Dear Simon, let me thank You for Your time and patience
On 29 Jul., 14:20, Simon King <[email protected]> wrote:
> Hi Thomas! (sorry, previously I made a wrong guess on your name)
>
> On Jul 29, 10:02 am, eggartmumie <[email protected]> wrote:
>
> > ...
> > Later I need the coefficients of the Goppa polynomial as linear
> > combination of monomials.
>
> I don't understand what that means. Please provide explicit examples!
a Goppa polynomial is any polynomial over a finite field F, in
practice over GF(2^m), in some indeterminant. In order to define a
Goppa polynomial I wanted to write something like
def goppapolynomial(F,z):
# returns a polynomial over F in indeterminant z
X = PolynomialRing(F,repr(z)).gen();
return (X^3+F(1))*X
expecting it the function would return a polynomial over F in
indeterminant z.
I thought the indeterminant essential, because the elements of F, e.g.
in case F = GF(2^2), are given as polynomials in another
indeterminant.
Then, in the following I need the coefficients of the Goppa
polynomial, in my contrived example above
I expect to get an coefficient array c=[0,1,0,0,1] of elements in F
with
y=var('y'); goppapolynomial(F,y) == c[0]*y^0+c[1]*y^1+c[2]*y^2+....
> Currently I can only jump from assumption to vague interpretation to
> guess; that makes it very difficult to provide any help.
> Thus, I think I should ask you more precise questions as well.
>
> So far, I thought we are talking about a univariate polynomial,
> something like
> sage: R.<x> = QQ[]
> sage: p = R.random_element()
> sage: p
> -x^2 + 4/9*x - 2
>
> Question 1:
> Is that your setting, or what exactly is the ring you are working
> with?
yes, only I need coefficients in finite fields, like e.g.
Phi.<x>=GF(2^4)
> Concerning coefficients:
> sage: p.coefficients()
> [-2, 4/9, -1]
>
> A "coefficient" is an element of the base ring (QQ, in my example).
> I know two conventions for monomials:
> 1) Some people say that a monomial is a power product of variables,
> like x^3.
> 2) Other people say that a monomial is the product of a coefficient
> with some power product of variables, like 4/9*x (the first sort of
> people would call this a "term") :/
I use coefficient to mean Your factor 4/9.
> In either case, I really don't see what it could mean to express "the
> coefficients as linear combinations of monomials". The best I can come
> up with are some associations.
sorry, I meant coefficients in the linear combination of monomials,
and not coefficients with a Newton base or a Lagrange base.
> First association:
> Get the part of the polynomial that is divisible by a certain monomial
> sage: p>>1
> -x + 4/9
sorry, in reading reference.pdf I never came across this operation,
I just did not know that You can shift a polynomial! Just to explain a
newbies situation,
I searched reference.pdf for >>, got 27 hits, would have looked at the
first which looks at least something like dealing with polynomials and
landed in a section about 'p-Adic ZZ_pX CR Element' (quote 'This file
implements elements of Eisenstein and unramified extensions of Zp and
Qp with capped relative precision.'). To be honest, I would not have
looked further, and all this assuming I'd known that SAGE (or Python
or ?) overloads the shift operator.
> Meaning: p == x*(-x+4/9)+remaining terms not divisible by x.
>
> Second association:
> Start with a multivariate polynomial
> sage: R.<x,y,z> = GF(3)[]
> sage: p = R.random_element(6)
> sage: p
> -x*y^5 + x^4*y*z + x^2*z^2 - y^2*z
> sage: p.coefficient(z)
> x^4*y - y^2
> So, here we have a coefficient that, itself, is a linear combination
> of monomials.
Goppa polynomials are univariate polynomials even though the
coefficients may introduce other indeterminants as shown above.
> Third association:
> A polynomial ring over a polynomial ring
> sage: R.<x> = ZZ['t'][]
> sage: R
> Univariate Polynomial Ring in x over Univariate Polynomial Ring in t
> over Integer Ring
> sage: p = R.random_element()
> sage: p
> (-3*t^2 - t - 1)*x^2 + (t^2 - t + 1)*x - 3*t - 2
> sage: p.coeffs()
> [-3*t - 2, t^2 - t + 1, -3*t^2 - t - 1]
> So, again, each coefficient is a polynomial (but in a different ring).
This is exactly what I intended to get by calling the function (!)
goppapolynomial with parameters ZZ['t'] and var('x'), expecting it to
return the polynomial
(-3*t^2 - t - 1)*x^2 + (t^2 - t + 1)*x - 3*t - 2
over ZZ['t'] in indeterminant x. I was confused by the different
result I get on f = goppapolynomial(F,z) when printing
type(f)
or
f.parent()
Even though I use tab completion quite a lot I did not find the
methods coefficient and coeffs because I only tried the tab completion
on the rings and not on their elements, the polynomials.
> Question 2:
> Is any of my associations close to what you mean? Otherwise, please
> provide an example.
see above
> > In order to be able to once define the Goppa polynomial in a function
> > I tried to compute the coefficients from the given function goppapolynomial.
>
> Question 3:
> What exactly is the "function" goppapolynomial supposed to do / to
> be?
> Is it a function that returns a polynomial? Or is it a polynomial
> function, itself?
> Please specify input and output. Is it really just returning X^(N-K)
> (what are N and K, anyway?)?
some global integer parameters
> Cheers,
> Simon
I did not want to bother everybody but saw no alternative.
Although I now know how to get the coefficients I'd still like to know
whether I can coerce some
polynomial to an symbolic expression so I can do differentiation or
integration an it and coerce
the result back to a polynomial in the original polynomial ring. Again
thank You for Your helpfulness, Th
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