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2017-03-15 15:17:06 +0100 asked a question Multivariate polynomial coefficients including zeros


I would like to get the coefficients of a multivariate polynomial including the zero coefficients (in their correct positions). I have found a similar answer as regards a polynomial f of two variables x,y

P.<x,y> = PolynomialRing(ZZ, 2, order='lex') 

Then using the following code:

coeffs = []
for i in range(, -1, -1):
  for j in range(, -1, -1):
      coeffs.append(f.coefficient({x:i, y:j}))

The result is as expected: [3, ,0 , 0, 1, 0, 0, 0, 0, 3]

Now, I would like to extend this solution for a multivariate polynomial of n variables [xo,x1,...xn-1] The polynomial is defined with the following code: (Note: q=next_prime(10000))

A1 = [(', '.join('x%i'%i for i in [0.. n-1]))]; ### construct a suitable multivariate ring
V = var(A1[0])

How can I do that? Any help will be much appreciated.

Regards, Natassa

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2016-12-08 11:40:08 +0100 commented answer Problem with the minus sign in symbolic expression

Indeed, now I see what the problem was. Thank you!

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2016-12-08 10:15:46 +0100 asked a question Problem with the minus sign in symbolic expression

Hi all!

My sage version is:

'SageMath Version 6.9, Release Date: 2015-10-10'

In my code, I have a multivariate polynomial with a sign symbol which doesn't return the correct result. I.e.

The code snippet is as follows:

A1 = [(', '.join('x%i'%i for i in [1.. n]))]; ### construct a suitable multivariate ring
 V = var(A1[0])                               ### define a str variable
x=vector(list(V))                                 ### convert to vector

a1 and b1 are returned from another function.
types of a1 and x are different:

 type(a1)=<type 'sage.modules.vector_modn_dense.Vector_modn_dense'>

type(x)=  <class 'sage.modules.vector_symbolic_dense.FreeModule_ambient_field_with_category.element_class'>

Could this be a problem?

a1 is a vector with values (76, 83), b1 is scalar with value 62, x is a vector with values (x1,x2). Thus, f1 should return 62-76 * x1-83 * x2; however it returns 25x1 + 18x2 + 62 which is not correct.

Please note that if I use addition in f1 as follows:

sage: f1=b1+a1.dot_product(x)

The result is correct,i.e. 76x1 + 83x2 + 62

type(f1) is sage.symbolic.expression.Expression

What could the problem be? I tried to convert the symbolic expression to multivariate polynomial, but this didn't work either? I also though there might be a problem with the minus symbol in my PC but couldn't solve it this way either.

Thank you for your responses! Regards, Natassa

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2016-12-03 12:23:41 +0100 answered a question Reduce multivariate polynomial coefficients to 1

Thank you all,

Indeed, considering sum of all monomials does the trick! As a reference, a slightly tweaked solution below that worked for me:

g # the polynomial 
list(g) # convert polynomial to list
g1=sum(mon for coeff, mon in g) # get the sum of all monomials
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2016-12-02 14:57:35 +0100 asked a question Reduce multivariate polynomial coefficients to 1

Hi all,

I'm new to Python and SAGE and would like to ask a question: I have a multivariate polynomial, of a degree 3, i.e:

g.<x1,x2,x3>=x1^2 + 2x1x2 + x2^2 + 2x1x3 + 2x2x3 + x3^2 + 2x1 + 2x2 + 2*x3 + 1 Its type is <type 'sage.rings.polynomial.multi_polynomial_libsingular.mpolynomial_libsingu\="" lar'="">

The question is how can I reduce all its leading coefficients to 1, i.e. transform ig to:

g.<x1,x2,x3>=x1^2 + x1x2 + x2^2 + x1x3 + x2x3 + x3^2 + *x1 + *x2 + *x3 + 1

I tried to get the coefficients of the polynomial (which is a list), iterate through its items, and set the values of the items to 1, i.e.

for s in g.coefficients(): if s==2: s=1

I think that the problem is that list items can't be cast to integer (TypeError).

How could I solve this? Is there a more efficient way to do this? Thank you for your responses! Regards, Natassa