Re: [sage-support] Minimum norm of an ideal lattice

2012-09-06 Thread Cindy
Let M denote the generator matrix of the lattice. Suppose M is a 2 by 2 
matrix. 

sage: var('x', domain=ZZ);
sage: var('y', domain=ZZ);
sage: v=vector((x,y));
sage: f=(M*v).norm();minimize(f,[1,1])

But the output is

Warning: Maximum number of iterations has been exceeded
 Current function value: 0.00
 Iterations: 400
 Function evaluations: 804
 Gradient evaluations: 804
(1.07117779258e-121, -7.20701845996e-121)



On Wednesday, September 5, 2012 7:31:48 PM UTC+8, David Loeffler wrote:

  how can I get the minimum norm for the 
  ideal lattice (J,\alpha) using sage? 

 What have you tried so far? 

 David 


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Re: [sage-support] Minimum norm of an ideal lattice

2012-09-06 Thread Cindy
BTW, the generator matrix I used for the previous example is 
[1 2]
[3 4]

Thanks.

Cindy

On Wednesday, September 5, 2012 7:31:48 PM UTC+8, David Loeffler wrote:

  how can I get the minimum norm for the 
  ideal lattice (J,\alpha) using sage? 

 What have you tried so far? 

 David 


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Re: [sage-support] Minimum norm of an ideal lattice

2012-09-06 Thread Cindy
Hi, David,

Thanks for your explanation about the minimize function in sage. I didn't 
realize it's only for differentiable functions.

For the stuff regarding lattice, I think there may be some misunderstanding 
here.

What I want is to find the minimum of a lattice.

A lattice L can be defined as

L={x=\lambda M| \lambda\in Z^m},

where M is the generator matrix of L and the gram matrix of L is equal to 
MM^T.

The matrix 
[1 2]
[3 4]
has determinant 4-2*3=-2, which is nonzero. Moreover, I use it as the generator 
matrix for lattice, not the gram matrix. Thus I don't think it needs to be 
symmetric or Hermitian.

As for the definition of minimum of a lattice, I assume it's defined for 
all lattice, thus there should be no other restrictions on the generator 
matrix (not gram matrix) except for it to be invertible.

 
According to the definition I found:

N(x)=x\cdot x=(x,x)=\sum x_i^2

for a vector x=(x_1,x_2,\dots,x_n) in a lattice and the minimum norm of 
lattice L is 

min{N(x): x\in L, x\neq  0}.

When I calculate Mv (M is the generator matrix and v is the vector (x,y) 
with x,y both integers) I get a vector in L. Then I find the norm of Mv, 
which is the norm of this vector in L.

What I need is the minimum of this value.

Did I get the wrong definition of the minimum of lattice?

Best Regards,
Xiaolu

On Thursday, September 6, 2012 6:50:00 PM UTC+8, David Loeffler wrote:

 Dear Cindy, 

 Without wishing to cause offence, I think your problem isn't a Sage 
 problem: it's that you don't understand the mathematical problem that 
 you're trying to solve. 

 Firstly, if V is an inner product space with basis v_1, ..., v_n and M 
 is its Gram matrix (the matrix whose i,j entry is v_i paired with 
 v_j), then the norm of the vector with coordinates x_1, .., x_n is not 
 the usual norm of (M * [x_1; ...; x_n]); it's [x_1, ..., x_n] * M * 
 [x_1; ...; x_n]. 

 Secondly, the matrix [1, 2; 3, 4] is not symmetric or Hermitian and 
 its determinant is 0, so it is not the Gram matrix of a positive 
 definite inner product space. 

 Thirdly, the minimize function does what it says on the tin: it 
 finds the minimum value of a function, and it does so by using 
 calculus, assuming the function is differentiable. The minimum value 
 of the norm of a vector in a positive definite inner product space is 
 0, the norm of the zero vector. You want the minimum value at a 
 non-zero integer point and calculus is not going to help you with 
 that. 

 May I ask what motivates this long string of questions? Are you a 
 student? If so, you should go back and read your undergraduate linear 
 algebra notes a bit more carefully. 

 Regards, David Loeffler 

 On 6 September 2012 10:38, Cindy cindy42...@gmail.com javascript: 
 wrote: 
  BTW, the generator matrix I used for the previous example is 
  [1 2] 
  [3 4] 
  
  Thanks. 
  
