Skip to main content

Questions tagged [groebner-bases]

Questions on the use of Gröbner basis techniques in Mathematica.

Filter by
Sorted by
Tagged with
2 votes
0 answers
154 views

Different Groebner bases in 12.1 and 13.3 with the same code

With exactly the same code I get different Groebner bases with different versions of Mathematica. Even worse, with exactly the same code I get a Groebner basis with a version of the program but I ...
Ramon's user avatar
  • 21
1 vote
1 answer
57 views

How to get the multiplicity of a certain root in a system of polynomial equation

Suppose I have a system of polynomial equations with base field $\mathbb{C}$ with $n$ equations and $n$ unknowns, how can I get the multiplicity of a certain solution? As an example, consider $$x^3=0\\...
Peter Wu's user avatar
  • 111
1 vote
1 answer
69 views

Grading and Groebner bases

How to specify the degree or weight of variables in the computation of Gröbner bases in Mathematica? I tried something like below, but 'Weights->degrees' part doesn't work. ...
Astro Naut's user avatar
1 vote
2 answers
137 views

How to get similar GroebnerBasis?

Near the bottom of the page, there is "The above system of equations has only a limited number of solutions and the ideal can be calculated". But I cannot get the same GroebnerBasis, four ...
Ytrewq's user avatar
  • 179
3 votes
0 answers
47 views

Can we ask Mathematica if a reduction of a system to one scalar equation in one variable+ rational representations of the other variables exists?

Nonlinear determined systems have typically several solutions involving square or higher order roots. Instead of solving them, it may be more profitable to reduce them to smaller systems with fewer ...
florin's user avatar
  • 1,900
1 vote
0 answers
85 views

Why `GroebnerBasis` fails to reduce a system to one scalar equation, when I can almost do it by hand?

I have a system of five equations (Behn's model of CD4+T/APC interaction). The last four yield rational solutions in x1, so I can plug these in the first equation ...
florin's user avatar
  • 1,900
1 vote
1 answer
226 views

Is there a function in Mathematica that computes the number of solutions to polynomial systems?

Given a system of polynomial equations in $\mathbb{C}$-coefficients, is there a tool in Mathematica that computes the number of solutions to this system, counted with multiplicity? (We may assume ...
Boyu Zhang's user avatar
4 votes
3 answers
442 views

Basis for multivariable polynomials

I have a bunch of two-variable polynomials and as part of a larger algorithm need to find a basis for them and express them in terms of this basis. As an illustrative example, for one case my ...
R.W's user avatar
  • 137
0 votes
1 answer
83 views

Transform Polynomial in Trigonometric Functions to AssociatedLegendrePolynomials

I want to transform a polynomial in Sin and Cos to AssociatedLegendrePolynomials. I have a working code, but the end result is ...
infinitezero's user avatar
  • 1,419
4 votes
1 answer
123 views

Monomial order when testing ideal membership with PolynomialReduce?

The documentation for PolynomialReduce points out that, to test membership in an ideal, the list of polynomials must be a Groebner basis for that ideal. But does ...
Lyle Ramshaw's user avatar
2 votes
2 answers
165 views

When 'GroebnerBasis' is used for eliminating the state variables in determinant of Jacobian, is the sign related to the signs of det at fix points?

Elimination is typically very difficult without Mathematica, for nonlinear dynamical systems, for example when one must find the Determinant at the fixed points. With Mathematica, this seems ...
florin's user avatar
  • 1,900
5 votes
2 answers
278 views

Is it possible to characterize the sign of the trace and det at the fixed points of a dynamical system using Gröbner, postponing computing the points?

Here is an example of a dynamical system for which the isoclines may intersect at two or more points (under certain numeric conditions). It is easy to compute the trace and determinant of the Jacobian ...
florin's user avatar
  • 1,900
0 votes
1 answer
94 views

Using a Groebner Basis to show Bijectivity

I have a set of n polynomials over n variables, like ...
Thomas Ahle's user avatar
2 votes
2 answers
227 views

Does Mathematica have build-in function to compute dimension of square polynomial system?

I'm working with square polynomial systems and wish to know if a (small) system has a finite number of solutions. That is, if it's zero-dimensional. I'm not aware of any built-in function to do this ...
Dominic's user avatar
  • 2,914
3 votes
1 answer
109 views

Eliminating a variable with GroebnerBasis

I have the following two hefty equations: ...
Aharon Naiman's user avatar
6 votes
1 answer
202 views

GroebnerBasis internals and runtime dependence on variable list ordering

I have a problem I have reduced to asking for a Gröbner basis. For some reason, Mathematica is able to solve this in minutes, while other programs more dedicated to these types of calculations run for ...
RedCurry's user avatar
1 vote
0 answers
115 views

Groebner Basis for Functions

We can use Reduce to solve the following Reduce[1 - Sqrt[a^(-1)] + Sqrt[2]*Sqrt[(a + a^2)^(-1)] - Sqrt[2 - 2/(1 + a)] == -1 + Sqrt[2], a, Reals] We can also do ...
Moo's user avatar
  • 3,388
4 votes
1 answer
547 views

Solving a system of polynomial equations - can I be sure that the number of solutions, and the solutions are correct?

