Hi William: Thank you for your post regarding my problem on solving the Groebner bases of a set of polynomial equation. The main problem is that not all of the variables are associated with the set of polynomials, i.e., in this case I have Q[x,y,ca,sa]. The rest of the variables are independent parameters which can be interpreted as rational functions. I already tried simpler polynomial equations and axiom gives me meaningful results. Just as an example, if you try the following code in Q(a2,a3,a4,rx,ry)[c3,s3,c4,s4];
n : List DMP([c3,s3,c4,s4], FRAC POLY INT) n := [a2*c2+a3*c3-a4*c4-rx,a2*s2+a3*s3-a4*s4-ry,c3^2+s3^2-1,c4^2+s3^2-1] g := groebner(n) You will obtain meaningful results. This is the solution of the kinematic equations of a simple four-bar linkage and the other parameters a2,a3,a4,rx,ry are just symbols, yet axiom is capable of giving me solution in this case. Here I am not talking about whether the Groebner bases is defined or not, I am obtaining a set of Groebner base solutions in terms of the symbolic variables. On my original question: I am obtaining [1] as the Groebner base solution which I interpreted as the solution set is the whole Q[ca,sa,x,y], which would imply no solution as a consequence of Nullstellensatz. Thus, I interpreted this as I have inconsistent (meaningless) equations. But the equations describe a physically meaningful system. And I can't catch an error on the equations. So, I thought I may misinterpret the [1] that I receive from axiom. To my understanding you think [1] may be coming from the symbolic variables; but this does not make sense to me as I already defined Q[ca,sa,x,y] and the rest is being taken as symbols as in the simple example I illustrated. I may yet try including the rest of the variables in the definition of the polynomial, i.e., Q[ca,sa,x,y,cb,sb,r1,r2,r3,..]. But I have my doubts as I have already shown that axiom can solve equations in Q(cb,sb,r1,r2,r3,..)[ca,sa,x,y]. I may be misinterpreting your suggestion, can you give more details of your recommendation? Thank you for including your study on linear systems as well. Generally, I am dealing with nonlinear polynomials in trigonometric/transcendental functions. Thus, I will not be able to use a solution in linear polynomials. Yet I will read to gain a bit more insight into the symbolic computation. Best Regards, On Wed, 2014-02-19 at 13:26 -0500, William Sit wrote: > 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/S0747717108801046 > I 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 <sahinsure...@yahoo.ca> 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 needs > > some 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 algebra > > system 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/SandBoxGroebnerBasis#msg20140219000533+0...@axiom-wiki.newsynthesis.org > >> > >> On 18 February 2014 11:23, sahin <sahinsure...@yahoo.ca> > >>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 > >> > Axiom-math@nongnu.org > >> > https://lists.nongnu.org/mailman/listinfo/axiom-math > > > > > > > > _______________________________________________ > > Axiom-math mailing list > > Axiom-math@nongnu.org > > 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/ _______________________________________________ Axiom-math mailing list Axiom-math@nongnu.org https://lists.nongnu.org/mailman/listinfo/axiom-math
