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symbolic polynomial euclidean algorithm

asked 13 years ago

manenir gravatar image

updated 13 years ago

Kelvin Li gravatar image

Hi there!

Suppose we have polynomials f(x),g(x) with coefficients in Q(a,b) for example f(x)=ax2,g(x)=x2b. How can we find polynomials f1(x),g2(x)Q(a,b)[x] such that f(x)f1(x)+g(x)g1(x)=g.c.d.(f(x),g(x))?

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answered 13 years ago

mmarco gravatar image

The answer x2 is correct. The number 2 is a unit in K.

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D'oh. It's been too long since my last encounter with algebra, I suppose.

cswiercz gravatar imagecswiercz ( 13 years ago )
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answered 13 years ago

Here's a simple example over Q with f(x)=x3+1 and g(x)=x21. First, define a polynomial ring over Q:

sage: R.<x> = PolynomialRing(QQ,'x')
sage: f = x^3+1; g = x^2-1
sage: f.gcd(g)
x+1

One can also do this with extension fields. (By the way, there's probably a faster way to do the following.) First, let's construct Q(2,3) using the ring R constructed above

sage: p = x^2-2        # using the x defined above
sage: q = x^2-3
sage: K.<a,b> = QQ.extension([p,q])
sage: print a^2, b^2   # testing variable assignment
2  3

Now create a polynomial ring over K, define some elements, and compute their GCD:

sage: S.<x> = PolynomialRing(K,'x')
sage: f = a*x^2 * (x-1); g = a*x^2 * (x-b)
sage: f.gcd(g)
x^2

Hrm, it seems like there's either a bit of a bug, here, or I constructed something incorrectly. (The result should be 2x2.) I'll submit this as a bug report. However, it seems like you'll at least get a common factor "modulo" the elements used to create the extension field.

sage: f/x^2
a*x - a
sage: g/x^2
a*x - b*a

So perhaps by inspection you can compute the GCD. Hopefully this will be resolved soon.

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Asked: 13 years ago

Seen: 1,932 times

Last updated: Apr 01 '11