# Polynomdivision with coefficients in Z_n Classic List Threaded 11 messages Open this post in threaded view
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## Polynomdivision with coefficients in Z_n

 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 _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math
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## Re: Polynomdivision with coefficients in Z_n

 > 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 _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math
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## Re: Polynomdivision with coefficients in Z_n

 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/SandBoxPolynomialOverFiniteFieldRalf _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math
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## Re: Polynomdivision with coefficients in Z_n

 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? _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math
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## Re: Polynomdivision with coefficients in Z_n

 > 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 _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math
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## Re: Polynomdivision with coefficients in Z_n

 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 _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math
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## Re: Polynomdivision with coefficients in Z_n

 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. _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math
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## Re: Polynomdivision with coefficients in Z_n

 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 _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math
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## Re: Polynomdivision with coefficients in Z_n

 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/SandBoxPolynomialOverFiniteFieldIf 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/LanguageDifferencesI guess, meanwhile you have already read the book http://fricas.github.io/book.pdfRalf _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math