Thank you for the reply. I am now trying the computation with a somewhat more involved example and I am getting the error. The example I am trying to solve is below:
R = QQ[sqrt(-1)] RI = R.gens()[0] S.<x1,x2,x3,x4,x5,x6,x7,x8> = PolynomialRing(R,order='lex') SI = S.ideal(4*RI*x4+2*x1-2.52*x3-8*RI,3*x4+2*x1-3.5*x2-6,-4*RI*x8+2*x5-2.52*x7+8*RI,3*x8+2*x5-3.5*x6-6,x1*x5-1,x2*x6-1,x3*x7-1,x4*x8-1) The last line ends up with an error: TypeError: unsupported operand parent(s) for '*': 'Real Field with 53 bits of precision' and 'Multivariate Polynomial Ring in x1, x2, x3, x4, x5, x6, x7, x8 over Number Field in I with defining polynomial x^2 + 1' I tried the other two methods discussed in this thread and I am obtaining similar type errors. I am not sure why this example is causing this error. Is there any solution to this error? Any help is well appreciated. Best Regards, On Monday, February 17, 2014 3:47:24 PM UTC-5, john_perry_usm wrote: > > The actual computation I had in mind requires a somewhat more convoluted: > > sage: R = QQ[sqrt(-1)] > sage: RI = R.gens()[0] # necessary, since Sage's I is symbolic, and causes > issues > sage: S.<x,y> = PolynomialRing(R,order='lex') > sage: SI = S.ideal((1+RI)*x+y,x+(1-RI)*y-(1-RI)) > sage: SI.groebner_basis() > [x + (-I + 1), y - 2] > > Sorry for my earlier imprecision: when I did it the first time, I didn't > realize that "I" was defined as something other than sqrt(-1) in my current > session. > > is the variable x in the first line a dummy one, i.e. has nothing to do >> with the >> x in the second line? >> > > I wouldn't say it's a dummy; it's used to define the number field. But, it > is unrelated to the second x. > > I don't use CC much myself, but it certainly has its uses (e.g., > approximation). > > john perry > > On Monday, February 17, 2014 12:39:46 PM UTC-6, sahi...@gmail.com wrote: >> >> Thank you, I get the solution by using >> >> N.<i> = NumberField(x^2+1) >> S.<x,y> = PolynomialRing(QQ,order='lex') >> >> is the variable x in the first line a dummy one, i.e. has nothing to do >> with the >> x in the second line? Sorry, I am new to Sage and sometimes I get >> confused. >> >> If CC is not appropriate for this kind of problems we are discussing, for >> what >> computational reason can CC be used in sage or any other computer algebra >> system? >> >> Best Regards, >> >> On Monday, February 17, 2014 1:08:30 PM UTC-5, luisfe wrote: >>> >>> >>> >>> On Monday, February 17, 2014 6:39:38 PM UTC+1, sahi...@gmail.com wrote: >>>> >>>> OK, I tried the following: >>>> >>>> S.<i,x,y> = PolynomialRing(QQ,order='lex') >>>> I = ideal(i^2+1,(1+i)*x+y,x+(1-i)*y-(1-i)) >>>> G = I.groebner_basis() >>>> G >>>> >>>> would give me >>>> >>>> [i - x - 1, x^2 + 2*x + 2, y - 2] >>>> >>>> which are the results. But I am confused; why I can't get the result when >>>> I try >>>> to get a polynomial ring in the field of complex numbers implemented by >>>> sage? Also, >>>> does adding i**2+1=0 really extend the rational numbers to complex number >>>> field? >>>> >>>> >>> The problem with CC is that it is an *inexact field. *If you do >>> computations with coefficients in CC, you will end up with roundup errors. >>> For instance, buchberger algorithm to compute Grobner basis would yield the >>> ideal (1) with high probability. >>> >>> In the case, you are not computing on the complex numbers, only on the >>> gaussian rationals. Essentially, you are working on QQ[i] without naming >>> it. In this case your solutions live on QQ[i] so it is not a problem. >>> >>> Consider the following example: >>> >>> system x^2+i+y^3, y^4-x >>> >>> sage: S.<i,x,y>=PolynomialRing(QQ,order='lex') >>> sage: I=Ideal(x^2+i+y^3, y^4-x) >>> sage: I.groebner_basis() >>> [i + y^8 + y^3, x - y^4] >>> >>> Then, y is any of the 8 roots of the polynomial *'i + y^8 + y^3*', and >>> for each one of these roots, *x=y^4*. So you get 8 pairs (x,y) of >>> solutions. >>> >>> By the way, the suggestion given by John Perry is to do: >>> >>> sage: N.<i> = NumberField(x^2+1) >>> sage: S.<x,y>=PolynomialRing(QQ,order='lex') >>> >>> if you do this, then >>> >>> sage: i^2 >>> -1 >>> >>> you are really working on QQ[i] >>> >> -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/groups/opt_out.
