Questions tagged [groebner-bases]
Questions on the use of Gröbner basis techniques in Mathematica.
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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 ...
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80
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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 ...
- 1,642
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1
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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 ...
- 13
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3
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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 ...
- 137
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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 ...
- 1,357
4
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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 ...
- 531
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2
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161
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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 ...
- 1,642
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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 ...
- 1,642
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Using a Groebner Basis to show Bijectivity
I have a set of n polynomials over n variables, like
...
- 503
2
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2
answers
208
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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 ...
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102
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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 ...
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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 ...
- 3,044
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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 ...
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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....
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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 ...
- 21
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How to make GroebnerBasis Work or to speed NSolve up [closed]
I consider a set of three polynomials of two variables
...
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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:
...
- 325
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Default weight matrix for EliminationOrder
For the computation of elimination ideals via Mathematica's GroebnerBasis method, e.g.
...
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1
answer
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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 ...
- 306
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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?
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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:
...
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164
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Finding a set of characteristics of a system of polynomial equations
we have an ODE system as following
...
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170
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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, ...
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How to automatically find sequence of linear transformations
I have a set of three polynomials in x and y, and 9 real coefficients a1, ...
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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 ...
- 1
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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 ...
- 83
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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 ...
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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 ...
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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, ...
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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 ...
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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 $C_{ij}$.
...
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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 ...
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Need help solving a polynomial system [closed]
I need help solving the following system of equations:
...
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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 ...
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How to speed up the calculations of the Gröbner basis of the following system of equations?
Consider the following ideal:
...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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2
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Finding common roots of a specific system of polynomials
Some of my colleges are interested in finding common zeros of the following four polynomials:
...
- 2,449
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2
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625
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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 ...
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784
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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 ...
- 1,229
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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 ...
- 1,522
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1
answer
424
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GroebnerBasis without specifying variables
All the examples in the Mathematica documentation specify that the syntax for the GroebnerBasis command is
...
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Gröbner basis on a particular set of equations
This question is very similar in gist to equation solving with GroebnerBasis, but hopefully when I say that I make the system a little larger I mean little. I have ...
- 884
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Equation solving with GroebnerBasis
I have a system of quadratic equations defined below
...
