Hello,
On Mar 20, 4:18 am, luisfe <lftab...@yahoo.es> wrote:
> Hi all,
> Mathematically, I have the following field:
> Q(x,y,z,t,a)
>
> Where x,y,z,t are indeterminates and "a" is an algebraic number over
> the
> rationals (lets say degree 4).
>
> If I have some elements, let say f,g,h in this field I would like to:
The best way to work with this object is to do like you did:
sage: K.<a>=NumberField(x^4+x+1)
sage: R.<x,y,z,t>=K['x,y,z,t']
Then, we can construct some elements of this field:
sage: f = (a^2*x + y)*(z+a*t); f
(a^2)*x*z + y*z + (a^3)*x*t + (a)*y*t
sage: g = f/(x+a)
sage: g
((a^2)*x*z + y*z + (a^3)*x*t + (a)*y*t)/(x + (a))
> (1) Obtain a representation "in base a", that is, write f as
> f0 + f1*a + f2*a^2 + f3*a^3
I can't think of a way to do this other than what the following
function does:
def powers_of_a(f):
powers = {}
for m, fc in zip(f.monomials(), f.coefficients()):
for i, c in enumerate(fc):
powers[i] = powers.get(i, 0) + c*m
for i, p in powers.items():
print i, ":", p
sage: powers_of_a(f)
0 : y*z
1 : y*t
2 : x*z
3 : x*t
You can return the dictionary and do more programmatic things if you'd
like.
> (3) Extract a valid numerator and denominator of f, that is, compute
> polynomials r,s in Q[x,y,z,t,a] such that f=r/s
sage: g
((a^2)*x*z + y*z + (a^3)*x*t + (a)*y*t)/(x + (a))
sage: g.numerator()
(a^2)*x*z + y*z + (a^3)*x*t + (a)*y*t
sage: g.denominator()
x + (a)
> (4) Construct ideals and compute a Grobner basis of a series of
> polynomials
> extracted as numerators of rational fuctions
sage: I = R.ideal(x+a + a*y*z, a^2*t+z); I
Ideal ((a)*y*z + x + (a), z + (a^2)*t) of Multivariate Polynomial Ring
in x, y, z, t over Number Field in a with defining polynomial x^4 + x
+ 1
sage: I.groebner_basis()
[y*t + (a^3 - a^2 + a + 1)*x + (-a^3 + a^2 - 1), z + (a^2)*t]
> (5) Factorize polynomials in Q[x,y,z,t,a] extracted from
> numerators/denominatos of rational functions.
We can do this via Maxima. First we convert f to Maxima and call the
factor command passing in the defining polynomial for the number
field. Then we extract each of the products and convert them back to
polynomials in Sage:
sage: ff = maxima(f).factor(K.defining_polynomial
().change_variable_name('a'))
sage: ff
(y+a^2*x)*(z+a*t)
sage: map(R, map(repr, ff.args()))
[(a^2)*x + y, z + (a)*t]
Hope that helps,
--Mike
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