rburing's profile - activity

Can you add the set of equations? Select your code and press the '101010' button to format it correctly.

I would re-evaluate whether using strings is really what you want to do; probably whatever problem you have can be solved in a better way. Nevertheless, it is possible:

function('P')
var('x1 x2 x3 y1 y2 y3')
st = eval('y1,0,1')
sum(P(x1,y1,y2,y3),*st)

Note number fields in SageMath are abstract extensions of $\mathbb{Q}$; they come with several embeddings into $\mathbb{C}$; see below.

Option 1 (number field from algebraic numbers):

sage: K, [A,B], emb = number_field_elements_from_algebraics([alpha,beta], minimal=True)
sage: K
Number Field in a with defining polynomial y^8 + 4*y^6 + 358*y^4 + 708*y^2 + 529
sage: A
-1/80*a^6 - 3/80*a^4 - 359/80*a^2 - 357/80
sage: B
1/920*a^7 + 1/230*a^5 + 381/920*a^3 + 607/460*a
sage: emb(A) == QQbar(alpha), emb(B) == QQbar(beta)
(True, True)

For other options, let's first find what polynomial we are talking about:

sage: f = alpha.minpoly(); f
x^4 + 11*x^2 + 11

Option 2 (splitting field):

sage: K.<c> = f.splitting_field()
sage: f_roots = f.roots(K, multiplicities=False)
sage: emb = K.embeddings(QQbar)[0]
sage: map(emb, f_roots)
[0.?e-16 + 1.054759596450271?*I,
 0.?e-16 - 1.054759596450271?*I,
 0.?e-16 - 3.144436705309246?*I,
 0.?e-16 + 3.144436705309246?*I]
sage: QQbar(alpha), QQbar(beta)
(1.054759596450272?*I, 3.144436705309246?*I)
sage: a = f_roots[0]; b = f_roots[3]
sage: emb(a) == QQbar(alpha), emb(b) == QQbar(beta)
(True, True)

Option 3 (successive extensions):

sage: K.<a> = NumberField(x^4 + 11*x^2 + 11)
sage: R.<y> = PolynomialRing(K)
sage: R(f).factor()
(y - a) * (y + a) * (y^2 + a^2 + 11)
sage: g = y^2 + a^2 + 11
sage: L.<b> = K.extension(g)
sage: f.change_ring(L).factor()
(x - b) * (x - a) * (x + a) * (x + b)
sage: emb = L.embeddings(QQbar)[4]
sage: emb(a) == QQbar(alpha), emb(b) == QQbar(beta)
(True, True)

You can also make an absolute extension from the relative one:

sage: M.<c> = L.absolute_field(); M
Number Field in c with defining polynomial x^8 + 110*x^6 + 3839*x^4 + 44770*x^2 + 43681
sage: from_M, to_M = M.structure()
sage: to_M(a)
7/2508*c^7 + 145/627*c^5 + 509/114*c^3 + 421/76*c
sage: to_M(b)
7/1254*c^7 + 290/627*c^5 + 509/57*c^3 + 459/38*c
sage: emb = M.embeddings(QQbar)[4]
sage: emb(to_M(a)) == QQbar(alpha), emb(to_M(b)) == QQbar(beta)
(True, True)

@user789 if that's all, then just substitute e.g. $e=a+b-d$ into your polynomial.

How can the first equation have any rational solutions?

Probably OP wants to know why taking the cube root in SageMath yields this particular root and not the real root.

When you make a vague request (e.g. 'simplify'), there is a risk that the answer will change if you ask it again.

Unfortunately I don't know what has changed or why. Maybe e.g. using git blame someone can find it out.

I am guessing that you are interested in the expression as a polynomial in $x,y,Z$.

