# 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 ...
1 vote
80 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 ...
1 vote
199 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 ...
305 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 ...
62 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 ...
72 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 ...
161 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 ...
267 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 ...
91 views

### Using a Groebner Basis to show Bijectivity

I have a set of n polynomials over n variables, like ...
208 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 ...
102 views

### Eliminating a variable with GroebnerBasis

I have the following two hefty equations: ...
172 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 ...
1 vote
112 views

### Groebner Basis for Functions

We can use Reduce to solve the following Reduce[1 - Sqrt[a^(-1)] + Sqrt*Sqrt[(a + a^2)^(-1)] - Sqrt[2 - 2/(1 + a)] == -1 + Sqrt, a, Reals] We can also do ...
453 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 ...
96 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....
178 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 ...
102 views

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

I consider a set of three polynomials of two variables ...
122 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: ...
231 views

### Default weight matrix for EliminationOrder

For the computation of elimination ideals via Mathematica's GroebnerBasis method, e.g. ...
144 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 ...
158 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?
164 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: ...
164 views

### Finding a set of characteristics of a system of polynomial equations

we have an ODE system as following ...
170 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, ...
1 vote
105 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, ...
146 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 ...
428 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 ...
760 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 ...
867 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 ...
90 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, ...
339 views

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### GroebnerBasis without specifying variables

All the examples in the Mathematica documentation specify that the syntax for the GroebnerBasis command is ...