  Cindy 
  
  
  On Wednesday, September 5, 2012 7:31:48 PM UTC+8, David Loeffler wrote: 
  
   how can I get the minimum norm for the 
   ideal lattice (J,\alpha) using sage? 
  
  What have you tried so far? 
  
  David 
  
  -- 
  You received this message because you are subscribed to the Google 
 Groups 
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Re: [sage-support] Minimum norm of an ideal lattice

2012-09-06 Thread Cindy
Hi, David,

Thanks for your explanation about the minimize function in sage. I didn't 
realize it's only for differentiable functions.

For the stuff regarding lattice, I think there may be some misunderstanding 
here.

What I want is to find the minimum of a lattice.

A lattice L can be defined as

L={x=\lambda M| \lambda\in Z^m},

where M is the generator matrix of L and the gram matrix of L is equal to 
MM^T.

The matrix 
[1 2]
[3 4]
has determinant 4-2*3=-2, which is nonzero. Moreover, I use it as the generator 
matrix for lattice, not the gram matrix. Thus I don't think it needs to be 
symmetric or Hermitian.

As for the definition of minimum of a lattice, I assume it's defined for 
all lattice, thus there should be no other restrictions on the generator 
matrix (not gram matrix) except for it to be invertible.

 
According to the definition I found:

N(x)=x\cdot x=(x,x)=\sum x_i^2

for a vector x=(x_1,x_2,\dots,x_n) in a lattice and the minimum norm of 
lattice L is 

min{N(x): x\in L, x\neq  0}.

When I calculate Mv (M is the generator matrix and v is the vector (x,y) 
with x,y both integers) I get a vector in L. Then I find the norm of Mv, 
which is the norm of this vector in L.

What I need is the minimum of this value.

Did I get the wrong definition of the minimum of lattice?

Best Regards,
Cindy

On Thursday, September 6, 2012 6:50:00 PM UTC+8, David Loeffler wrote:

 Dear Cindy, 

 Without wishing to cause offence, I think your problem isn't a Sage 
 problem: it's that you don't understand the mathematical problem that 
 you're trying to solve. 

 Firstly, if V is an inner product space with basis v_1, ..., v_n and M 
 is its Gram matrix (the matrix whose i,j entry is v_i paired with 
 v_j), then the norm of the vector with coordinates x_1, .., x_n is not 
 the usual norm of (M * [x_1; ...; x_n]); it's [x_1, ..., x_n] * M * 
 [x_1; ...; x_n]. 

 Secondly, the matrix [1, 2; 3, 4] is not symmetric or Hermitian and 
 its determinant is 0, so it is not the Gram matrix of a positive 
 definite inner product space. 

 Thirdly, the minimize function does what it says on the tin: it 
 finds the minimum value of a function, and it does so by using 
 calculus, assuming the function is differentiable. The minimum value 
 of the norm of a vector in a positive definite inner product space is 
 0, the norm of the zero vector. You want the minimum value at a 
 non-zero integer point and calculus is not going to help you with 
 that. 

 May I ask what motivates this long string of questions? Are you a 
 student? If so, you should go back and read your undergraduate linear 
 algebra notes a bit more carefully. 

 Regards, David Loeffler 

 On 6 September 2012 10:38, Cindy cindy42...@gmail.com javascript: 
 wrote: 
  BTW, the generator matrix I used for the previous example is 
  [1 2] 
  [3 4] 
  
  Thanks. 
  
  Cindy 
  
  
  On Wednesday, September 5, 2012 7:31:48 PM UTC+8, David Loeffler wrote: 
  
   how can I get the minimum norm for the 
   ideal lattice (J,\alpha) using sage? 
  
  What have you tried so far? 
  
  David 
  
  -- 
  You received this message because you are subscribed to the Google 
 Groups 
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Re: [sage-support] Minimum norm of an ideal lattice

2012-09-06 Thread Cindy
Thanks.

Cindy

On Thursday, September 6, 2012 9:03:47 PM UTC+8, David Loeffler wrote:

 On 6 September 2012 13:28, Cindy cindy42...@gmail.com javascript: 
 wrote: 
  Hi, David, 
  
  Thanks for your explanation about the minimize function in sage. I 
 didn't 
  realize it's only for differentiable functions. 
  
  For the stuff regarding lattice, I think there may be some 
 misunderstanding 
  here. 
  