A precise way to request the expression in that form is as follows (using polynomial rings):

sage: var('Z,x,y,a,w1,w2,w3,G')
sage: eqn = (Z*a + Z*w2 - a + w2)*(Z*a + Z*w3 - a + w3)*y == (Z*a + Z*w1 - a + w1)*G*(Z + 1)*x
sage: A = PolynomialRing(QQ, names='a,w1,w2,w3,G')
sage: B = PolynomialRing(A, names='Z,x,y')
sage: SR(B(eqn.lhs())) == SR(B(eqn.rhs()))
(a^2 + a*w2 + a*w3 + w2*w3)*Z^2*y - 2*(a^2 - w2*w3)*Z*y + (a^2 - a*w2 - a*w3 + w2*w3)*y == (G*a + G*w1)*Z^2*x + 2*G*Z*w1*x - (G*a - G*w1)*x

You could also factor each coefficient:

sage: sum(SR(C).factor()*SR(X) for C,X in B(eqn.lhs())) == sum(SR(C).factor()*SR(X) for C,X in B(eqn.rhs()))
Z^2*(a + w2)*(a + w3)*y - 2*(a^2 - w2*w3)*Z*y + (a - w2)*(a - w3)*y == G*Z^2*(a + w1)*x + 2*G*Z*w1*x - G*(a - w1)*x

You're welcome. That it takes a long time doesn't mean SageMath is not able to do it. Your original code works on my machine; it took 1 hour and 12 minutes.

This can be done (with some effort) in many cases, depending on the type of condition. Are the conditions linear equations, polynomial equations, trigonometric equations? An example would be great.

The error depends on the information you've omitted, such as the definition of r, l, and x. As a general rule, please give an unambiguous self-contained example that produces the error. Probably the problem is that l cannot be lifted to an integer.

It is working, but there seems to be some kind of bug in displaying the algebraic numbers. For example if you define S = Q*M[0]*P then set(S.diagonal()) == set(M[0].eigenvalues()) is True, and the workaround S.apply_map(lambda z: z.interval(CIF)) shows the matrix correctly.

Since $\bar{\mathbb{Q}}$ is a little big, try instead working in a number field $K$ that's big enough but as small as possible:

M = [matrix(QQ,l) for l in L]
K,_,_ = number_field_elements_from_algebraics(sum([m.eigenvalues() for m in M], []), minimal=True)
print(K)
p = identity_matrix(K,8)
q = ~p
for m in M:
    a = q*m*p
    da, pa = a.change_ring(K).jordan_form(transformation=True)
    p = p*pa
    q = ~p

Here $K$ is an abstract number field of degree $48$ with several (i.e. $48$) different embeddings into $\bar{\mathbb{Q}}$.

sage: set([sigma(p.det()) for sigma in K.embeddings(QQbar)])
{-97929.96732359303?, 97929.96732359303?}

For a given sigma in K.embeddings(QQbar), you can embed p and q by p.apply_map(sigma) and q.apply_map(sigma); those matrices are defined over QQbar and they simultaneously diagonalize all m in M.

Those look like dependencies, yes. What is the issue with them? What is invalid?

Please be more precise about the dependency issue (for the .deb file).

For one, because you can't transpose lists. Secondly, you should take the (single) entry of c*aa.

aa = vector(var('delta_%d' % i) for i in (0..2))
c = n(matrix(1,3,(-0.5,0.2,.3)),2)
U = e^((c*aa)[0])
show(U)

$$e^{\left(-0.50 \delta_{0} + 0.19 \delta_{1} + 0.25 \delta_{2}\right)}$$

(Note that you specified 2 bits of precision with n.)

Assuming that the rest of the code is correct, the issue is that you use the text "GGG" to refer to the variable GGG. Instead you should use something like GGG.name(). I don't have Magma so I am not sure of the exact syntax; try dir(GGG) to find the right method name.

This is not a bug.

sage: var('x,y,z')
sage: ((x+y+z)^5).expand()
x^5 + 5*x^4*y + 10*x^3*y^2 + 10*x^2*y^3 + 5*x*y^4 + y^5 + 5*x^4*z + 20*x^3*y*z + 30*x^2*y^2*z + 20*x*y^3*z + 5*y^4*z + 10*x^3*z^2 + 30*x^2*y*z^2 + 30*x*y^2*z^2 + 10*y^3*z^2 + 10*x^2*z^3 + 20*x*y*z^3 + 10*y^2*z^3 + 5*x*z^4 + 5*y*z^4 + z^5

Note the divisibility by 5 of all terms which are "missing" in your characteristic $p=5$ calculation.