I need to solve a system of two polynomials with integer coefficients in two variables, $\{Q_1(w,z)=0,\,Q_2(w,z)=0\}$, and want to compute all real solutions. I'm able to run ...
ilan's user avatar
  • 53
3 votes
0 answers
97 views

Why are some symbols recognized by Mathematica but nowhere in the documentation? [duplicate]

I was trying to calculate a Groebner basis for a given set of polynomials, and I did it successfully. But I wanted to calculate the conversion matrix between the original basis and the Groebner basis....
Bob Jones's user avatar
  • 131
2 votes
0 answers
190 views

Monitoring PolynomialReduce/Alternatives for other CAS packages

I have a large generic polynomial That looks like $N = \sum_{i_1,i_2\cdots}c_{i_1,i_2,i_3,\cdots} {x_{1}}^{i_1}{x_{2}}^{i_2}\cdots $ This could have anywhere between 3000-9000 terms with a maximum ...
2010jetta's user avatar
3 votes
0 answers
103 views

How to make GroebnerBasis Work or to speed NSolve up [closed]

I consider a set of three polynomials of two variables ...
Anna Veselovska's user avatar
3 votes
0 answers
143 views

Same problem, different GroebnerBasis

Consider the following system of equations: $$ \begin{cases} d_0^2+d_1^2=2 \\ d_0 d_1 = -1 \end{cases} $$ Here's Mathematica solving it and producing the Gröbner Basis: ...
Džuris's user avatar
  • 325
6 votes
1 answer
241 views

Default weight matrix for EliminationOrder

For the computation of elimination ideals via Mathematica's GroebnerBasis method, e.g. ...
Kathryn's user avatar
  • 63
2 votes
1 answer
146 views

Why is my GroebnerBasis = 0 in this case?

I recently asked a question: Finding Intersections Between Arbitrary Surface and A Line whose solution required me to use the GroebnerBasis function. I successfully ...
TribalChief's user avatar
2 votes
0 answers
165 views

Grobner basis calculation while keeping track of the change of basis [closed]

Can one compute a Groebner basis for a polynomial ideal in Mathematica, while simultaneously getting the change of basis matrix?
baltazar's user avatar
  • 121
5 votes
1 answer
172 views

How to stop GroebnerBasis from automatically normalizing?

I need to compute the Groebner basis modulus a set of prime without normalizing the Groebner basis output. Is this possible? Example: ...
VulcanEconomist's user avatar
0 votes
1 answer
173 views

Finding a set of characteristics of a system of polynomial equations

we have an ODE system as following ...
ali's user avatar
  • 5
0 votes
1 answer
176 views

Define a new CoefficientDomain in PolynomialReduce function

If we have some parameter like $q_1, q_2,...,q_n$, I would like to define a new CoefficientDomain for PolynomialReduce function such that get these parameters as integers, In the other words, ...
ali's user avatar
  • 5
1 vote
1 answer
107 views

How to automatically find sequence of linear transformations

I have a set of three polynomials in x and y, and 9 real coefficients a1, ...
QuantumDot's user avatar
  • 19.7k
0 votes
0 answers
147 views

NSolve command for underdetermined systems in Mathematica 11

I used to have Mathematica 8 previously and whenever I ran the NSolve command on underdetermined systems, NSolve[x^2 + y^2 + z^2 == 1 && x y z == 2, {x, y, z}] it used to give me a ...
dbm's user avatar
  • 1
6 votes
1 answer
465 views

How to express Groebner Basis in terms of original elements?

I have polynomials {A1,B1} and then I find their Groebner basis to be {g1,g2,g3,g4}. Is there a way for me to express each polynomial g1,g2,g3,g4 in terms of A1, B1? (ie have g1= h1 A1 + h2 B1 where ...
Andy Nguyen's user avatar
9 votes
1 answer
907 views

How to use Mathematica's GroebnerBasis to automate solving system of polynomials

Currently, Only way I know to use GroebnerBasis to solve system of polynomials require lots of manual steps. I'd like to know how to automate this process and if ...
Nasser's user avatar
  • 147k
8 votes
1 answer
927 views

Using a Grobner basis to calculate common roots of a system of polynomial equations

