ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Fri, 05 Jun 2020 09:41:45 +0200Factorization/Irreducibility of multivariate polynomials over function fieldshttps://ask.sagemath.org/question/51769/factorizationirreducibility-of-multivariate-polynomials-over-function-fields/I want to do this:
L.<t> = FunctionField(GF(5))
R.<x,y> = PolynomialRing(L)
p = x^2-t^2*y^2
p.factor()
As a result, I would like to get something like `(x+t*y)(x-t*y)` or `(x+t*y)(x+4*t*y)`.
But what I get is:
sage: L.<t> = FunctionField(GF(5))
/software/sagemath/9.0/SageMath/local/lib/python3.7/site-packages/sage/rings/function_field/ideal.py:2217: DeprecationWarning: invalid escape sequence \g
"""
sage: R.<x,y> = PolynomialRing(L)
sage: p = x^2-t^2*y^2
sage: p.factor()
---------------------------------------------------------------------------
NotImplementedError Traceback (most recent call last)
<ipython-input-5-f448610e9d77> in <module>()
----> 1 p.factor()
/software/sagemath/9.0/SageMath/local/lib/python3.7/site-packages/sage/rings/polynomial/multi_polynomial_element.py in factor(self, proof)
1842 proof = get_flag(subsystem="polynomial")
1843 if proof:
-> 1844 raise NotImplementedError("Provably correct factorization not implemented. Disable this error by wrapping your code in a `with proof.WithProof('polynomial', False):` block.")
1845
1846 R._singular_().set_ring()
NotImplementedError: Provably correct factorization not implemented. Disable this error by wrapping your code in a `with proof.WithProof('polynomial', False):` block.
However, this works, but (as already indicated in the parentheses) I get a wrong result:
sage: p.factor(false)
[(x + t*y, 1), (-x + t*y, 1)]
Unlike the univariate case, there is also no option to check whether the polynomial is irreducible, i.e. `p.is_irreducible()` doesn't exist.
ThrashFri, 05 Jun 2020 09:41:45 +0200https://ask.sagemath.org/question/51769/FunctionField with more than 1 variablehttps://ask.sagemath.org/question/36170/functionfield-with-more-than-1-variable/I have tried the following to construct an appropriate function field:
F.<a,b>=FunctionField(QQ,2)
as a result I got
TypeError: create_key() got multiple values for keyword argument 'names'
after that I looked for the documentation but only find one variable examples. I also tried a few more versions like
F.<a,b>=FunctionField(QQ,names=('a','b'))
but did not work. Thank for any advice in this direction in advance.castorWed, 04 Jan 2017 15:41:55 +0100https://ask.sagemath.org/question/36170/the same function appears twice in the documentationhttps://ask.sagemath.org/question/11056/the-same-function-appears-twice-in-the-documentation/The functions base_field()/constant_base_field()/constant_field()/vector_space()/maximal_order()/random_element appear twice in the following document:
http://sagemath.org/doc/reference/function_fields/sage/rings/function_field/function_field.html
and what different with RationalFunctionField() and FunctionField()?cjshThu, 20 Feb 2014 00:16:02 +0100https://ask.sagemath.org/question/11056/Compute Galois closure of an extension of a function fieldhttps://ask.sagemath.org/question/7875/compute-galois-closure-of-an-extension-of-a-function-field/Say I want to look at the field extension $Quot(\mathbb{Q}[x,y]/y^7-x)$ over $\mathbb{Q}(x)$ and then compute its Galois closure. How do I do that?
Ideally it could be done on the scheme-level (to define the scheme-morphism: (the projectivization of the affine plane curve $y^7-x$) mapping to (the projective $x$-line); and then compute its Galois closure -- a scheme!). But I don't know how to implement either version.OliverSat, 15 Jan 2011 21:11:48 +0100https://ask.sagemath.org/question/7875/is exist a field is FunctionField() but not RationalFunctionField()?https://ask.sagemath.org/question/10855/is-exist-a-field-is-functionfield-but-not-rationalfunctionfield/is exist a field is FunctionField() but RationalFunctionField()?cjshMon, 23 Dec 2013 04:55:36 +0100https://ask.sagemath.org/question/10855/is_RationalFunctionField cannot run?https://ask.sagemath.org/question/10849/is_rationalfunctionfield-cannot-run/from sage.rings.function_field.function_field import is_RationalFunctionFieldcjshMon, 23 Dec 2013 07:15:13 +0100https://ask.sagemath.org/question/10849/hot to get a functionfield'galois_group?https://ask.sagemath.org/question/10821/hot-to-get-a-functionfieldgalois_group/is there a table about functionfield'galois_group?
