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2024-01-20 09:06:08 +0100 commented question Evaluate polynomial over extension ring

thanks a lot!

2024-01-20 09:05:42 +0100 marked best answer Evaluate polynomial over extension ring

Given a ring and polynomial like:

_.<I> = RR[]
K.<i> = RR.extension(I^2 + 1)
L.<a, b> = K[]
f = a + i*b

then I would expect that f(i=0) would give me a. But instead it just returns the same value again.

Any ideas how to do this? Thanks

2024-01-20 09:05:38 +0100 commented answer Evaluate polynomial over extension ring

thanks a lot! Gives me a load of stuff to look into now. Good day

2024-01-19 09:53:01 +0100 edited question Evaluate polynomial over extension ring

Evaluate polynomial over extension ring Given a ring and polynomial like: _.<I> = RR[] K.<i> = RR.extension

2024-01-19 09:52:40 +0100 asked a question Evaluate polynomial over extension ring

Evaluate polynomial over extension ring Given a ring and polynomial like: _.<I> = RR[] K.<i> = RR.extension

2023-12-03 10:05:51 +0100 answered a question Sage convert SymbolicRing equation to symbolic expression

Here's my solution in the end: Convert equation to symbolic form. Make it homogenous. Then use the expr.coefficient(),

2023-12-03 09:37:05 +0100 edited question Sage convert SymbolicRing equation to symbolic expression

Sage convert SymbolicRing equation to symbolic expression Often I want to group terms in different ways. Say for example

2023-12-03 08:58:57 +0100 edited question Sage convert SymbolicRing equation to symbolic expression

Sage convert SymbolicRing equation to symbolic expression Often I want to group terms in different ways. Say for example

2023-12-03 08:58:23 +0100 asked a question Sage convert SymbolicRing equation to symbolic expression

Sage convert SymbolicRing equation to symbolic expression Often I want to group terms in different ways. Say for example

2023-02-11 22:57:39 +0100 marked best answer Manually grouping symbolic terms

Given a symbolic expression like:

sage: var("a b c x y")
(a, b, c, x, y)
sage: a*x^2 + a*y^2 + b*y^2 + c*y^2 + (2*a*y + b*y)*x

How can I manually collect terms together? I want to represent this equation in the form:

$$ Ax^2 + Bxy + Cy^2 $$

Where $A = a, B = 2a + b, C = a + b + c$. Desired output should be:

a*x^2 + (2*a + b)*x*y + (a + b + c)*y^2

And then is there any way to read off these coefficients? Thanks

2023-02-01 09:03:56 +0100 asked a question Manually grouping symbolic terms

Manually grouping symbolic terms Given a symbolic expression like: sage: var("a b c x y") (a, b, c, x, y) sage: a*x^2 +

2023-01-15 17:15:53 +0100 asked a question Assumptions giving wrong boolean answer for simple expression

Assumptions giving wrong boolean answer for simple expression Let $a, b \in \mathbb{Z}$ with $a < 0$, then we can pla

2022-08-25 19:33:16 +0100 received badge  Self-Learner (source)
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2022-08-25 16:08:03 +0100 answered a question Polynomial is in ideal of a coordinate ring

This works: sage: K.<x, y> = QQ[] sage: I = Ideal(y^2 - x^3 - x) sage: L.<X, Y> = K.quotient(I) sage: I = I

2022-08-25 15:57:19 +0100 asked a question Polynomial is in ideal of a coordinate ring

Polynomial is in ideal of a coordinate ring I am getting an error with the code below. Please advise how I can do this.

2022-08-25 15:55:02 +0100 answered a question how to import a function in another file

For importing sage files, try the functions load() and attach()

2022-08-04 08:59:44 +0100 received badge  Supporter (source)
2022-08-04 08:58:30 +0100 marked best answer Evaluation map and ideal from polynomial ring K[X] -> K

How can I define a localized evaluation map $\phi_{a_1, \dots, a_n} : K[x_1, \dots, x_n] \rightarrow K$ by $\phi_{a_1, \dots, a_n}(p(x_1, \dots, x_n)) = p(a_1, \dots, a_n)$?

K = GF(47)
R.<x, y> = K[]
H = R.hom_eval(K(10), K(20))
assert H((x - 10)*(y - 20)) == 0

And also how can I construct the ideal of this map?

