I'm not sure if this helps, but you can create a polynomial
of the type you want a bit simpler:
sage: var("x,y")
(x, y)
sage: Inds = CartesianProduct(range(5), range(4))
sage: sum([var("a"+str(i)+str(j))*x^i*y^j for i,j in Inds])
a43*x^4*y^3 + a33*x^3*y^3 + a42*x^4*y^2 + a23*x^2*y^3 + a32*x^3*y^2 +
a41*x^4*y + a13*x*y^3 + a22*x^2*y^2 + a31*x^3*y + a40*x^4 + a03*y^3 +
a12*x*y^2 + a21*x^2*y + a30*x^3 + a02*y^2 + a11*x*y + a20*x^2 + a01*y
+ a10*x + a00
Now you can reference them on the fly like this:
sage: eval("a"+str(3)+str(2))
a32
On Wed, Jun 3, 2009 at 10:51 AM, James Parson <par...@hood.edu> wrote:
>
> Dear sage-support group,
>
> I am completely new to computer algebra systems and to computer
> programming, and I hope you'll indulge the following beginner's
> question. I was wondering if there is a simple way to create a
> polynomial of degree d in x and y with symbolic coefficients in Sage.
> Here is what I mean: if I were at the board in class, I might write
> (in LaTeX transcription) something like
>
> P(x,y) = \sum_{i+j\leq d} a_{ij} x^i y^j,
>
> which I would view as an element of \mathbf{Z}[a_{ij},x,y]. I might
> then impose some linear conditions on the a_{ij} by insisting that P
> (x_t,y_t) = 0 for a list of points
>
> (x_1,y_1), (x_2,y_2), ... .
>
> Finally, I might solve the resulting system of linear equations.
>
> How would you recommend that I set up something like the a_{ij} and P
> (x,y) in Sage? In order to make the question more definite, I
> illustrate it with an example that I took from a lecture of Doron
> Zeilberger on experimental mathematics. He proposed the question of
> finding a polynomial of degree d in x and y that vanishes when x and y
> are specialized to consecutive Fibonacci numbers. The lines below are
> my attempt at a Sage version of his suggested computer search for a
> likely solution (originally written in Maple). The program should take
> the degree d as an input and then provide a parameterized family of
> polynomials of degree d that are likely candidates.
>
> Here is what I came up with, after an enlightening afternoon of
> studying computer manuals:
>
> d = 4
> e = d+1
> L = []
> M = []
> for i in range(e):
> for j in range(e-i):
> L.append('a_%s_%s' %(i,j))
> M.append([i,j])
> V = var(' '.join(L))
> P = sum(V[j]*x^(M[j][0])*y^(M[j][1]) for j in range(len(L)))
> E = [P(x=fibonacci(n),y=fibonacci(n+1)) for n in range(1,len(V)+6)]
> P.substitute(solve(E,V,solution_dict = True)[0])
>
> I could not figure out how to create and reference the variables a_
> {ij} conveniently, and so I ended up with the strange lists V and M
> above. Even though I got my polynomial P and solved the original
> problem to my satisfaction, I still don't think I know how I would
> have Sage do something like sum the a_{i.i+1} for 2i+1<=d. Is there a
> better way to do this sort of thing?
>
>
> Thanks for your help and indulgence,
>
> James Parson
>
> >
>
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