Clemens Heuberger's profile - activity

SageMath code is written in Python (with a few extensions by the preparser). Your code does not resemble Python. Indentation matters, no end is required, forloops are written as for b in s:, functions are defined as def bruteforce():, write print instead of puts.

How to I convert an element of the fraction field of QQ[X] to the fraction field of ZZ[X]?

sage: R.<x> = ZZ[]
sage: F = R.fraction_field()
sage: e = (1/2)/(x+1)
sage: e.parent()
Fraction Field of Univariate Polynomial Ring in x over Rational Field
sage: F(e)
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer

In my use case, e is the result of a lengthy computation where at some point beyond my control the result was coerced into QQ[X]. In my particular case, I prefer the fraction field of ZZ[X] because the output is nicer.

Thank you, the work-around works for me. Is the underlying problem a known bug or shall I create a ticket?

Let

R.<a, b, X> = ZZ[]
f = 1 - a*X^2
g = 1 - b*X^3

I need to compute polynomials u and v such that u f + v g = r where r = f.resultant(g, X).

Pari has a function polresultantext for that purpose, so one solution for my problem is

(u, v, r) = (R(p) for p in f._pari_().polresultantext(g._pari_(), 'X'))

Nevertheless, I have a few questions:

• Is there a method in Sage which directly does this?
• Is the detour via Pari the recommended solution? Or is there an alternative, e.g., in singular?
• If there is currently no method in Sage for doing this directly, what name would you recommend? resultant_ext ?

Using SageMath 7.4, I do the following:

sage: var('u R omega')
sage: hs = (1+u^R).series(u==omega, 2); hs
(omega^R + 1) + (R*omega^R/omega)*(-omega + u) + Order((omega - u)^2)
sage: hs.subs({omega^R: -1}).simplify()
R*(omega - u)/omega + Order((omega - u)^2) + _SAGE_VAR_(0)

I am wondering what _SAGE_VAR_(0) shall be and how to get rid of it.

I have a matrix in $\mathbb{Q}[Y]/(Y-1)$ and want to lift it to $\mathbb{Q}[Y]$, however, I get an error:

sage: D.<Y> = QQ[]
sage: B = matrix(D, [[Y, 0]])
sage: Dbar = D.quotient(Y-1)
sage: Bbar = B.change_ring(Dbar)
sage: Bbar.lift()
Traceback (most recent call last):
...
TypeError: unable to convert 1 to a rational

Lifting single elements instead of a matrix works:

sage: Dbar(Y^2).lift()
1

Lifting a matrix from the integers modulo a prime works also:

sage: B = matrix(ZZ, [[7, 0]])
sage: Dbar = ZZ.quotient(5)
sage: Bbar = B.change_ring(Dbar)
sage: Bbar.lift()
[2 0]

So how do I lift the matrix? Building a new matrix by hand und lifting componentwise seems to be an option; however, I think that it is somewhat ugly.

I now have performance problems with my above solution:

R.<x> = AA[]
v = QQbar.polynomial_root(AA.common_polynomial(x^5 - 2), CIF(RIF(RR(0.35496731310463009), RR(0.35496731310463014)), RIF(RR(1.0924770557774537), RR(1.0924770557774539))))
rho = 1/60*v^4 + 1/30*v^3 + 1/60*v^2 + 1/30*v + 1/15
mp = rho.minpoly()
%time NumberField(mp, 'rho', embedding=rho)
CPU times: user 1min 5s, sys: 64 ms, total: 1min 5s
Wall time: 1min 5s
Number Field in rho with defining polynomial x^5 - 1/3*x^4 + 1/30*x^3 - 1/600*x^2 + 1/24000*x - 1/2400000

Any ideas?

It worked once, but then my code came accross another example and UniqueRepresentation gave me problems. So I ended up avoiding as_number_field_element alltogether and replacing it by

alpha_qqbar = QQbar(sqrt(2))
nf = NumberField(alpha_qqbar.minpoly(), 'alpha', embedding=alpha_qqbar)
hom = nf.coerce_embedding()
x1 = polygen(nf, 'x')
x2 = polygen(QQbar, 'x')
1/x1 - 1/x2

Thank you, the shortcut was exactly what I need.

I have a homomorphism from a number field nf to the field of algebraic numbers QQbar:

nf, alpha, hom = QQbar(sqrt(2)).as_number_field_element()

I now work in the polynomial ring R over nf:

R.<x> = nf[]

f = x - alpha; f

How do I get a homomorphism from R to the polynomial ring over QQbar extending hom? For the moment, I can use

f.map_coefficients(hom)

Same question about the fraction field of R, e.g.,

g = f/(x+1)

Is there a more elegant way than calling

g.numerator().map_coefficients(hom)/g.denominator().map_coefficients(hom)

So basically, I'd like to extend my homomorphism hom to the polynomial ring and its field of fractions.