azerbajdzan's profile - activity

I think the documentation is written quite well for this topic.

@rburing: Thanks. Done.

Congruence of two non-positive definite matrices Method here works only with positive definite matrices. How to adapt i

Why it only works when matrices are positive definite? Any idea how to solve it if we have quadratic form that its Gram

Is this the only way to do it or there is some built-in function? QuadraticForm(gm + gm.transpose()).coefficients()

How to convert Gram matrix to quadratic form? How to do the opposite? Given gm, how to compute coefficients of quadratic

Equivalent of Magma's LatticeWithGram and IsIsometric Is there equivalent of LatticeWithGram and IsIsometric so that the

R(g).roots() [(82764486716702815285605477501188164702466527314352175978120539775788537185277, 1)] You computed roo

I hoped for a more simple an easier to read code, but probably it is not that simple. And yes, it partly answers the que

@John Palmieri: The second half of the magma code does not use CanonicalDivisor but explicit rational point so no need t

I think that if there was someone with such abilities that he/she would have already answered it in my original question

The error tells you to set option... when done the equation is solved. y = function('y')(x) de = diff(y,x)*(4*x+2*y-3)

Autocompletion function on version 9.2 I had version 8.9 and decided to install version 9.3 - the latest version that do

Transforming genus zero curve to conic The following code outputs rational parametrization of genus 0 curve of degree 5:

Transforming genus zero curve to conic The following code outputs rational parametrization of genus 0 curve of degree 5:

Transforming genus zero curve to conic The following code outputs rational parametrization of genus 0 curve of degree 5:

That is what I am interested in. Does not seem people here are much interested in it. I do not have a code, I did it hal

The following code outputs rational parametrization of genus 0 curve of degree 5:

x, y = QQ['x,y'].gens()
C=Curve(2*x^5 + x^2*y - 4*x^3*y + 2*x*y^2 + 2*x*y^3 + 1*y^5)
print(C.is_smooth())
print(C.genus())
C.rational_parameterization()

False
0
Scheme morphism:
  From: Affine Space of dimension 1 over Rational Field
  To:   Affine Plane Curve over Rational Field defined by 2*x^5 + y^5 - 4*x^3*y + 2*x*y^3 + x^2*y + 2*x*y^2
  Defn: Defined on coordinates by sending (t) to
        ((-2*t^2 - t)/(4*t^5 + 1), (4*t^4 + 2*t^3)/(4*t^5 + 1))

Is there a function that instead of parametrization outputs transformation x -> f1(u, v), y -> f2(u, v) so that the curve is transformed into a conic section curve (any curve of degree 2)?

The curve would look like this with given coefficients a1..a6.

a1*u^2 + a2*v^2 + a3*u*v + a4*u + a5*v + a6

UPDATE:

Here is an example of a curve of degree 3 and genus 0 that I was able to transform to conic - specifically parabola (with forward and backward transformations):

$$y^2=x^3-x^2,\left(x\to \frac{v}{u^2},y\to \frac{v}{u^3}\right)\\ v=u^2+1,\left(u\to \frac{x}{y},v\to \frac{x^3}{y^2}\right)$$

UPDATE2:

$$2 x^5-4 x^3 y+x^2 y+2 x y^3+2 x y^2+y^5=0,\left(x\to -\frac{8 u v^3-u+4 v^3-2 v}{16 v^5-1},y\to \frac{2 v \left(8 u v^3-u+4 v^3-2 v\right)}{16 v^5-1}\right)\\v-u^2=0,\left(u\to \frac{-2 x^5+2 x^3 y-x y^3-y^5}{2 x \left(x^3+y^3\right)},v\to -\frac{y}{2 x}\right)$$

$$2 x^5-4 x^3 y+x^2 y+2 x y^3+2 x y^2+y^5=0,\left(x\to -\frac{8 u v^3-u+4 v^3-2 v}{16 v^5-1},y\to \frac{2 v \left(8 u v^

@dan_fulea: Hello, I updated question with an example of degree 3 curve that was transformed to parabola. Can you do the

Transforming genus zero curve to conic The following code outputs rational parametrization of genus 0 curve of degree 5:

Transforming genus zero curve to conic The following code outputs rational parametrization of genus 0 curve of degree 5:

Transforming genus zero curve to conic The following code outputs rational parametrization of genus 0 curve of degree 5:

Why I got those ugly _SAGE_VAR_u and _SAGE_VAR_x instead of straight u and x? I am using SageMath 8.9.

var('x, t, u')
maxima.eliminate([x == -((t*(1 + 2*t))/(1 + 4*t^5)), u == -((2*t)/(1 + t^2))],[t])

[(17*_SAGE_VAR_u^5-40*_SAGE_VAR_u^4+160*_SAGE_VAR_u^2-128)*_SAGE_VAR_x^2+(4*_SAGE_VAR_u^5+46*_SAGE_VAR_u^4-56*_SAGE_VAR_u^3-64*_SAGE_VAR_u^2+64*_SAGE_VAR_u)*_SAGE_VAR_x+5*_SAGE_VAR_u^5-4*_SAGE_VAR_u^4]

How do I get rid off it? Substituting does not seem to work.

Update:

Eliminating without maxima - why it does not wok?

R.<x,t,u> = PolynomialRing(QQ)
gens = [x == -((t*(1 + 2*t))/(1 + 4*t^5)), u == -((2*t)/(1 + t^2))]
J = R.ideal(gens)
J.elimination_ideal([t])

Ideal (0) of Multivariate Polynomial Ring in x, t, u over Rational Field

While this works:

R.<x,y,z> = PolynomialRing(QQ)
gens = [ x^2 + y^2 + z^2 - 14, x*y + y*z + z*x -11, x*y*z - y^2 -2]
J = R.ideal(gens)
J.elimination_ideal([x,y])

Ideal (z^12 + 2*z^11 - 25*z^10 - 40*z^9 + 329*z^8 - 4*z^7 - 1763*z^6 + 3984*z^5 + 2475*z^4 - 43722*z^3 + 75942*z^2 - 60588*z + 23409) of Multivariate Polynomial Ring in x, y, z over Rational Field

Thanks. Can you also answer the update in my question?

@FrédéricC: I edited my question... maybe you can answer also the update.

Maxima eliminate with ugly _SAGE_VAR_ Why I got those ugly _SAGE_VAR_u and _SAGE_VAR_x instead of straight u and x? I am

Thanks, this could be an answer. Yes I know, but I have not in plan to install Linux to use the latest version, I heard

Maxima eliminate with ugly _SAGE_VAR_ Why I got those ugly _SAGE_VAR_u and _SAGE_VAR_x instead of straight u and x? I am

Maxima eliminate with ugly _SAGE_VAR_ Why I got those ugly _SAGE_VAR_u and _SAGE_VAR_x instead of straight u and x? I am

It is likely that the function is already implemented in sage or that combination of two or three functions would produc

It is likely that the function is already implemented in sage or that combination of two or three functions would produc

If there does not exist rational parametrization of a curve then it is impossible to transform it birationally to conic

Transforming genus zero curve to conic The following code outputs rational parametrization of genus 0 curve of degree 5:

Transforming genus zero curve to conic The following code outputs rational parametrization of genus 0 curve of degree 5:

Transforming genus zero curve to conic The following code outputs rational parametrization of genus 0 curve of degree 5:

Cygwin takes 200MB on my HDD, How much space needs WSL together with Ubuntu? Is it around 25GB?