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2011-03-25 20:03:25 +0100 asked a question symbolic polynomial euclidean algorithm

Hi there!

Suppose we have polynomials $f(x),g(x)$ with coefficients in $\mathbb{Q}(a,b)$ for example $f(x)=ax^2, g(x)=x^2-b.$ How can we find polynomials $f_1(x),g_2(x) \in \mathbb{Q}(a,b)[x]$ such that $f(x)f_1(x)+g(x)g_1(x)=g.c.d.(f(x),g(x))$?

Thanks

2011-03-18 01:40:51 +0100 answered a question symbolic calculus on doubling points in elliptic curves

Dear John, your help is greatly appreciated :) One further question : What i want to do is exactly what you have descibed at the end, but with an expression of the form $\frac{f(a)}{g(a)}$ where $f,g$ are polynomials in $a$ (and with coefficients in $Z[m])$ instead of just a polynomial $f(a)$. So when I define say $$f=\frac{(a^2-m^2)^2}{a^3-m}$$ what Sage notebook gives is

1/256a^13/(a^2 - 4m)^2 - 1/8a^11m/(a^2 - 4m)^2 + 7/4a^9m^2/(a^2 - 4m)^2 - 14a^7m^3/(a^2 - 4m)^2 + 70a^5m^4/(a^2 - 4m)^2 - 1/2a^7m/(a^2 - 4m)^2 - 224a^3m^5/(a^2 - 4m)^2 + 8a^5m^2/(a^2 - 4m)^2 + 448am^6/(a^2 - 4m)^2 - 48a^3m^3/(a^2 - 4m)^2 - 512m^7/((a^2 - 4m)^2a) + 128am^4/(a^2 - 4m)^2 + 256m^8/((a^2 - 4m)^2a^3) - 128m^5/((a^2 - 4m)^2a) + 16am^2/(a^2 - 4m)^2

That seems rather incomprehensible as it computes the final expression in disctinct fractions. Is there any way to format this in a nice latexed expression of the form ''polynomial over (other)polynomial'' where both polynomials will be factored ?

thanx !

2011-03-18 01:23:08 +0100 commented answer symbolic calculus on doubling points in elliptic curves

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2011-03-18 01:22:23 +0100 marked best answer symbolic calculus on doubling points in elliptic curves

How are you defining f? This works for me:

sage: R = GF(2)['y']
sage: R.inject_variables()
Defining y
sage: f = y^2 + y + 1
sage: f(f(y))
y^4 + y + 1

Edit: if you want to do something like f(y) = y^2 + ny, then you need two variables, and you could make one a polynomial variable, one a symbolic variable. Make sure that the symbolic one comes after the polynomial one, alphabetically:

sage: var('m')
sage: R = QQ['a']
sage: R.inject_variables()
sage: f = a^2 + m*a
sage: f(f(a))
(a^2 + a*m)^2 + (a^2 + a*m)*m
sage: f(f(a)).expand()
a^4 + 2*a^3*m + a^2*m^2 + a^2*m + a*m^2

Then

sage: f(f(f(f(f(a))))).expand()

works, but gives a very long expression.

(You need the polynomial generator to come alphabetically before the symbolic variable because f depends on two variables, and when you call f(3), for example, it chooses to substitute the 3 for the first of the variables. You want to substitute f(a) for a by default, so make sure a comes before m.)

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2011-03-17 22:06:08 +0100 asked a question symbolic calculus on doubling points in elliptic curves

Suppose I have a polynomial $f(x)$ and i need to compute $f(f(x))$

how do i do this ??

i also need to compute $f(f(f(x)))$ and so on, is there an easy way of doing this ?

Example: If $f(x)=x^2+nx$ then $f(f(x))=...=x^4 + 2n x^3+(n^2+n) x^2+ n x$ and the computations become difficult when trying to compute say $f(f(f(f(f(x)))))$