I had some library problems in creating a Sage sandbox but resolved
them with the help of ask.sagemath.org. So now i'm back into
submitting to Sage a function that checks whether a list of
polynomials is algebraically independent. Since my function uses an
elimination ideal, i took Simon's advice and added it as a method of
the class
sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular.
Then i ran into trouble, because that class is written in Cython, i
only know Python, and my code is written in Sage. Man, this is
getting difficult, and yet my function is so simple!
So do i need to convert my code to Cython? For concreteness, here's
the code, which is short. The examples are kind of bogus, because,
while the answers are correct ---i got them by running a non-object-
oriented version of the method--- the syntax of the calls currently
doesn't work.
Alex
====================================
def algebraic_dependence(self,fs):
r"""
Returns an irreducible annihilating polynomial for the polynomials
in the
list `fs`, if those polynomials are algebraically dependent.
Otherwise returns the zero polynomial.
INPUT:
- ``fs`` - A list of polynomials `f_1,\ldots,f_r' from self.
OUTPUT:
If the polynomials in the list `fs` are algebraically dependent,
then the
output is an irreducible polynomial `g` in `K[T_1,\ldots,T_r]`
such that
`g(f_1,\ldots,f_r) = 0`.
Here `K` is the coefficient ring of self and `T_1,\ldots,T_r` are
indeterminates different from those of self.
If the polynomials in `fs` are algebraically independent, then the
output
is the zero polynomial.
EXAMPLES::
sage: R.<x>= PolynomialRing(QQ)
sage: fs= [x-3, x^2+1]
sage: g= R.algebraic_dependence(fs); g
10 + 6*T0 - T1 + T0^2
sage: g(fs)
0
::
sage: R.<w,z>= PolynomialRing(QQ)
sage: fs= [w, (w^2 + z^2 - 1)^2, w*z - 2]
sage: g= R.algebraic_dependence(fs); g
16 + 32*T2 - 8*T0^2 + 24*T2^2 - 8*T0^2*T2 + 8*T2^3 + 9*T0^4 -
2*T0^2*T2^2 + T2^4 - T0^4*T1 + 8*T0^4*T2 - 2*T0^6 + 2*T0^4*T2^2 + T0^8
sage: g(fs)
0
sage: g.parent()
Multivariate Polynomial Ring in T0, T1, T2 over Rational Field
::
sage: R.<x,y,z>= PolynomialRing(GF(7))
sage: fs= [x*y,y*z]
sage: g= R.algebraic_dependence(fs); g
0
AUTHORS:
-Alexander Raichev (2011-01-06)
"""
r= len(fs)
R= self
B= R.base_ring()
Xs= list(R.gens())
d= len(Xs)
# Expand R by r new variables.
T= 'T'
while T in [str(x) for x in Xs]:
T= T+'T'
Ts= [T + str(j) for j in range(r)]
RR= PolynomialRing(B,d+r,tuple(Xs+Ts))
Vs= list(RR.gens())
Xs= Vs[0:d]
Ts= Vs[d:]
# Find an irreducible annihilating polynomial g for the fs if
there is one.
J= ideal([ Ts[j] -RR(fs[j]) for j in range(r)])
JJ= J.elimination_ideal(Xs)
g= JJ.gens()[0]
# Shrink the ambient ring of g to include only Ts.
# Using the negdeglex because i find it convenient in my work.
RRR= PolynomialRing(B,r,tuple(Ts),order='negdeglex')
return RRR(g)
--
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