------- Forwarded message -------
From: "Douglas Lind" <lind@xxxxxxxxxxxxxxxxxxx>
To: "Michael Abshoff" <Michael.Abshoff@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>
Cc: "William Stein" <wstein@xxxxxxxxxxxxxxxxxxx>, boothby@xxxxxxxxxxxxxxxx
Subject: [sage-devel] Fwd: Gröbner Basis computation over ZZ (fwd)
Date: Thu, 08 Feb 2007 19:03:29 -0700
Dear Michael,
The recent question of Groebner bases over ZZ probably
originated with me, so let me tell you why this is interesting to me
(and others). Given a polynomial in d variables with ZZ coefficients, or
more generally an ideal in the polynomial ring ZZ[x_1,...,x_d], you can
obtain directly from Pontryagin duality an action of ZZ^d on the compact
abelian dual group by continuous automorphisms. The use of integral
coefficients is crucial, since the dual of ZZ is the unit circle, and
allowing the standard fields for coefficients breaks things down and you
lose compactness among other things. There is a wonderful interplay
between dynamics on the one hand, and the commutative algebra with ZZ
coefficients on the other. For example, entropy (a dynamical concept) is
computed in these cases using Mahler measure of polynomials.
There is an entire book about this by Klaus Schmidt in Vienna,
called "Dynamical Systems of Algebraic Origin".
If one wants to compute with these dynamical systems, the one
immediately starts to think in terms of Groebner bases over ZZ. It would
be very convenient to have a package which did these computations.
Let me mention a typical problem, which has dynamical origins
but can be stated completely inside the algebra. Let P be a prime ideal
in ZZ[x_1^{\pm 1}, ..., x_d^{\pm 1}] be the ring of Laurent polynomials.
Is there an algorithm to decide whether or not P contains a "binomial",
i.e. a polynomial of the form x_1^{n_1}...x_d^{n_d} - 1, where the
n_1, ..., n_d are integers, not all zero?
In any event, should your team implement Groebner bases over ZZ,
I'd be very grateful.
Best regards,
Doug Lind
---------- Forwarded message ----------
Date: Thu, 08 Feb 2007 17:45:16 -0700
From: William Stein <wstein@xxxxxxxxx>
Reply-To: sage-devel@xxxxxxxxxxxxxxxx
To: "sage-devel@xxxxxxxxxxxxxxxx" <sage-devel@xxxxxxxxxxxxxxxx>
Subject: [sage-devel] Fwd: Gröbner Basis computation over ZZ
Can anybody who cares about Groebner Basis over ZZ pipe up!
------- Forwarded message -------
From: "Michael Abshoff" <Michael.Abshoff@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>
To: wstein@xxxxxxxxx
Cc:
Subject: Gr�bner Basis computation over ZZ
Date: Thu, 08 Feb 2007 16:18:33 -0700
Dear William,
I hope you got my last EMail - I am sure you are quite busy at the moment.
We had some discussion about Gr�bner Bases over ZZ here:
http://cocoa.mathematik.uni-dortmund.de/forum/viewtopic.php?t=624
John's time estimate is probably realistic since we (at least the 'we' in
Dortmund :)) couldn't find an application for that sort of computation, so
could you please enlighten us why one would need to compute GBases over
rings.
I don't think it is particularly difficult to implement, but without an
application it is rather low on the list of priorities.
Cheers,
Michael
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