Using computers 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.

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-1
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0answers
31 views

How can I put elements of a GL(2,2) in diagonals of identity matrix [closed]

I want to put the elements of GL(2,2) in diagonal of I_3 (3*3 identity matrix). I mean the result is a block matrix its blocks are three arbitrary elements of GL(2,2). For this aim I used kronecker ...
6
votes
1answer
228 views

Faster multiplication with a restricted set of multiplicands?

Let $A$ be a set of $k>1$ distinct elements from a semigroup. We wish to compute the product $$ p=b_1 b_2 \cdots b_n$$ where each $b_i\in A$. Clearly $n-1$ multiplications suffice to compute $p$; ...
-2
votes
0answers
93 views

Given a set of generators of a group G, is there a method to find a presentation for G using those generators? [migrated]

Suppose I have a group $G$, which I know is finitely presentable and infinite. (In particular, I have a presentation for it, though not the one I want). Suppose I have a small list of generators ...
30
votes
20answers
7k views

Open source mathematical software.

I want some recomendation on which software I should install on my computer, an open source program for general abstract mathematical purposes (as opposed to applied mathematics). I would likely use ...
2
votes
2answers
307 views

Numerical Determination of Generating Functions from Recursion Relations

Are there computer packages which calculate coefficients of generating functions, such as $$D_n(q)=\sum_m d_{m,n}q^m= \frac{1}{\prod_{i=1}^n (1-q^i)^2} \text{ or}$$ $$S_d(q)=\sum_m s_{m,d}q^m = ...
3
votes
0answers
129 views

Dimension of a commuting nilpotent variety

Fix $k$ an algebraically closed field, $n$ a natural number, and $\lambda=(\lambda_1,\ldots,\lambda_m)$ a partition of $n$. Let $A$ be any $n\times n$ nilpotent matrix with entries in $k$ whose ...
9
votes
4answers
688 views

Is Gauss-Seidel guaranteed to converge on *semi* positive definite matrices?

I know that the Gauss-Seidel method is guaranteed to converge given that the matrix you want to solve is positive definite. I've looked at the proofs of convergence, and it appears that one cannot ...
6
votes
1answer
276 views

Using math software to show that the following groups are infinite?

I would like to show that the following finitely presented group in 3 generators $P, Q, R$ is infinite in certain cases: $$P^p, Q^q, R^r, (PQ)^2, (QR)^2, (PQR)^2, (QR^{r/2+1})^a (RQR^{r/2})^b$$ For ...
0
votes
1answer
51 views

Software for noncommutative Groebner bases over rational function fields

I am wondering whether there is any software package that can compute Groebner bases for noncommutative algebras defined over the field of rational functions $\mathbb{Q}(q)$. I have tried the GAP ...
3
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0answers
122 views

What is the function like when its Mobius inversion is $\sum_{w|r, (w,t)=1}\mu(w)q^{r/w}$?

Everyone, I am now reading a paper named The Irreducible Factors of $(cx+d)x^{q^m}-(ax+b)$ over $GF(q)$, http://qjmath.oxfordjournals.org/content/14/1/61.extract. And I’m confused with one of its ...
9
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9answers
2k views

Is there a non self-referencing non-computable function?

I've seen in college that some functions are not computable. The proof for that was the case of Halt(x,y) function. The thing is, the proof used a very artificial (IMHO) case which is evaluating ...
9
votes
5answers
855 views

Multipolynomial resultants

We know that the resultant of two polynomials can be computed as the determinant of their Sylvester matrix ( http://en.wikipedia.org/wiki/Sylvester_matrix ). How do we compute the resultant of more ...
78
votes
66answers
13k views

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 ...
1
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0answers
86 views

Benchmark problems for computing rational points on varieties

Are there standard benchmark problem sets used for empirically evaluating algorithms designed for computing rational points on (various classes of) algebraic varieties? If so, could you please point ...
3
votes
1answer
123 views

Is the ideal membership problem solvable for differential ideals? Is there a good notion of a Gröbner basis?

Let $K$ be a field of characteristic zero. Let $\Omega = K[x_1, \dots, x_n, dx_1, \dots, dx_n]$ be the differential ring of algebraic differential forms over $K[X_1, \dots, X_n]$. Is there an ...
2
votes
1answer
197 views

efficient arithmetic with (short) Conway games?