2022-08-01 09:37:43 +0100 edited question Evaluation map and ideal from polynomial ring K[X] -> K

Evaluation map and ideal from polynomial ring K[X] -> K How can I define a localized evaluation map $\phi_{a_1, \dots

2022-08-01 09:37:05 +0100 edited question Evaluation map and ideal from polynomial ring K[X] -> K

Evaluation map and ideal from polynomial ring K[X] -> K How can I define a localized evaluation map $\phi_{a_1, \dots

2022-08-01 09:35:35 +0100 asked a question Evaluation map and ideal from polynomial ring K[X] -> K

Evaluation map and ideal from polynomial ring K[X] -> K How can I define a localized evaluation map $\phi_{a_1, \dots

2022-08-01 08:12:51 +0100 answered a question Construct local ring of function field variety

https://math.stackexchange.com/questions/294644/basis-for-the-riemann-roch-space-lkp-on-a-curve?rq=1

2022-07-31 07:44:58 +0100 commented question Construct local ring of function field variety

Maybe I should have excluded the part about the ideal. I just want to construct the local ring, which is $K[V]_P$ aka th

2022-07-30 08:24:34 +0100 asked a question Construct local ring of function field variety

Construct local ring of function field variety Hello sage community, I want to localize a variety's field at a certain

2022-07-30 07:45:56 +0100 marked best answer How to test element is in multivariate function field's ideal

I'm trying to calculate the valuation of a function in a the function field of a coordinate ring $K(V) = { f / g : f, g \in K[V] }$.

My first attempt is to construct the coordinate ring $K[V] = K[x, y] / \langle C(x, y) \rangle$

sage: K.<x, y> = Integers(11)[]
sage: S = K.quotient(y^2 - x^3 - 4*x)
sage: S
Quotient of Multivariate Polynomial Ring in x, y over Ring of integers modulo 11 by the ideal (10*x^3 + y^2 + 7*x)

Now I want to see if a function $f = y - 2x$ lies in the ideal $I = \langle u \rangle$ where $u = x - 2$.

sage: I = S.ideal(x - 2)
sage: S(y - 2*x) in I
False

But sage is wrong. $y - 2x \in \langle x - 2 \rangle$ as shown by the following code:

sage: f1 = S(y - 2*x)
sage: f1
9*xbar + ybar
sage: f2 = S(
....:     (x - 2) * (
....:         (x - 2)^2*(y + 4) - 5*(x - 2)*(y + 4) - 2*((x - 2)^3 - 5*(x - 2)^2 + 5*(x - 2)
....:     ))) / S((y + 4)^2)
sage: f2
9*xbar + ybar
sage: bool(f1 == f2)
True

As plainly visible, f2 has the factor $(x - 2)$.

How can I calculate this in sage without having to factor the polynomial myself?

I assume this is because we need an actual function field in sage, but I get singular errors when I attempt to turn S into a fraction field.

sage: K.<x, y> = Integers(11)[]
sage: S = K.quotient(y^2 - x^3 - 4*x)
sage: R = FractionField(S)
# ...    
RuntimeError: error in Singular function call 'primdecSY':
ASSUME failed:   ASSUME(0, hasFieldCoefficient(basering) );
error occurred in or before primdec.lib::primdecSY_i line 5983: `  return (attrib(rng,"ring_cf")==0);`
leaving primdec.lib::primdecSY_i (5983)

Is there a way to construct local rings and maximal ideals? Or a way to calculate valuations (order of vanishing for poles and zeros) on elliptic curves?

Thanks

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2022-07-30 07:33:26 +0100 commented answer How to test element is in multivariate function field's ideal

For example: sage: K.<x, y> = GF(11)[] sage: S = K.quotient(y^2 - x^3 - 4*x) sage: I = S.ideal(x - 2) sage: I Ide

2022-07-29 18:23:32 +0100 commented answer How to test element is in multivariate function field's ideal

But why now does it say True for all powers of the ideal? The valuation of this function on the curve is 2, so the outpu

2022-07-28 10:54:23 +0100 received badge  Student (source)
2022-07-28 08:05:29 +0100 edited question How to test element is in multivariate function field's ideal

How to test element is in multivariate function field's ideal I'm trying to calculate the valuation of a function in a t

2022-07-28 08:04:37 +0100 edited question How to test element is in multivariate function field's ideal

How to test element is in multivariate function field's ideal I'm trying to calculate the valuation of a function in a t

2022-07-28 07:27:10 +0100 edited question How to test element is in multivariate function field's ideal

How to test element is in multivariate function field's ideal I'm trying to calculate the valuation of a function in a t

2022-07-28 07:26:37 +0100 edited question How to test element is in multivariate function field's ideal

How to test element is in multivariate function field's ideal I'm trying to calculate the valuation of a function in a t

2022-07-28 07:24:20 +0100 edited question How to test element is in multivariate function field's ideal

Incorrect result for x in ideal. Sage reports false, but I provide a correct factorization I'm trying to calculate the v

2022-07-28 07:24:14 +0100 received badge  Editor (source)
2022-07-28 07:24:14 +0100 edited question How to test element is in multivariate function field's ideal

Incorrect result for x in ideal. Sage reports false, but I provide a correct factorization I'm trying to calculate the v

2022-07-28 07:20:00 +0100 asked a question How to test element is in multivariate function field's ideal

Incorrect result for x in ideal. Sage reports false, but I provide a correct factorization I'm trying to calculate the v

2022-07-14 11:48:04 +0100 asked a question Elliptic Curve divisor from polynomial and vice versa

Elliptic Curve divisor from polynomial and vice versa I want to construct the Picard group of an elliptic or hyperellipt