  What I want is to find the minimum of a lattice. 
  
  A lattice L can be defined as 
  
  L={x=\lambda M| \lambda\in Z^m}, 
  
  where M is the generator matrix of L and the gram matrix of L is equal 
 to 
  MM^T. 

 OK, I've never heard of this definition but if you want to take that 
 to be the definition that's up to you -- apparently for you all 
 lattices come with a fixed embedding into Euclidean space. But that 
 then changes the interpretation of your previous question, because in 
 your previous thread I assumed you wanted a Gram matrix, and that is 
 what the code I suggested calculates; the lattices coming from trace 
 pairings on number fields won't have any preferred embedding into 
 Euclidean space. To get *a* generator matrix (in your sense) from the 
 Gram matrix, you could use Cholesky decomposition, for example. But to 
 do this you will have to introduce square roots all over the place and 
 hence the computation becomes inexact; it is far simpler to just work 
 with the Gram matrix, which will be integer-valued in the examples 
 you've mentioned so far. 

 To find the shortest vector, you might want to use some of the 
 routines in Sage's quadratic forms module. 

 David 


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Re: [sage-support] Dual of an ideal

2012-09-05 Thread Cindy
Hi, David,

Could you please explain a little bit about the code?

For the example you use, it seems I is an ideal above 17, what does [0] 
mean?

In the end do we get a basis of the dual of I? Why do we need to put 
I.basis() in the bracket of trace_dual_basis?

Thanks a lot.

Cindy

On Wednesday, September 5, 2012 4:21:22 PM UTC+8, David Loeffler wrote:

 On 5 September 2012 02:41, Cindy cindy42...@gmail.com javascript: 
 wrote: 
  Hi, David, 
  
  Yes, that's what I mean. Can I find it using sage? 
  
  Thanks. 
  
  Cindy 

 sage: K.z = NumberField(x^3 - x + 17) 
 sage: I = K.primes_above(17)[0] 
 sage: K.trace_dual_basis(I.basis()) 
 [4/132583*z^2 + 6/7799*z + 2597/132583, -153/7799*z^2 - 2/7799*z + 
 102/7799, -6/7799*z^2 - 153/7799*z + 4/7799] 

 hth, David 


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Re: [sage-support] Dual of an ideal

2012-09-05 Thread Cindy
Hi, David,

Thanks a lot. I tried trace_dual_basis? to find out the meaning. I didn't 
realize I should use K.trace_dual_basis?

Thanks. :)

Cindy

On Wednesday, September 5, 2012 5:15:19 PM UTC+8, David Loeffler wrote:

 On 5 September 2012 09:34, Cindy cindy42...@gmail.com javascript: 
 wrote: 
  Hi, David, 
  
  Could you please explain a little bit about the code? 

 Sure, but you should make a little effort to play with it yourself for 
 a bit first. 

  For the example you use, it seems I is an ideal above 17, what does [0] 
  mean? 

 The command K.primes_above(...) returns a list of the prime ideals 
 above the given rational prime. The [0] selects the first (zeroth?) 
 from the list. So yes, I is an ideal above 17 which I am just using as 
 an example (any number field ideal, except the zero ideal, would work 
 here). There are lots of examples like this in the Sage documentation. 

  In the end do we get a basis of the dual of I? 

 Yes, that's the whole point of the exercise :-). Did you read the 
 documentation for trace_dual_basis? You should know that you can get 
 documentation on any method of any Sage object by typing its name then 
 ?, e.g. 

 sage: K.trace_dual_basis? 

 will tell you lots more about this method. 

  Why do we need to put 
  I.basis() in the bracket of trace_dual_basis? 

 Because trace_dual_basis takes a list of generators as its argument -- 
 it can calculate the trace dual of any Z-submodule of K, it needn't be 
 an ideal. 

 Regards, David 


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Re: [sage-support] Generator matrix of ideal lattice

2012-09-05 Thread Cindy
Hi, David,

Thanks a lot! It works.^^

Cindy

On Wednesday, September 5, 2012 4:30:40 PM UTC+8, David Loeffler wrote:

 On 5 September 2012 02:56, Cindy cindy42...@gmail.com javascript: 
 wrote: 
  Hi, 
  
  Let K be a number field and O_k denote its ring of integers. For an 
 ideal, J 
  of O_k, we can have an ideal lattice (I,b_\alpha), where 
  
  b_\alpha: J\times J \to Z, b_\alpha(x,y)=Tr(\alpha xy), \forall x,y \in 
 J 
  
  and \alpha is a totally positive element of K\{0}. 
  