We consider "games" in the sense of ONAG. Conway's definition of a game $G$ as a pair $G = \{L \mid R \}$ of sets of games, together with the definitions of inequality and the arithmetic operations ...
2
votes
0answers
95 views

Finding relations between invariant polynomials

Suppose I have an action of a linear reductive group ($GL(2,\mathbb{C})^2$ in this case) on a complex vector space (of dimension $16$) and I want to compute explicitly the ring of invariants of this ...
2
votes
1answer
108 views

Finding particular reduced words for Weyl group elements

I am studying cluster algebra structures on the coordinate rings of partial flag varieties, as defined in the paper Partial flag varieties and preprojective algebras by Geiss, Leclerc and Schröer. One ...
4
votes
0answers
191 views

Algorithm/denominators of elements of a rational affine space

I hope it's not a trivial question... Suppose I have a finite dimensional vector space $V$ over $\mathbb{Q}$ with a distinguished basis (in my case it's the $k$th graded piece of the free associative ...
1
vote
0answers
139 views

Finding a generator of an ideal in an algebraic function field

I have an algebraic function field $\mathbb{Q}(x,y)$, where $y$ satisfies $$ (y^2-1)^2 = x^2(1+x^2), $$ and I need to find a rational function that has a first order root at $x=0,y=1$ a first order ...
4
votes
1answer
195 views

The Representation of $\mathrm{Sp}_{2n}$ of Dimension $2^n$ in characteristic 2

Let $G:=\mathrm{Sp}_{2n}$ be the simple algebraic group of simply connected type with root-system $C_n$. Is there a way, to explicitly construct the highest weight representation ...
1
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0answers
66 views

Testing functional equivalence

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 ...
17
votes
1answer
689 views

Main open computational problems in quantifier elimination?

A language is said to have quantifier elimination if every first-order-logic sentence in the language can be shown to be equivalent to a quantifier-free sentence, i.e., a sentence without any ...
2
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0answers
177 views

Using the Affine Maxima Package

The Maxima computer algebra system has a package called Affine for doing the calculations implicit in Bergman's diamond lemma for rings. It can be viewed as a kind of noncommutative analogue of ...
4
votes
4answers
695 views

computer algebra system for polynomial algebras over finite fields

Is there a computer algebra system that can do arithmetic over polynomial algebras over finite fields where I can specify the extension? Exempli gratia, if $f(x), g(x) \in ...
1
vote
1answer
217 views

Serre's conditions under blow-ups, Blowup and normalization

Suppose $X = \mathbb{Z}[x, y, z]/(f,g)$ is a 2-dimensional Cohen-Macaulay surface. In particular, $X$ satisfies Serre's condition $S_2$. Suppose it is irreducible, reduced but not normal. ...
4
votes
1answer
215 views

Algorithm to decide if ideal is principal

Suppose $R = \mathbb{Q}[x_1, ..., x_n]/I$, and $J \subset R$ is a given height one ideal. Is there a quick algorithm one could write to determine if $J$ is a principal ideal or necessarily not ...
2
votes
1answer
284 views

Computer algebra system (CAS) with good re-presenting or transformation support

Such heavy-weight transformations as expanding or factoring are provided by most of CAS-es, but what about light-weight, but a useful transformations, like "reorder some terms to make expression more ...
47
votes
30answers
14k views

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 ...
1
vote
1answer
120 views

Finding reducible polynomials with restricted factors

Given $f(x),g(x) \in \mathbb{Z}[x]$, two irreducible polynomials, is there a polynomial $h(x) \in \mathbb{Z}[x]$ coprime to $f(x)$ such that $f(x) + g(x)h(x)$ is reducible over $\mathbb{Z}[x]$ with ...
0
votes
1answer
292 views

What program/language can I use to handle large numbers (over 10^300) [closed]

I'm trying to multiply together a set of numbers. However, the programs I have tried (Excel, R, FreeMat) do not allow numbers above around 300 digits. This limits me to sets of a few hundred numbers ...
3
votes
3answers
486 views

Good Computer Package for Calculating Inverse of a Formal Power Series?

This might be a question people already asked or is obvious to experts, or is not appropriate for this forum, if so, I apologize. I am trying to calculate things like $z/(e^z-1)$, or find the inverse ...
1
vote
1answer
203 views

Sage or Magma Implementation of Nilpotent Orbit Varieties

For a given partition $[n_{1},...,n_{k}]$ of $N \in \mathbb{N}$ there exists a corresponding nilpotent orbit variety $O_{[n_{1},...,n_{k}]}$ in $\mathfrak{gl}(N)$ which can be represented by a set of ...
8
votes
2answers
198 views

Ideal Membership without Certificate?