This is a consequence of the Freshman's dream: $(x+y)^p \equiv x^p + y^p \pmod p$; just apply it twice.

It looks too good to be true, but it really is true in characteristic $p$ (the proof is in the linked article).

I didn't ask the question, but yes that's right. The binary ordering was just a (natural) suggestion. Of course the ordering is irrelevant, but probably some ordering is necessary for the implementation.

$|A|$ means the number of elements in $A$. To have $\underline{\alpha}$ be a vector one should choose an ordering of the (nonempty) subsets of $[5]$, e.g. by identifying them with binary strings of length 5 (not equal to $00000$).

The relation $r^2u^4 - (a^2 + r^2 + t^2)u^2 + t^2 = 0$ holds.

You have the right idea: do f.change_ring(R) with the argument R set to one of K.embeddings(RR):

var('w')
K.<t> = NumberField(w^2-3)
R.<x,y> = PolynomialRing(K)
f = y - t*x
implicit_plot(f.change_ring(K.embeddings(RR)[0]),(x,-3,3),(y,-3,3))

You might also like:

implicit_plot(f.change_ring(K.embeddings(RR)[0]),(x,-3,3),(y,-3,3), color='blue') + implicit_plot(f.change_ring(K.embeddings(RR)[1]),(x,-3,3),(y,-3,3), color='red')

Here is a way:

sage: var('a,b,c,x')
sage: sinx = function('sinx',nargs=1)
sage: pol =  3*a*x^(-b)*log(x)*b^2 - 6*a*b*c*sinx(x*b) + 3*a*c^2 + 5
sage: R = PolynomialRing(SR, names='a,c')
sage: f = pol.polynomial(ring=R)
sage: dict(zip(f.monomials(), f.coefficients()))
{1: 5, a: 3*b^2*log(x)/x^b, a*c: -6*b*sinx(b*x), a*c^2: 3}

Note the generators of the ring R are not the same as the symbolic variables a and c.

To get the coefficient of c as in your example, you can do:

sage: f.monomial_coefficient(R.gen(1))
0

Or maybe more conveniently, something like:

sage: A,C = R.gens()
sage: f.monomial_coefficient(C)
0

This may depend on the backend used, so can you give an example of a Polyhedron for which you want to know the answer?

You can do this:

sage: T.<t> = FunctionField(QQ)
sage: R.<u,v> = T[]
sage: f = u^3+v^3+t
sage: h = Jacobian(f, curve=Curve(f)); h
Scheme morphism:
  From: Affine Plane Curve over Rational function field in t over Rational Field defined by u^3 + v^3 + t
  To:   Elliptic Curve defined by y^2 = x^3 + (-27/4*t^2) over Rational function field in t over Rational Field
  Defn: Defined on coordinates by sending (u, v) to
        ((-t^3)*u^4*v^4 + (-t^4)*u^4*v + (-t^4)*u*v^4 : 1/2*t^3*u^6*v^3 + (-1/2*t^3)*u^3*v^6 + (-1/2*t^4)*u^6 + 1/2*t^4*v^6 + 1/2*t^5*u^3 + (-1/2*t^5)*v^3 : t^3*u^3*v^3)

To avoid constructing the list you could use next on the iterator.

Please add a code sample including a definition of f.

Not sure if it's the best way, but you can do the following:

sage: for n in range(10):
sage:    c = (x2^n*y2^n*v).coefficients()[0]
sage:    print('c_{} = {}'.format(n, c))
c_0 = 1
c_1 = -3
c_2 = 24
c_3 = -360
c_4 = 8640
c_5 = -302400
c_6 = 14515200
c_7 = -914457600
c_8 = 73156608000
c_9 = -7242504192000

Entering digraphs.nauty_directg?? into a SageMath session will show you the source code of the function.