  Suppose now I know J and \alpha, how can I get the generator matrix for 
 the 
  ideal lattice (J,\alpha) using sage? 
  
  Thanks a lot. 
  
  Cindy 

 The first thing I tried was this, and it seems to work fine: 

 sage: K.z = NumberField(x^3 - x + 17) 
 sage: I = K.primes_above(17)[1] 
 sage: alpha = 13*z + 4 
 sage: matrix([[(u*v*alpha).trace() for u in I.basis()] for v in 
 I.basis()]) 
 [  3468646 -11339] 
 [   646   -591   -871] 
 [-11339   -871225] 

 David 


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[sage-support] Minimum norm of an ideal lattice

2012-09-05 Thread Cindy
Hi, 

Let K be a number field and O_k denote its ring of integers. For an ideal, 
J of O_k, we can have an ideal lattice (I,b_\alpha), where

b_\alpha: J\times J \to Z, b_\alpha(x,y)=Tr(\alpha xy), \forall x,y \in J

and \alpha is a totally positive element of K\{0}.

Suppose now I know J and \alpha, how can I get the minimum norm for the 
ideal lattice (J,\alpha) using sage?

Thanks a lot.

Cindy 

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Re: [sage-support] Generator matrix of ideal lattice

2012-09-05 Thread Cindy
Hi, David,

BTW, do you know how to find the minimum norm of the lattice? I posted a 
question regarding this in this group. Do you know which function I should 
use?

Thanks.

Cindy

On Wednesday, September 5, 2012 4:30:40 PM UTC+8, David Loeffler wrote:

 On 5 September 2012 02:56, Cindy cindy42...@gmail.com javascript: 
 wrote: 
  Hi, 
  
  Let K be a number field and O_k denote its ring of integers. For an 
 ideal, J 
  of O_k, we can have an ideal lattice (I,b_\alpha), where 
  
  b_\alpha: J\times J \to Z, b_\alpha(x,y)=Tr(\alpha xy), \forall x,y \in 
 J 
  
  and \alpha is a totally positive element of K\{0}. 
  
  Suppose now I know J and \alpha, how can I get the generator matrix for 
 the 
  ideal lattice (J,\alpha) using sage? 
  
  Thanks a lot. 
  
  Cindy 

 The first thing I tried was this, and it seems to work fine: 

 sage: K.z = NumberField(x^3 - x + 17) 
 sage: I = K.primes_above(17)[1] 
 sage: alpha = 13*z + 4 
 sage: matrix([[(u*v*alpha).trace() for u in I.basis()] for v in 
 I.basis()]) 
 [  3468646 -11339] 
 [   646   -591   -871] 
 [-11339   -871225] 

 David 


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Re: [sage-support] Dual of an ideal

2012-09-04 Thread Cindy
Hi, David,

Yes, that's what I mean. Can I find it using sage?

Thanks.

Cindy

On Tuesday, September 4, 2012 3:12:25 PM UTC+8, David Loeffler wrote:

 What exactly do you mean by the dual of an ideal? Do you mean dual 
 with respect to the trace pairing, so the dual of the ideal (1) is the 
 inverse different? 

 David 

 On 4 September 2012 04:15, Cindy cindy42...@gmail.com javascript: 
 wrote: 
  Hi, 
  
  How can I calculate the dual of an ideal using sage? 
  
  Thanks. 
  
  Cindy 
  
  -- 
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 Groups 
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Re: [sage-support] Dual of an ideal

2012-09-04 Thread Cindy
Hi, Vijay,

Let K be a number field and O_k be its ring of integers. Given an ideal J 
of O_k, I want to find the dual of J, which is defined as the O_k-module:

J^*={x\in K| Tr(xJ)\subset Z}.

Thanks.

Cindy

On Tuesday, September 4, 2012 3:20:35 PM UTC+8, Vj wrote:

 Cindy,

 Could you elaborate little more, what is precisely you need.