I have a homogeneous ideal $I=\langle f_1,\ldots,f_r\rangle$ of the polynomial ring $\mathbb C[X_1,\ldots,X_n]=:R$ where each of the $f_i$ is actually over $\mathbb Z$. My computations are usually ...
3
votes
1answer
346 views

Algorithm to find exponential map of differential operators acting on function

I am trying to write a computer program which computes the action of the exponential of a differential operator on a function, for any given differential operator. Examples: $\exp(\varepsilon ...
2
votes
1answer
156 views

How to maximize the determinant of a matrix of the form VDV^H

Hi, I have a matrix of the form $A=VDV^H$, where $V$ is a $M \times 2M$ complex matrix, $D$ is a $2M \times 2M$ diagonal real matrix, thus the dimension of $A$ is $M \times M$. My problem is how ...
35
votes
0answers
503 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 ...
2
votes
1answer
339 views

Is there an algorithm to decide if an ideal contains monomials?

Let $I\subset k[x_1,\dots,x_n]$ be an ideal in a polynomial ring in commuting variables. Is there a procedure to decide if $I$ contains a monomial and possibly to find one? Gröbner bases come to ...
20
votes
5answers
6k views

The unification of Mathematics via Topos Theory

When the paper The unification of Mathematics via Topos Theory by Olivia Caramello, says "one can generate a huge number of new results in any mathematical field without any creative effort." is this ...
3
votes
1answer
180 views

The existential theory of the reals

Some definitions of the existential theory of the reals (ETR) allow a real closed field and some definitions allow only rational numbers as coefficients of polynomials. Which one is correct? Will the ...
0
votes
0answers
132 views

Computer package to compute HOMFLY polynomial?

I apologize of already asked by someone else, but what (in your opinion) is the best package for computing HOMFLY polynomials?
1
vote
1answer
111 views

Recommendations for binomial system solver

I am interested in solving binomial systems of the form $$ \begin{cases} a_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} + b_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} &= 0 \\\\ ...
1
vote
2answers
184 views

My output of a group and inverse-closed subset in MAGMA is no longer inverse-closed when entered as input to GAP.

In MAGMA, I input the following: G:=SmallGroup(20,3); G; E:=[xx:xx in G]; S:=[E[6],E[7],E[13],E[20]]; S; S[1]^2; S[2]^2; S[3]*S[4]; This gives the output: GrpPC : ...
10
votes
5answers
2k views

“Oldest” bug in computer algebra system?

The goal of this question is to find an error in a computation by a computer algebra system where the 'correct' answer (complete with correct reasoning to justify the answer) can be found in the ...
10
votes
3answers
428 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 ...
1
vote
0answers
135 views

Randomized alternative to Buchberger's algorithm

Richard Lipton's blog describes a A New Way To Solve Linear Equations by Prasad Raghavendra. Can the ideas in this algorithm be generalized to systems of polynomial equations to provide a randomized ...
7
votes
2answers
319 views

Computing determinants of matrices of linear forms

Suppose we have three $n \times n$ matrices $A$, $B$, $C$ with floating point entries. We would like to compute the polynomial $\det (xA+yB+zC)$. At least in Mathematica, and I think in all computer ...
12
votes
1answer
433 views

How can I tell if a variety is normal?

Suppose $R$ is a subalgebra of ${\mathbb C}[x_1,...,x_N]$ generated by polynomials $p_1,...,p_k.$ I know that ${\mathbb C}[x_1,...,x_N]$ is the integral closure of $R$. Is there an algorithm to ...
3
votes
0answers
248 views

On finding A-polynomials

I have two questions to obtain the explicit forms of A-polynomials. Takata used the mathematica pacage qMultisum.m to obtain the recursion relation of the colored Jones polynomials for twist knots. ...
1
vote
2answers
117 views

Non-Uniform Root of Polynomial in Open Cube

I'm looking to find a root $(x_1,\dots,x_n)$ of a polynomial $p \in {\mathbb R}[x_1,\dots,x_n]$ such that $0 \leq x_i < 1$ for all $i$. Further, I know in advance that setting $x_1 = \cdots = x_n$ ...