On 16 Jul., 15:20, Bjarke Hammersholt Roune <bjarke.ro...@gmail.com>
wrote:
> Frobby is currently an optional component of Sage, which performs
> computations related to monomial ideals. In particular, it can compute
>
> * Multigraded Hilbert series
> * Alexander dual of monomial ideals
> * Maximal standard monomials of monomial ideals
> * Irreducible decomposition of monomial ideals
> * Optimization of any linear function over the maximal standard
> monomials of a monomial ideal using branch-and-bound.
>
> Sage currently is not able to do any of this. Sage does have the less
> general (1,..., 1)-graded Hilbert series using Singular, but it
> doesn't support arbitrarily large exponents as Frobby does.
>
> Applications of the last item above include Frobenius numbers for
> instances with very large numbers (with 4ti2), as demonstrated at Sage
> Days 16, and described in (1) below. Another application is the
> integer programming gap (also with 4ti2) of a matrix where the right-
> hand-side is allowed to vary as described in (2).
>
> This is put up for a vote now since I wrote a cython interface to
> Frobby at Sage Day 16, and I'm told this requires Frobby to be a
> standard component of Sage.
>
> Frobby is fastest at items 2-4 listed above as documented in (3)
> below, by factors of up to 1000x, with the exception of specially-
> constructed inputs (in particular taking the dual of a dual to recover
> the original ideal). Item 1 is Hilbert series, where CoCoALib might be
> faster right now, since the algorithm I use is for now unpublished,
> and I haven't compared it to CoCoALib yet. In any case I will also
> implement the Bigatt et.al. algorithm that CoCoALib uses, though this
> is not done yet.
>
> Frobby has an extensive test-suite, which includes running Frobby
> under valgrind to detect memory leaks, and is supported for Mac OS
> 10.5, Linux and Cygwin. It compiles using MS Visual Studio Express,
> though I haven't tested it on that platform since I couldn't get GMP
> to build on Windows. GMP is the only dependency Frobby has other than
> a C++ compiler. The build system is make-based. I am the upstream
> contact, and Frobby is licensed as GPL version 2.0 or later.
>
> Cheers
> Bjarke Hammersholt Rounewww.broune.com
>
> (1) Bjarke H. Roune, Solving Thousand Digit Frobenius Problems Using
> Grobner Bases
> Journal of Symbolic Computation (January 2008), volume 43, issue 1
> Seehttp://www.broune.com/papers/index.html
>
> (2) Hocsten, S., Sturmfels, B., 2007. Computing the integer
> programming gap.
> Combinatorica 27 (3).
> Seehttp://arxiv.org/abs/arXiv:math/0301266
>
> (3) Bjarke H. Roune, The Slice Algorithm For Irreducible Decomposition
> of Monomial Ideals
> Journal of Symbolic Computation (April 2009), volume 44, issue 4
> Seehttp://www.broune.com/papers/index.html
Hi,
+1
to essentially add Frobby to Sage. There might be some issues left to
be weeded out, but all these should be easily addressable (famous last
words).
Cheers,
Georg
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