The old Google Groups will be going away soon, but your browser is incompatible with the new version.
Quotients of polynomials
 There are currently too many topics in this group that display first. To make this topic appear first, remove this option from another topic. There was an error processing your request. Please try again. Standard view   View as tree
 7 messages

From:
To:
Cc:
Followup To:
Subject:
 Validation: For verification purposes please type the characters you see in the picture below or the numbers you hear by clicking the accessibility icon.

More options Oct 11 2011, 9:28 am
Date: Tue, 11 Oct 2011 06:28:16 -0700 (PDT)
Local: Tues, Oct 11 2011 9:28 am
Subject: Quotients of polynomials
Hi,

I have the following problem. Given a function f in the complex
variable s. I know that f can be written as the quotient of two
polynomials in s with real coefficients, but it is not possible to
determine the coefficients manually. Thus my question is whether one
can calculate the coefficients using sage.

Urs

To post a message you must first join this group.
You do not have the permission required to post.
More options Oct 11 2011, 10:44 am
From: John Cremona <john.crem...@gmail.com>
Date: Tue, 11 Oct 2011 15:44:35 +0100
Local: Tues, Oct 11 2011 10:44 am
Subject: Re: [sage-algebra] Quotients of polynomials
How is f given -- as a powers series?  If so, and you have enough
coefficients relative to the denominator degree, then you can
reconstruct f, by detrermining the linear recurrence satisfied by the
coefficients of the power series.

John Cremona

On Tue, Oct 11, 2011 at 2:28 PM, Urs Hackstein

To post a message you must first join this group.
You do not have the permission required to post.
More options Oct 11 2011, 11:36 am
Date: Tue, 11 Oct 2011 08:36:17 -0700 (PDT)
Local: Tues, Oct 11 2011 11:36 am
Subject: Re: Quotients of polynomials
Unfortunately, I can't recognize any nice mathematical structure in
the form f is given. It is given not exactly as an continous fraction,
but it is given as a big (in the sense of many variables and terms)
fraction where both denominator and numerator are "complicated"
functions containing big fractions where both denominator and
numerator are .. and so on.
f is not an object of pure mathematical interest, it arise from some
calculation in electronical engineering. But engineers told me that it
is known by theoretical results that it could be written as the
quotient of polynomials.

On Oct 11, 4:44 pm, John Cremona <john.crem...@gmail.com> wrote:

To post a message you must first join this group.
You do not have the permission required to post.
More options Oct 11 2011, 12:12 pm
From: John Cremona <john.crem...@gmail.com>
Date: Tue, 11 Oct 2011 17:12:28 +0100
Local: Tues, Oct 11 2011 12:12 pm
Subject: Re: [sage-algebra] Re: Quotients of polynomials
In that case I cannot help.

I suggest that you ask again on sage-support.  Most people who read
sage-algebra are pure algebraists, while you need people with more
expertises in symbolic computer algebra (which is different!)

John Cremona

On Tue, Oct 11, 2011 at 4:36 PM, Urs Hackstein

To post a message you must first join this group.
You do not have the permission required to post.
More options Oct 12 2011, 5:45 am
From: luisfe <lftab...@yahoo.es>
Date: Wed, 12 Oct 2011 02:45:48 -0700 (PDT)
Local: Wed, Oct 12 2011 5:45 am
Subject: Re: Quotients of polynomials

Urs Hackstein wrote:
> Unfortunately, I can't recognize any nice mathematical structure in
> the form f is given. It is given not exactly as an continous fraction,
> but it is given as a big (in the sense of many variables and terms)
> fraction where both denominator and numerator are "complicated"
> functions containing big fractions where both denominator and
> numerator are .. and so on.
> f is not an object of pure mathematical interest, it arise from some
> calculation in electronical engineering. But engineers told me that it
> is known by theoretical results that it could be written as the
> quotient of polynomials.

Umm, the solution depends a lot on the exact form that f is given and
what the "complicated" functions are.

Do you have a previous bound of the numerator and denominator of f?
without a theoretical bound on numerator and denominator I guess you
are doomed. You will be able to approximate f though.

Can you evaluate f in many point? Is this evaluation exact or
approximate? That is, if I compute f(2) will it be a rational number,
a float point?

This question is better placed in sage-support.

To post a message you must first join this group.
You do not have the permission required to post.
More options Oct 12 2011, 8:27 am
Date: Wed, 12 Oct 2011 05:27:02 -0700 (PDT)
Local: Wed, Oct 12 2011 8:27 am
Subject: Re: Quotients of polynomials
My membership in sage-support ist still pending.

"Complicated" means here that the whole term is more than three pages
long. Starting from a fraction with products of constants, values and
variables in the numerator and denominator you subsitute some of the
values in the numerator and the denominator by fractions with products
of constants, values and variables, then you substitute some of the
variables in all of the numerators and denominators by fractions
with ... again and so on. Thus the only pattern of the form f is given
is that there are only basic arithmetic operations in its definition.

Of course, I could choose a set of numerical values for our constants
and evaluate f in some points (this gives us float points) and get the
coefficients by interpolation. But we are looking for a description of
the coefficients as functions of some other variables, so numerical
values for the coefficients don't help us.

By the way, there is another question more elementary: How can I
compute in \mathbb{C}(t) (i.e. the quotient field of the polynomial
ring) using sage?

On Oct 12, 11:45 am, luisfe <lftab...@yahoo.es> wrote:

To post a message you must first join this group.
You do not have the permission required to post.
More options Oct 12 2011, 8:50 am
From: Florent Hivert <Florent.Hiv...@lri.fr>
Date: Wed, 12 Oct 2011 14:50:15 +0200
Local: Wed, Oct 12 2011 8:50 am
Subject: Re: [sage-algebra] Re: Quotients of polynomials

Nothing particular to do: just divide polynomials, the fraction field is
automatically created.

sage: P.<t> = CC[]
sage: P
Univariate Polynomial Ring in t over Complex Field with 53 bits of precision
sage: F = (t/t).parent()
sage: F
Fraction Field of Univariate Polynomial Ring in t over Complex Field with 53 bits of precision

You can alternatively define

sage: P.fraction_field()
Fraction Field of Univariate Polynomial Ring in t over Complex Field with 53
bits of precision

Cheers,

Florent