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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