We are trying to solve the inverse kinematics of a robot with $3$ revolute joints and one prismatic arm, link $4$, able to take lengths between $1$ and $2$. We assume all other lengths are $1$. Image ...
K.Power's user avatar
  • 183
0 votes
0 answers
92 views

to reduce the unknowns of a set of equations with several symbolic coefficients, e.g. using the GroebnerBasis

I have a set of $4$ equations with $4$ unknowns $\theta_1$ to $\theta_4$, and would like to eliminate the number of unknowns, to get a smaller set of equations, e.g. $n$ equations with $n$ unknowns, ...
larry's user avatar
  • 735
3 votes
0 answers
395 views

Operations on ideals of polynomial rings

There is GroebnerBasis to compute a Gröbner basis of an ideal in a polynomial ring, but I am looking for a package to perform operations on ideals $I,J\subseteq\Bbb ...
Jesko Hüttenhain's user avatar
0 votes
0 answers
35 views

Groebner bases with symbolic exponents [duplicate]

I'm trying to eliminate variables using Groebner bases or polynomial resultants with symbolic exponents. I'm wondering if it is possible to do it in general case for non-negative integers $C_{ij}$. ...
Caims's user avatar
  • 432
2 votes
2 answers
163 views

Crash after long GroebnerBasis calculation

I am running a long computation of a GröbnerBasis and after some hours the kernel crashes. The memory usage increases enormous, and it crashes, when it reaches somewhat 4 GB RES, however it is ...
BeniBela's user avatar
  • 131
1 vote
0 answers
118 views

Need help solving a polynomial system [closed]

I need help solving the following system of equations: ...
Дмитрий Саморядов's user avatar
5 votes
0 answers
319 views

How to find an algebraic representation

I want to find matrix representation of some algebra (Clifford algebra Cl(3,1) in this case). Here is an example, which I would like to get and then extend to higher dimensional matrices. Suppose we ...
Acus's user avatar
  • 3,680
1 vote
0 answers
237 views

How to speed up the calculations of the Gröbner basis of the following system of equations?

Consider the following ideal: ...
Ed Mendes's user avatar
  • 618
3 votes
1 answer
865 views

How to convert a rational parametric plane curve into implicit form?

This problem is generated from another Green's theorem related question of mine. And here is a forward of the same problem in math.stackexchange. The original equation of the plane curve is not in ...
LCFactorization's user avatar
4 votes
1 answer
1k views

Producing a minimal Gröbner base

Mathematica has the command GroebnerBasis[{p_1,...},{x_1,...}] that returns a Gröbner base for some set of polynomials. I want to know if there is a command to do ...
Pedro's user avatar
  • 141
2 votes
0 answers
88 views

Finding related roots to a polynomial

I posted this in math.stackexchange, but this might be a better place. Assume that $f(X,Y,Z,V,W)\in \mathbb{Z}[X,Y,Z,V,W]$ is some polynomial and assume that $f(x,y,z,v,w)=0$. I would like to know if ...
JT1's user avatar
  • 121
0 votes
1 answer
395 views

PolynomialReduce inconsistent results

I was playing around with gröbner basis and s-polynomials and I fell upon the PolynomialReduce, and I was wondering why it gives different results when I move around the polynomials in its second ...
DrLime2k10's user avatar
4 votes
2 answers
332 views

Number of complex roots for the system of polynomials

I have a system of algebraic equations. For example: $$ \left\{ \begin{aligned} x^2 y + 2 x y + 2 &= 0,\\ y^3 + 2 x + 1 &= 0. \end{aligned} \right. $$ (In my problem the system contains 16 ...
bcp's user avatar
  • 781
6 votes
2 answers
419 views

Finding common roots of a specific system of polynomials

Some of my colleges are interested in finding common zeros of the following four polynomials: ...
Per Alexandersson's user avatar
5 votes
2 answers
660 views

Dimension of an algebraic variety

I would like to compute, using Groebner bases, the dimension of the variety defined by a set of polynomials in several variables. In the wikipedia page the method is described, and an implementation ...
Ion Nechita's user avatar
5 votes
1 answer
851 views

Multivariate resultant in Mathematica?

Is there any command in Mathematica 7 which can compute the (McCaulay) resultant of a parametric system of multivariate polynomial equations? In fact, it would be great if there is also a way to ...
dbm's user avatar
  • 1,249
11 votes
1 answer
1k views

How to express the original ideal elements in the Groebner basis?

Suppose I call GroebnerBasis[{f1, f2, ...}, {x1,x2, ...}] The output is a list {g1,g2,...} For each $g_j$, there should be ...
David E Speyer's user avatar
9 votes
1 answer
442 views

GroebnerBasis without specifying variables

All the examples in the Mathematica documentation specify that the syntax for the GroebnerBasis command is ...
mboratko's user avatar
  • 884