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 |
> 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 |
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 _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math |
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 |
> 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 |
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 |
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 |
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 |
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 _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math |
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 _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math |
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? _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math |
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