9
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Using a Grobner basis to calculate common roots of a system of polynomial equations
I will suggest setting this up in a way that is, to my view, slightly simpler. have each angle be measured from the positive horizontal axis. Place the origin at juncture of the vertical bar and the ...
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How to express Groebner Basis in terms of original elements?
bas = {x^2 + y^2 + z^2 - 1, x - 2 y^3 - 3};
{gb, mat} = GroebnerBasis`BasisAndConversionMatrix[bas, {x,y,z}, {}]
mat.bas == gb // Simplify
(* True *)
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How to use Mathematica's GroebnerBasis to automate solving system of polynomials
As noted in comments, Solve does this. An efficient "by hand" approach is to solve for the first equation, plug the solutions into the next, solve, join ...
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6
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Basis for multivariable polynomials
You can use CoefficientRules to convert polynomial to a vector of coefficients. Below I use CoefficientRules on generic linear ...
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Basis for multivariable polynomials
Not sure if this is what you want, but here is a way to get a vector space basis. We need to extract the different exponent vectors, create a matrix with each row representing a polynomial, augment ...
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6
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Default weight matrix for EliminationOrder
It's not actually a block ordering. EliminationOrder constructs the weight matrix as follows. Order variables so that those to be eliminated precede (are to the ...
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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?
I do not know what OP means by "right sign". But to get a canonical normalization, one can rescale the polynomial and set the leading coefficient equal to $1$. For example, define
...
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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?
My answer does not answer your doubts regarding the Groebner basis, but consider that it is an option to the problem of obtaining non-trivial equilibria.
A viable option is to ...
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Eliminating a variable with GroebnerBasis
How about using Resultant? The result is very large so not posted (noticed I removed the =0 from both expressions):
...
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GroebnerBasis internals and runtime dependence on variable list ordering
From the help of GroebnerBasis: "The Gröbner basis in general depends on the ordering assigned to monomials. This ordering is affected by the ordering of the $...
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4
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Why is my GroebnerBasis = 0 in this case?
Let me show another way to derive implicit Cartesian equations from a given set of parametric equations with trigonometric components. It consists of applying the Weierstrass substitution to the ...
4
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How to stop GroebnerBasis from automatically normalizing?
One cannot change the way the polynomial coefficients are normalized in GroebnerBasis. To do recovery over Q when working with modulo-prime bases one can use ...
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4
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Is there a function in Mathematica that computes the number of solutions to polynomial systems?
Here is code I cribbed from the NSolve method based on computing an eigensystem from a Groebner basis.
...
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4
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Basis for multivariable polynomials
I was going to tell you to use PolynomialReduce, but unfortunately, it's not so simple. In the code below, it expresses the first polynomial as the sum of the ...
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How to get the multiplicity of a certain root in a system of polynomial equation
I would expect the multiplicity of (0,0) to be 12.
sol = SolveValues[{x^3 == 0, y^4 == 0}, {x, y}]
Tally[sol]
Regarding comment, this is what I get for the new ...
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Grading and Groebner bases
This is an addendum to march's comment. You can use a non-singular "weight matrix" to specify weights to the variables. Although how the weight matrix works is not explained in the Document, ...
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How to get similar GroebnerBasis?
You cannot get the same Groebner basis because them basis shown in the paper is not correct. Which maybe is not so surprising, given that this is a 2022 paper about a method that was known as far back ...
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3
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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 believe you missed a lot of solutions, namely, infinitely many of them.
As Daniel points out in the comment below, there is a multiple root at (0,0). Apart from ...
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Monomial order when testing ideal membership with PolynomialReduce?
First question. Yes, the monomial orders must agree in order to have the guarantees that come with Groebner bases, for example to conclusively check membership in an ideal.
Let me take the following ...
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2
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Does Mathematica have build-in function to compute dimension of square polynomial system?
I wish to follow up with this post: The reference above: idealDimension
written by Daniel does answer this question in terms of Ideal Dimension and I tested it in ver. 12.0.0. All credit goes to ...
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Transform Polynomial in Trigonometric Functions to AssociatedLegendrePolynomials
The functions:
LegendreP[n,x] Sqrt([2n+1)/2]
are an orthogonal system of functions over -1.. 1 with the weight function: 1.
Therefore, you can get the expansion ...
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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?
For me, the conclusion of these discussions is that:
The easiest quantities to obtain symbolically for bifurcation analysis of dynamical systems are ...
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vote
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Using a Groebner Basis to show Bijectivity
It's slightly more complicated I think. You have to be able to solve for c1...c4 but also you need to show the injectivity and surjectivity of these maps. I will ...
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Does Mathematica have build-in function to compute dimension of square polynomial system?
@Daniel:
Regarding your comments above. I am working with iterated polynomial systems in which idealDimension quickly become CPU-bound after only a few iterations. Here is a benchmark running on a 4....
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Define a new CoefficientDomain in PolynomialReduce function
To make this self contained I will copy the somewhat large input from the question.
...
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How to automatically find sequence of linear transformations
These commands do not produce what you want since you said that a3 or b2 can be zero. Still they might be a good alternative of ...
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Crash after long GroebnerBasis calculation
I encountered this problem too. You can restart kernel periodically by code. Search for this post "Self-restarting MathKernel - is it possible in Mathematica?"
https://stackoverflow.com/questions/...
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