Polynomdivision with coefficients in Z_n

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Polynomdivision with coefficients in Z_n

panAxiom
Hi,


how to divide polynoms, which have coefficients from Z_2, Z_3 and so on?

Normal polynoms can be divided just by dividing them.

But if the coefficients are modulo n, how to specify this?


TIA,
   pan



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Re: Polynomdivision with coefficients in Z_n

Ralf Hemmecke-4
> how to divide polynoms, which have coefficients from Z_2, Z_3 and so on?

Please give an example of what your input is and what you expect as output.

Ralf

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Re: Polynomdivision with coefficients in Z_n

Ralf Hemmecke-4
In reply to this post by panAxiom
> But if the coefficients are modulo n, how to specify this?

Does that help?

http://axiom-wiki.newsynthesis.org/SandBoxPolynomialOverFiniteField

Ralf

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Re: Polynomdivision with coefficients in Z_n

panAxiom

Quoting Ralf Hemmecke <[hidden email]>:

>> But if the coefficients are modulo n, how to specify this?
>
> Does that help?
>
> http://axiom-wiki.newsynthesis.org/SandBoxPolynomialOverFiniteField
[...]

It looks, like if it is going into the right directions

FiniteField looks good, maybe FiniteRing also available?


What I'm looking for, is  polynoms with modulo-coefficients.


Say Z_2 = { 0, 1 }
And the coefficients a_i of a polynom   e.g.  a2 * t^2 + a1 * t + a0
can only have values of 0 an 1, which are the representations of  
aequivalence classes.

So, -1 = 1, -2 = 0,  2 = 0, 3 = 1,  4 = 0, and so on

Aequivalence classes in Z2 (LaTeX:   \mathbb{Z}_2)
...
-4 -3
-2 -1
  0  1
  2  3
  4  5
  6 7
....

And 0 and 1 are the usual representations of these classes.

And the coefficients can have only values from these classes.
So, it bahaves as if they can have only values 0 and 1.


In Z_3, it would be 0,1,2
in Z_4, it would be 0,1,2,3
...

any ideas about that?




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Re: Polynomdivision with coefficients in Z_n

Ralf Hemmecke-4
> It looks, like if it is going into the right directions
>
> FiniteField looks good, maybe FiniteRing also available?

Well... don't you have HyperDoc->Browse available?
Sorry, I cannot point you to an online API for AXIOM, but I'm
maintaining an API for FriCAS (which is relatively close.

http://fricas.github.io/

You are probably looking for

http://fricas.github.io/api/IntegerMod.html

> What I'm looking for, is  polynoms with modulo-coefficients.

Huh? For primes p that's exactly what I've given you. Since then Z_p is
a field.

For non-prime-powers use IntegerMod.

Ralf

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Re: Polynomdivision with coefficients in Z_n

Ralf Hemmecke-4
On 11/25/2014 12:17 AM, Ralf Hemmecke wrote:
> For non-prime-powers use IntegerMod.

Oh, but wait... you still have not specified what you meant by "to
divide polynoms".

Do you know that Z_6[x] has zero divisors. Try to compute
(2*x-4)*(3*x+3). So what is 0/(3*x+3)? Is it 2 or 4 or 2*x+2 or 2*x^2+4
or ...?

Ralf


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Re: Polynomdivision with coefficients in Z_n

panAxiom

Quoting Ralf Hemmecke <[hidden email]>:

> On 11/25/2014 12:17 AM, Ralf Hemmecke wrote:
>> For non-prime-powers use IntegerMod.
>
> Oh, but wait... you still have not specified what you meant by "to
> divide polynoms".


I meant something like


  (t^2 + t) / t  = t + 1



or in fricas:


(6) -> (t^2+t) / t

    (6)  t + 1
                                           Type: Fraction(Polynomial(Integer))
(7) ->


But the modulo part was, what I did not knew how to specify.




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Re: Polynomdivision with coefficients in Z_n

panAxiom
In reply to this post by Ralf Hemmecke-4

Quoting Ralf Hemmecke <[hidden email]>:

>> It looks, like if it is going into the right directions
>>
>> FiniteField looks good, maybe FiniteRing also available?
>
> Well... don't you have HyperDoc->Browse available?
> Sorry, I cannot point you to an online API for AXIOM, but I'm
> maintaining an API for FriCAS (which is relatively close.
>
> http://fricas.github.io/
>
> You are probably looking for
>
> http://fricas.github.io/api/IntegerMod.html
>
>> What I'm looking for, is  polynoms with modulo-coefficients.
>
> Huh? For primes p that's exactly what I've given you. Since then Z_p is
> a field.
>
> For non-prime-powers use IntegerMod.


OK, thanks.

I hope I can find out how to use this functionality.
(Examples would be fine btw.)



pan


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Re: Polynomdivision with coefficients in Z_n

Ralf Hemmecke-4
Hi Oliver,

> I hope I can find out how to use this functionality.
> (Examples would be fine btw.)

I thought it would be too easy to replace

  K ==> PrimeField p

by

  K ==> IntegerMod p

and change the p to your liking. But OK, I've added a bit more stuff for
you.

http://axiom-wiki.newsynthesis.org/SandBoxPolynomialOverFiniteField

If you are interested in the language, you should perhaps take a look
into http://www.aldor.org/docs/aldorug.pdf. Aldor is not exactly like
SPAD or the input language of AXIOM, but it's pretty close to learn the
principles.

http://axiom-wiki.newsynthesis.org/LanguageDifferences

I guess, meanwhile you have already read the book
http://fricas.github.io/book.pdf

Ralf


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Re: Polynomdivision with coefficients in Z_n

panAxiom

Quoting Ralf Hemmecke <[hidden email]>:

> Hi Oliver,
>
>> I hope I can find out how to use this functionality.
>> (Examples would be fine btw.)
>
> I thought it would be too easy to replace
>
>   K ==> PrimeField p
>
> by
>
>   K ==> IntegerMod p
>
> and change the p to your liking. But OK, I've added a bit more stuff for
> you.

I hope it's not only for me...


Regarding documentation, it's always potentially misleading, if it's  
not clear, how it is meant.


And what is meant with "==>" ?

When looking up the documentation, it first pops up at page 935 in  
bookvol0.pdf
and it's not explained there, just used.

So I think some potential confusion can pop up here.


Regarding your pointers to further documentation I thank you.
I will look it up.


pan




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Re: Polynomdivision with coefficients in Z_n

Ralf Hemmecke-4
Hi Oliver,

> I hope it's not only for me...

Well, I wrote it for you. If others find it useful, then it's even better.

> Regarding documentation, it's always potentially misleading, if it's not
> clear, how it is meant.

True. The original Axiom-Book is not very precise in defining the
language SPAD. For learning the language you should rather read the
Aldor User Guide. I've given you the link in my last mail. If you refuse
to read, i.e. do the homework, I'll stop responding.

> And what is meant with "==>" ?

http://www.aldor.org/docs/HTML/chap12.html

> When looking up the documentation, it first pops up at page 935 in
> bookvol0.pdf and it's not explained there, just used.

> So I think some potential confusion can pop up here.

Please be precise about the confusion. If I don't know what you are
complaining about, I cannot help.

Ralf

PS: May I ask how you came to AXIOM and what you intend to use it for?


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