Hi Florent! On Nov 26, 9:24 am, Florent Hivert <florent.hiv...@univ-rouen.fr> wrote: [...] > I don't understand why what you say here is an answer to the following > sentence of mine: > > Is there a problem in Symmetric Ideals if you have unused variables ?
You will always have an infinity of unused variables, of course: In each actual computation, you will use only finitely many of them. And then the problem is, how to implement it? Since part of the Aschenbrenner-Hillar algorithm involves *classical* (finite) Groebner basis computation, it seemed reasonable to have finite polynomial rings as underlying structure, rather than symbolic variables. The underlying finite polynomial ring should of course contain all variables that are used. In the sparse implementation, the underlying ring will contain *only* the used variables. In the dense implementation, it will contain all variables that have at most the highest used index. Example: sage: X.<x,y> = InfinitePolynomialRing(RR,implementation='sparse') sage: t=x[5] sage: t.polynomial().parent() Univariate Polynomial Ring in x5 over Real Field with 53 bits of precision sage: Y.<a,b> = InfinitePolynomialRing(RR) sage: s=a[5] sage: s.polynomial().parent() Multivariate Polynomial Ring in b5, b4, b3, b2, b1, b0, a5, a4, a3, a2, a1, a0 over Real Field with 53 bits of precision But when you start doing arithmetic, the underlying ring will, in general, change. And this is also the case when you let the symmetric group over the natural numbers act on the ring: sage: p=Permutation((5,10)) sage: (t^p).polynomial().parent() Multivariate Polynomial Ring in x10, x5 over Real Field with 53 bits of precision sage: (s^p).polynomial().parent() Multivariate Polynomial Ring in b10, b9, b8, b7, b6, b5, b4, b3, b2, b1, b0, a10, a9, a8, a7, a6, a5, a4, a3, a2, a1, a0 over Real Field with 53 bits of precision You see why the sparse implementation is called "sparse". __Consequences:__ - The sparse implementation has the disadvantage that, on average, you will more often change the underlying finite polynomial ring. That slows it down. - The dense implementation in some applications has the disadvantage that the underlying ring is too big. Does this answer your question? Cheers, Simon -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
