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Science Forum Index » Math - Symbolic Forum » factoring lists in Singular...
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| ... |
 Posted: Tue May 13, 2008 8:30 pm |
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c.ro... at (no spam) espero.org.uk schrieb:
Quote:
Finally, two general questions:
(1) in my simple example (7 May, 8:40pm), the factorize(resultant())
command yields four solutions with a multiplicity of 1,7,1 and 1,
respectively (10 May, 9:31pm). Summing these yields 10, which is one
more than what I would have expected, by Bezout's Theorem, from the
intersection of two third order polynomials.
There are only three distinct solutions to your simple example
[v11h=0, v21h=0]:
[v11h := b^3+v^3-2*b*v, v21h := v^3-3*b*v^2+b^2*v+2*b*v-b^3]
GROEBNER_BASIS([v11h,v21h],[b,v])
[9*v^6-13*v^5+4*v^4, 10*b*v^2+63*v^5-73*v^4, 10*b^2*v+189*v^5-~
219*v^4+20*v^3, b^3-2*b*v+v^3]
" the 1st term has only three distinct solutions "
FACTOR(9*v^6-13*v^5+4*v^4,Rational)
v^4*(v-1)*(9*v-4)
" accordingly: "
SOLVE([v11h=0,v21h=0],[b,v])
[b=0 AND v=0, b=1 AND v=1, b=8/9 AND v=4/9]
What is your fourth solution?
Martin (the other one). |
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| Posted: Tue May 13, 2008 9:52 pm |
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On May 14, 7:30 am, cliclic... at (no spam) freenet.de wrote:
Quote: ... There are only three distinct solutions to your simple example
[v11h=0, v21h=0]: ...
[b=0 AND v=0, b=1 AND v=1, b=8/9 AND v=4/9]
What is your fourth solution?
Sorry Martin: that was careless of me. You're absolutely right. I
had included the "-2", without realising the obvious, namely that it
does not have implications for b or v.
How is a 'solution' of this sort to be interpreted? If it implies "-2
= 0" then it is clearly inadmissible.
Best,
Colin |
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| Posted: Wed May 14, 2008 1:01 am |
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On May 14, 8:52 am, c.ro... at (no spam) espero.org.uk wrote:
Quote: How is a 'solution' of this sort to be interpreted? If it implies "-2
= 0" then it is clearly inadmissible.
Apologies - another careless remark by me: "-2" is just a
multiplicative factor.
Using my more "complicated example" suggests, though, that one can't
simply add the (non-constant) multiplicities and compare the result to
the product of the degrees (a la Bezout). In that case, the
multiplicities (excluding the constant) come to 3 + 15 + 1 + 2 + 6 + 7
+ 9 + 13 = 56, which is not equal to 11 x 11.
Is there any relationship between Bezout's Theorem and the
multiplicities of the resultants at this stage, or must one perform
the whole elimination/extension procedure first?
Thank you again,
Colin |
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| Waldek Hebisch... |
| Posted: Wed May 14, 2008 5:44 pm |
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c.rowat at (no spam) espero.org.uk wrote:
Quote:
Walter: why is it easier to generate Groebner bases of the ideals
generated by <v21l, fi> than it is to work with <v21l, v11l>?
After factoring we have polynomials of low degree, which are easy to
handle.
I am not aware of a text explicitely comparing FriCAS and Singular.
I can offer you may biased opinion (I work on FriCAS, while all
know about Singular comes from reading about it): FriCAS is a
general purpose system while Singular is oriented towards
algebraic geometry.
FriCAS has pretty wide coverage -- look at introductory chapter
in Axiom book (you can find .pdf for online reading at
http://fricas.sourceforge.net/doc/book.pdf) for a sample of possibilities.
When solving polynomial equations FriCAS offers classical resultants,
Groeber bases and triangular systems. Another possibility is to
use power (or Laurent or Puiseux) series to get approximate solutions.
Singular is pretty good at computing Groeber bases, and more
general operating on multivariate polynomials. It probably has
more functions to compute some specific properties of ideals.
However, a it is hard to give a simple and accurate statement.
For example, I found on the net a few examples of polynomials
that take quite a lot of time to factor using Singular --
I have tried them in FriCAS and pretty quickly obtained the
anwer. This is _not_ to claim that FriCAS is faster, rather,
to disprove claim "Singular has very fast polynomial
multiplication, so it must be faster at factoring polynomials"
Quote: Even more complicated examples, though, may yield higher order terms,
preventing explicit expressions. Can I still use the subst()
command? If not, are there workarounds?
It really depends on your data: sometimes solutions of polynomial
systems have "nice" parametrization. However, you may have
singularities or you may have parametrization which is regular
from analytic point of view, but does not admit simple
formula. Technics like resultunts, Groebner bases or
triangular system can be used to decompose solution set into
irreducible pieces, find dimension, qualitatively analyse singularites
etc.
--
Waldek Hebisch
hebisch at (no spam) math.uni.wroc.pl |
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| William stein... |
| Posted: Thu May 15, 2008 3:55 am |
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On May 14, 8:44 pm, Waldek Hebisch <hebi... at (no spam) math.uni.wroc.pl> wrote:
Quote: Singular is pretty good at computing Groeber bases, and more
general operating on multivariate polynomials.
Overall Singular is probably the best free open source math software
system for computing Groebner basis over fields. It is slow in
some cases, of course. Cocoalib is also rumored to be pretty
good, but I haven't benchmarked it. Singular has unusually good
support for a wide range of term orders. If you have a *lot* of RAM,
Magma is much much faster than Singular currently at many Groebner
basis computations (Allan Steel spent about five years
implementing and heavily tuning a variant of Faugere's F4 in Magma).
Quote: It probably has
more functions to compute some specific properties of ideals.
It has *tons* of such functions, motivated by a wide range of
research applications of algebraic geometry.
Quote: However, a it is hard to give a simple and accurate statement.
For example, I found on the net a few examples of polynomials
that take quite a lot of time to factor using Singular --
You might be referring to all the threads about this on sage-devel?
Quote: I have tried them in FriCAS and pretty quickly obtained the
anwer. This is _not_ to claim that FriCAS is faster, rather,
to disprove claim "Singular has very fast polynomial
multiplication, so it must be faster at factoring polynomials"
Multivariate polynomial factorization in Singular is very
sketchy. In fact, as far as I can tell, general multivariate
factorization is comparably bad still in all open source math
software. I say "bad", because Magma blows everything
else away speedwise at polynomial factorization. See these
trac tickets for some discussion of major efficiency issues
with Singular as compared to Magma:
"singular factorize is randomly slow"
http://trac.sagemath.org/sage_trac/ticket/1343
"multivariate polynomial factorization over GF(p) (sucks)"
http://trac.sagemath.org/sage_trac/ticket/2152
I've been told some of the Singular group is actively
working on improving their polynomial factorization; I hope
they succeed. There is also at least one Sage developer
(Joel Mohler) working on polynomial factorization for Sage.
I don't know what the latest status of this is.
-- William |
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