Polynomial identity

Question

I'm pretty interested in solving the following kind of problem using Sage: ``Let R be a polynomial ring in, say, x,y,z as variables over a field k. I'd like to find field-elements a,b,c such that

a(x^2+y^2)+b(xy+zx)+c(xyz)==0, if they exist (I know they do)''

so that Sage returns (a,b,c)=(0,0,0). That seems to be an easy matter if one can traduce the polynomial identity into a vector space identity. I proved to be unable to do so.

I've to say that my polynomial identities are quite more cumbersome and include up to 7 variables so working them by-hand is almost impossible in a finite ammount of time or patience.

Answer 1

Here is some code which "linearizes" your problem:

def polynomial_identity(polys):
    monomials = set()
    for p in polys:
        monomials.update(p.monomials())
        base_ring = p.base_ring()
    monomial_order = list(monomials)
    m = []
    for p in polys:
        row = []
        for monomial in monomial_order:
            try:
                row.append(base_ring(p.coefficient(monomial)))
            except TypeError:
                row.append(base_ring(0))
        m.append(row)
    m = matrix(m)
    return m.kernel()

The result is a vector space which gives you all of your solutions. Your example

sage: polynomial_identity([x^2 + y^2, x*y + x*z, x*y*z])
Vector space of degree 3 and dimension 0 over Rational Field
Basis matrix:
[]

shows that (0,0,0) is the only answer. Another example:

sage: polynomial_identity([x+y, 2*x+2*y, 3*(x+y)])        
Vector space of degree 3 and dimension 2 over Rational Field
Basis matrix:
[   1    0 -1/3]
[   0    1 -2/3]

First of all I thank you for your interest. I knew nothing about Sage till today's evening. I tried to type your code directly into the console, but something happens at line 8. Is any way I can include this function in the core of the program so that I don't need to define it everytime from console (just in case I am able to do so...). I mainly understand your code even I know nothing about this language nor its functions. Any comment that lead to understanding would be really appreciated.
Angel (May 27 '12)
You can put this in ~/.sage/init.sage and it will be loaded each time you start Sage. You can also run "%cpaste" from the command line, paste the code in, and then put in "--" to finish the paste.
Mike Hansen (May 27 '12)
Cool! I actually managed myself to write a slightly different version of your code. Great. I really thank you for your help ^^'.
Angel (May 28 '12)

Polynomial identity

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