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From: | William Sit |
Subject: | Re: [Axiom-math] Groebner bases of a set of equations |
Date: | Wed, 19 Feb 2014 13:26:41 -0500 |
Dear Sureyya:I suggest you put the parameters in DMP and set the main equation variables in DMP or POLY. That way, you won't have any ambiguity.
A groebner basis consisting of just 1 means the ideal generated by the given set of polynomials is the whole ring, but you need to know which ring your original set of parametric polynomials are in to interpret the result. Moreover, a system of parametric equations is DIFFERENT from a system of equations whose coefficient ring is another polynomial ring, because you are then solving the parametric system generically, rather than parametrically.
As an example, if you solve bx=1, where b is a parameter and x the unknown in DMP([x], FRAC POLY INT) (pseudo code only, as I don't remember the correct syntax), you will get x = 1/b, and clearly this is not valid when b = 0 (but of course, b is NOT zero in FRAC POLY INT---it is an indeterminate.
Solving a parametric algebraic system [1] should produce a covering of the parametric space where each "regime" in the cover is defined by a set of equations and inequations in the parametric variables and a solution in the main variables under the parametric conditions of the regime. In the simple example above, the cover consists of two regimes: b = 0 (no solution) and b \neq 0 (solution x = 1/b).
[1] Not to be confused with "parametric equations", which is parametrization or parametric representation of solutions of an algebraic equation, such as using x = cos theta, y = sin theta for x^2 + y^2 = 1.
For LINEAR systems in the main variables, I have a package called PLEQN (Parametric linear equations) that you may want to check out. See my paper on the subject:
http://www.sciencedirect.com/science/article/pii/S0747717108801046I am not aware of any package to solve general parametric algebraic equations in Axiom (but I have not kept up-to-date on Axiom). Mathematica has a built-in function called Reduce that seems to do that (it is even more general as it deals with inequalities as well, but the algorithm appears to be proprietary and it does not produce a Groebner basis).
William On Wed, 19 Feb 2014 08:15:41 -0500 Sureyya Sahin <address@hidden> wrote:
Thank you for the help. I tried your suggestion and I am indeed getting some results. But if we extend the number of variables by defining a few of the parameters (in this case cb,sb) while keeping the equations unchanged, would this effect the polynomial equations and therefore the solutions? Also, is this a standard way to attack this kind of a problem, i.e. equations with parameters, in axiom? I guess this needssome experimentation to obtain the solution.I was initially thinking that if I get [1] as a Groebner bases, then it means that there is no solution to the system of equations. But given this example, I guess I was wrong in my interpretation. What does getting [1] as a Groebner bases in axiom or another computer algebrasystem mean? Best Regards, On Tue, 2014-02-18 at 18:39 -0500, Bill Page wrote:Try a larger set of variables (generators). Other unlisted symbols default to being parameters (from FRAC POLY INT). For example[ca,cb,sa,sb,x,y] gives a basis of 12 polynomials. See http://axiom-wiki.newsynthesis.org/address@hiddenOn 18 February 2014 11:23, sahin <address@hidden> wrote:> Hello, >> I am trying to obtain Groebner bases of a system of equations. Below is my> code > > (1) -> m : List DMP([ca,sa,x,y],FRAC POLY INT) > (2) -> m :=> [x^2+y^2-r1^2,(x+lab*ca)^2+(y+lab*sa)^2-r2^2,(x+lac*(ca*cb-sa*sb))^2+(y+lac*(sa*cb+ca*sb))^2-r3^2,ca^2+sa^2-1]> > asking for groebner bases is leading to > (3) -> groebner(m) > > (3) [1] > Type:> List(DistributedMultivariatePolynomial([ca,sa,x,y],Fraction(Polynomial(Integer))))>> which does not make sense to me. The equations are based on a physical > system and I can't see any reason that would lead to an inconsistency. Why > am I getting [1] as the result? Any help or insight would be> well-appreciated. > > Best Regards, > > > > --> View this message in context: http://nongnu.13855.n7.nabble.com/Groebner-bases-of-a-set-of-equations-tp179213.html > Sent from the axiom-math mailing list archive at Nabble.com.> > _______________________________________________ > Axiom-math mailing list > address@hidden > https://lists.nongnu.org/mailman/listinfo/axiom-math_______________________________________________ Axiom-math mailing list address@hidden https://lists.nongnu.org/mailman/listinfo/axiom-math
William Sit, Professor Emeritus Mathematics, City College of New York Office: R6/291D Tel: 212-650-5179 Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/
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