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thanks a lot!

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

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

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

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

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

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

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

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

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

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 +

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

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

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

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

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?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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