In this case it simply calls an external program directg which is part of nauty.

You can download and edit that program's source code, compile your new version, and call that one instead.

To edit the source code you should start by looking at the file directg.c. Good luck!

It's not very nice to change your question when it has received a correct answer, even if you suffered from the XY problem. Anyway, I updated the answer. In the future, please ask the new question separately (and link to the previous question).

Define R2 as a multivariate polynomial ring in 1 variable:

sage: R2.<z> = PolynomialRing(QQ, 1)
sage: Q = z + 5
sage: for c,t in Q: print c,t
1 z
5 1

This ring (and its elements) will have different methods than the ordinary univariate ring (elements).

Alternatively: keep your original ring, define R3 = PolynomialRing(QQ, 1, name='z') and use R3(Q).

An answer to the new question:

def to_multivariate_poly(expr, base_ring):
    vars = expr.variables()
    if len(vars) == 0:
        vars = ['x']
    R = PolynomialRing(base_ring, len(vars), names=vars)
    return R(expr)

Then you can do:

sage: x,y,z = var('x,y,z')
sage: P = to_multivariate_poly(x + 2*y + x*y + 3, QQ)
sage: for c,t in P: print c,t
1 x*y
1 x
2 y
3 1
sage: Q = to_multivariate_poly(z + 5, QQ)
sage: for c,t in Q: print c,t
1 z
5 1

Ah, there is a subtlety with srange (explained in the documentation) that by default the output consists of numbers of the type you put in. You can override that by passing universe=ZZ to srange; I've added this to the answer.

It is not a bug. One should be aware of the difference between int and Integer. The second part of your code uses int division, there is no magic Sage can do to change that. Just be careful to write srange when you want Integers.

In the second part you are using Python ints instead of SageMath Integers.

Use srange (with universe=ZZ if you're not passing an Integer) instead of range, or convert to an Integer before the division, e.g. Integer(i[0])/len(a).

If you want to make a symbolic sum then all the terms should be symbolic. Your example does not work because qbin(n,k) is not defined for symbolic n.

What you can do is to make qbin a symbolic function with custom typesetting:

var('q')
def qbin_latex(self, n, k):
    return '{' + str(n) + ' \choose ' + str(k) + '}_{' + str(q) + '}'
qbin = function('qbin', nargs=2, print_latex_func=qbin_latex)
var('k,n')
show(sum(qbin(n,k),k,0,n))

$$\sum_{k=0}^{n} {n \choose k}_{q}$$

To check the identities that you are interested in, you will probably have to do some substitutions by hand, and/or pass an evalf_func parameter to function in the definition of qbin. Have a look at the documentation for symbolic functions.

It seems that you want the following:

var('x')
y = function('y')(x)
f = function('f')(x,y)
f.diff(x)

Output:

D[1](f)(x, y(x))*diff(y(x), x) + D[0](f)(x, y(x))

The matrix you constructed by hand doesn't look like a Hessian to me. Here is the Hessian:

sage: H = L.function(x,y,l).hessian()(x,y,l)
sage: H

$$\left(\begin{array}{rrr} A {\left(\alpha - 1\right)} \alpha x^{\alpha - 2} y^{\beta} & A \alpha \beta x^{\alpha - 1} y^{\beta - 1} & -p_{x} \\ A \alpha \beta x^{\alpha - 1} y^{\beta - 1} & A {\left(\beta - 1\right)} \beta x^{\alpha} y^{\beta - 2} & -p_{y} \\ -p_{x} & -p_{y} & 0 \end{array}\right)$$

Polynomial in one variable? Over which ring?

Sure, you can take a point from the ambient projective space and do a membership test:

sage: PP.<x,y,z,w,u,v> = ProductProjectiveSpaces([3,1],QQ)
sage: W = PP.subscheme([y^2*z-x^3,z^2-w^2,u^3-v^3])
sage: PP.point([1,1,1,1,1,1]) in W
True
sage: PP.point([1,1,1,1,1,2]) in W
False

Internally this does a try - except block around W.point(...).