 Regards,
 Vijay

 On Tue, Sep 4, 2012 at 12:42 PM, David Loeffler 
 d.a.lo...@warwick.ac.ukjavascript:
  wrote:

 What exactly do you mean by the dual of an ideal? Do you mean dual
 with respect to the trace pairing, so the dual of the ideal (1) is the
 inverse different?

 David

 On 4 September 2012 04:15, Cindy cindy42...@gmail.com javascript: 
 wrote:
  Hi,
 
  How can I calculate the dual of an ideal using sage?
 
  Thanks.
 
  Cindy
 
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Re: [sage-support] Dual of an ideal

2012-09-04 Thread Cindy
Hi,

BTW, the ideals I am dealing with are ideals of the ring of integers of a 
number field.

Cindy

On Tuesday, September 4, 2012 3:12:25 PM UTC+8, David Loeffler wrote:

 What exactly do you mean by the dual of an ideal? Do you mean dual 
 with respect to the trace pairing, so the dual of the ideal (1) is the 
 inverse different? 

 David 

 On 4 September 2012 04:15, Cindy cindy42...@gmail.com javascript: 
 wrote: 
  Hi, 
  
  How can I calculate the dual of an ideal using sage? 
  
  Thanks. 
  
  Cindy 
  
  -- 
  You received this message because you are subscribed to the Google 
 Groups 
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[sage-support] Generator matrix of ideal lattice

2012-09-04 Thread Cindy
Hi, 

Let K be a number field and O_k denote its ring of integers. For an ideal, 
J of O_k, we can have an ideal lattice (I,b_\alpha), where

b_\alpha: J\times J \to Z, b_\alpha(x,y)=Tr(\alpha xy), \forall x,y \in J

and \alpha is a totally positive element of K\{0}.

Suppose now I know J and \alpha, how can I get the generator matrix for the 
ideal lattice (J,\alpha) using sage?

Thanks a lot.

Cindy 

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[sage-support] Dual of an ideal

2012-09-03 Thread Cindy
Hi,

How can I calculate the dual of an ideal using sage?

Thanks.

Cindy

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Re: [sage-support] Calculate discrimanent of relative number fields

2012-08-31 Thread Cindy
Thanks a lot David! It works ^^


Cindy


On Thursday, August 30, 2012 4:39:36 PM UTC+8, David Loeffler wrote:

 On 29 August 2012 12:54, Cindy cindy42...@gmail.com javascript: 
 wrote: 
  Hi, 
  
  Given a cyclotomic field Q(zeta_n), where zeta_n is a primitive nth root 
 of 
  unity, with maximal real subfield F, how can I calculate the 
 discriminant of 
  K/F? 

 You need to use the relativize command to create the field extension 
 K / F. Here's an example for the 13th cyclotomic field: 

 -- 
 | Sage Version 5.2, Release Date: 2012-07-25 | 
 | Type notebook() for the browser-based notebook interface.| 
 | Type help() for help.| 
 -- 
 sage: K.zeta = CyclotomicField(13) 
 sage: Krel = K.relativize(zeta + zeta^(-1), w) 
 sage: Krel 
 Number Field in w0 with defining polynomial x^2 - w1*x + 1 over its base 
 field 
 sage: Krel.base_field() 
 Number Field in w1 with defining polynomial x^6 + x^5 - 5*x^4 - 4*x^3 
 + 6*x^2 + 3*x - 1 

 # so now Krel is QQ(zeta13) as an extension of F = QQ(zeta13 + 
 zeta13^(-1)) 

 sage: Krel.relative_discriminant() 
 Fractional ideal (w1^5 - 5*w1^3 + 4*w1) 
 # (an ideal of F) 

 sage: Krel.relative_different() 
 Fractional ideal ((w1^3 - 2*w1)*w0 - w1^2) 
 # (an ideal of Krel) 

 Regards, David 


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[sage-support] Calculate discrimanent of relative number fields

2012-08-29 Thread Cindy
Hi,

Given a cyclotomic field Q(zeta_n), where zeta_n is a primitive nth root of 
unity, with maximal real subfield F, how can I calculate the discriminant 
of K/F?

Thanks.

Cindy

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[sage-support] Find the generator matrix and minimum norm of ideal lattice

2012-08-29 Thread Cindy
Hi,

Given an ideal I, I want to find the following properties of the ideal 
lattice (I,b), where b is the trace form.

1. generator matrix
2. minimum norm

How can I do this in Sage?

Thanks.

Cindy

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