Hi David, I think "list" function is enough for me if I extend the ring in two steps: first with free variables and then I extend it algebraically. Your solution is working for me in this way:
sage: k = GF(2); sage: WRBasePolyRing = MPolynomialRing(k, 8, x); x = WRBasePolyRing.gens() sage: S = WRBasePolyRing['w']; w = S.gen() sage: WRPolyRing = S.quotient(w^4 + w^3 + w^2 + w + 1, 't'); t = WRPolyRing.gen() sage: X = x[0]*t + x[1]*t^2 + x[2]*t^4 + x[3]* t^8; sage: Y = x[4]*t + x[5]*t^2 + x[6]*t^4 + x[7]* t^8; sage: MyCurve = Y^2+X*Y+X^3+1 sage: print MyCurve.list() [x0^2*x1 + x0*x1^2 + x0*x2^2 + x1*x2^2 + x0^2*x3 + x2*x3^2 + x3^3 + x2*x4 + x3*x4 + x1*x5 + x3*x5 + x5^2 + x0*x6 + x0*x7 + x1*x7 + 1, x0^2*x1 + x1^3 + x0^2*x2 + x0*x2^2 + x2^2*x3 + x3^3 + x3*x4 + x1*x5 + x2*x5 + x5^2 + x1*x6 + x0*x7 + x3*x7 + x7^2, x0^2*x1 + x0*x2^2 + x2^3 + x1^2*x3 + x0*x3^2 + x3^3 + x0*x4 + x3*x4 + x4^2 + x1*x5 + x5^2 + x3*x6 + x0*x7 + x2*x7, x0^3 + x0^2*x1 + x1^2*x2 + x0*x2^2 + x1*x3^2 + x3^3 + x1*x4 + x3*x4 + x0*x5 + x1*x5 + x5^2 + x2*x6 + x6^2 + x0*x7] And after that as you mentioned, I can multiply the list by the basis change matrix. Thank you again (More question is on the way ... :">) Bests, Ahmad On Dec 1, 3:27 am, Ahmad <[EMAIL PROTECTED]> wrote: > Hi David, > > Thank you very much for your solution. I think it is enough for me as > finally I want to see the result in normal basis. However, I changed > the code a little to suite my purpose but it failed. Could you please > tell me what should I use instead of "list" property in multivariate > case (maybe we need some ordering like grobner basis): > > sage: k = GF(7) # any coefficient ring here is okay > sage: R = MPolynomialRing(k,2,x) # create polynomial ring over k > in variable x > sage: x = R.gens() > sage: g = x[0]^3 + 2*x[1] + 5 # create some polynomial > sage: print g.list() # list of coefficients of polynomial > sage: [2*u for u in g.list()] # multiplies every elements of > g.list() by 2 (mod 7), returns result as a list > Traceback (most recent call last): > ... > AttributeError: 'sage.rings.polynomial.multi_polynomial_libsingular' > object has no attribute 'list' > > I hope that you don't get the feeling that I just naging and > challenging you for nothing. It is almost exactly the computation I > need for my thesis as I'm working on GHS attack. I should look at a > curve when its parameters and variables is represented in base field > (so one equation become many equations) and then I should pass some > hyper-plane through it in its base field. > > Thanx again. > Ahmad > > On Nov 30, 8:20 am, David Harvey <[EMAIL PROTECTED]> wrote: > > > On Nov 30, 2007, at 2:45 AM, Ahmad wrote: > > > > Dear Sage Supporters, > > > > As nobody continued to pay attention to the question I asked in sept 3 > > > about how I want to change the field basis "permanently", I am using > > > john Cremona's idea to ask my question in another way, in hope to > > > attract more attention: > > > > Suppose k is a field. Let define ring k[x]. I extend this ring by > > > adding variable 't' and taking quotient by polynomial 't^4 + t^3 + t^2 > > > + t + 1'. So, I have the ring k[x][t]/(t^4 + t^3 + t^2 + t + 1) which > > > is a free module over k[x]. But again sage use default basis (1, t, > > > t^2, t^3) to represent this ring over k[x]: > > > > sage: k = GF(2); > > > sage: R = k['x']; x = R.gen() > > > sage: S = R['t']; t = S.gen() > > > sage: SBar = S.quotient(t^4 + t^3 + t^2 + t + 1, 'a'); a = SBar.gen() > > > sage: print x*a^4 > > > x*a^3 + x*a^2 + x*a + x > > > > How can I change this basis to normal basis, so I get: > > > > sage: print x*a^4 > > > x*a^4 > > > Hi Ahmad, > > > I looked over the september thread, and the problem is that William's > > solution won't work in this more general case, since you can't create > > a polynomial ring with coefficients in a vector space, it just > > doesn't make sense. > > > But if all you need is a list of the coordinates, then maybe we can > > make this work. All you need to be able to do is apply a function to > > each coefficient of a polynomial. Here's how you do that: > > > sage: k = GF(7) # any coefficient ring here is okay > > sage: R.<x> = PolynomialRing(k) # create polynomial ring over k > > in variable x > > sage: g = x^3 + 2*x + 5 # create some polynomial > > sage: g.list() # list of coefficients of polynomial > > [5, 2, 0, 1] > > sage: [2*u for u in g.list()] # multiplies every elements of g.list > > () by 2 (mod 7), returns result as a list > > [3, 4, 0, 2] > > > So you just need to replace "2*u" with whatever function William gave > > you to change basis representation. > > > Of course, what you *really* want is to be able to create a field > > that prints elements of itself with respect to a different basis, but > > I don't think this is implemented in sage (yet). That sounds like it > > could be a useful thing to have. > > > david --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-support URLs: http://sage.math.washington.edu/sage/ and http://sage.scipy.org/sage/ -~----------~----~----~----~------~----~------~--~---