only numberfield'galois_group at lmfdb.org
K.<x> = FunctionField(QQ); R.<y> = K[];L.<y> = K.extension(y^5 - x);L;LL.<z> = L[];LL.<z> = K.extension(z - x-y);LL
Function field in y defined by y^5 - x
Function field in z defined by z - y - x
K.maximal_order(); K.galois_group()
Maximal order in Rational function field in x over Rational Field
Traceback (most recent call last):cjshWed, 11 Dec 2013 04:25:02 +0100https://ask.sagemath.org/question/10821/Elliptic curves over global function fields.https://ask.sagemath.org/question/10644/elliptic-curves-over-global-function-fields/Hi all,
I would like to know how much of elliptic curves over global function fields have been implemented in SAGE. Specifically, I'm interested in torsion subgroups. Any hint on where I can find documentations, if there are any, on this specific subject would be very appreciated.AndryTue, 22 Oct 2013 07:09:28 +0200https://ask.sagemath.org/question/10644/conversions from/to FunctionField(SR) and symbolic expressionhttps://ask.sagemath.org/question/10633/conversions-fromto-functionfieldsr-and-symbolic-expression/Hello,
read the following session OR if you won't please go directly to the question below
$ sage
----------------------------------------------------------------------
| Sage Version 5.6, Release Date: 2013-01-21 |
----------------------------------------------------------------------
sage: a,b,s = var('a b s')
sage: expr1 = (a^2*s + 2)/(s^3 + s + 3) + s
sage: expr1.denominator()
s^3 + s + 3
sage: type(s)
<type 'sage.symbolic.expression.Expression'>
sage: FF.<s> = FunctionField(SR)
sage: FF(expr1)
s + (a^2*s + 2)/(s^3 + s + 3)
sage: FF(expr1).denominator()
1
# s in expr1 is NOT recognized as the s in the definition of the
# function field.
sage: type(s)
<type 'sage.rings.function_field.function_field_element.FunctionFieldElement_rational'>
# BUT:
sage: x = var('x')
sage: expr2 = x + (45^2 + 2)/(x^3 + x + 3)
sage: FF2.<x> = FunctionField(RR)
# now RR instead of SR and x as the variable.
sage: FF2(expr2)
(x^4 + x^2 + 3.00000000000000*x + 2027.00000000000)/(x^3 + x + 3.00000000000000)
sage: FF2(expr2).denominator()
x^3 + x + 3.00000000000000
# x is correctly recognized in expr2 but not in expr1 !
sage: type(x)
<type 'sage.rings.function_field.function_field_element.FunctionFieldElement_rational'>
QUESTION: a FunctionField over RR with the variable x correctly recognizes
expressions where x=var('x') appears in the expression (see above), and the computation
of denominator is correct; FunctionField over SR with the variable s do not
recognizes expressions with s=var('s'); instead in this case the s is treated
like a coefficient (denominator=1 in example above in the first part).
How can i adjust this behavior, so that I obtain the same answer in both following cases:
sage: expr1
s + (a^2*s + 2)/(s^3 + s + 3)
# here s = var('s') is symbolic expression
sage: FF(expr1).denominator()
1
# I DO NOT WANT THIS ANSWER
sage: FF(s + (a^2*s+2)/(s^3 + s + 3)).denominator()
s^3 + s + 3
# but this answer (that works if the expression is constructed by hand) with
sage: type(s)
<type 'sage.rings.function_field.function_field_element.FunctionFieldElement_rational'>
Any suggestion ?
THANK YOU VERY MUCH !
alessandroSun, 20 Oct 2013 07:16:23 +0200https://ask.sagemath.org/question/10633/Extension degree over function fieldhttps://ask.sagemath.org/question/9706/extension-degree-over-function-field/Hello!
I would like to compute the extension degree over a function field.
So I use the commend "degree()".
But I found it didn't work well.
It produces always just the degree of the polynomial.
For example,
sage: K.<x> = FunctionField(QQ)
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 - (x^2)); L
Function field in y defined by y^2 - x^2
sage: L.degree()
2
In fact, the extension degree [L:K] = 1.
What's wrong?
I'd appreciate it if you could let me know how to compute the extension
degree over a function field?
Thank you!
JeonSun, 13 Jan 2013 02:26:34 +0100https://ask.sagemath.org/question/9706/Elliptic curves over function fieldshttps://ask.sagemath.org/question/8937/elliptic-curves-over-function-fields/Let $E$ be an elliptic curve over a function field $K=\mathbb{F}_q(t)$.
How do we compute the height pairing matrix for a set of points $P_1,\ldots,P_n\in E(K)$? or the height of a single point?RPCSun, 29 Apr 2012 23:26:22 +0200https://ask.sagemath.org/question/8937/Working with function field extensionshttps://ask.sagemath.org/question/8753/working-with-function-field-extensions/Let $K=k(t)$, where $k$ is a finite field. Consider a rational function $F(t)\in K$ and a simple finite extension $L=K(u)$. For instance, take $L=k(u,t)$, where $u^5=t$.
My first question is: How do we evaluate $F(u)$?
The following code I am using produces an error (...
NotImplementedError)
k.<a>=GF(19^2)
Rk=k['t']
K.<t>=Frac(Rk)
Rx.<x>=PolynomialRing(K)
L.<u>=K.extension(Rx(x^5-t),'u')
F=(t^25+t^5)/(t^5+1)
F(u)
F.subs(t=u)
Notice that in my example $F(u)=(t^5+t)/(t+1)$ is a function of $t$, since $u^5=t$.
My second question is: How would I coerce $F(u)=g(t)$ back into $k(t)$?RPCMon, 27 Feb 2012 23:53:35 +0100https://ask.sagemath.org/question/8753/