expressions (elementary, Calculus-type functions) are equivalent (or if an expression is equivalent to ...
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507 views

Computing Subgroup Lattices

Let $G$ be a finite group, and let $L(G)$ be the lattice of subgroups, partially ordered by inclusion. For example, below is $L(D_8)$. $\quad\qquad\quad\qquad\quad\quad\qquad$ I have two questions: ...
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Minimal set of algebraically independent numbers

Suppose we have a set of polynomials $f_1, f_2, \ldots, f_n \in \mathbb{Q}[x]$. Consider the set $$S := \{\alpha \in \mathbb{C} \; | \; f_i(\alpha) = 0 \text{ for some } i \}$$ of complex roots of ...
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Computer algebra system for category theory

There are many different computer algebra systems which allows to perform computer aided computation in symbolic object like polynomials and function and also ordinals. Now being a category theory ...
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Postfix notation in MAPLE

I usually use MATHEMATICA as a computer algebra system and there I heavily employ the postfix notation (achieved by "//" at the end of some command), e.g. ...
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133 views

Some questions about SAGE

I search a programmable calculator like PARI/GP. In the net, I came across SAGE. I have some questions about the download of SAGE Is it really absolutely free ? Is it faster than PARI/PG ? Has it ...
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Tools for optimizing asymptotic bounds.

Is there any tool for this task ? Given the asymptotic bound in term of $n$ and other paramaters $t_1,\dots,t_r$, then return the value for each $t_i$ which optimizes the expression in term of $n$, ...
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338 views

Plotting parametric equations in Maple

I have the following problem. Let $\alpha:=\alpha(v)$ and $\beta:=\beta(v)$, where $v\in\mathbb{R}$. Using MAPLE, I would like to plot $\alpha$ and $\beta$ in the $(\alpha,\beta)$ space as $v$ varies ...
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102 views

Computing Resultant

The resultant of two polynomials is defined as the determinant of the Sylvester matrix. If the polynomials are of degree $n$ and $m$, than the Sylvester matrix will be of dimension $(m+n)\times ...