#5453: [with patch, needs review] Create a ring for working with polynomials in
countably infinitely many variables
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Reporter: mhansen | Owner: mhansen
Type: enhancement | Status: assigned
Priority: major | Milestone: sage-3.4.2
Component: commutative algebra | Keywords:
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Comment(by SimonKing):
Hi!
I just had a look at the abstract of Aschenbrenner's and Hillar's
articles. The title of one article is "An Algorithm for Finding Symmetric
Gröbner Bases in Infinite Dimensional Rings". They talk about "symmetric
ideals", but again they say "infinite dimensional polynomial ring", not
"symmetric polynomial ring".
This convinces me that Mike's InfinitePolynomialRing is better than
SymmetricPolynomialRing, and that the word "symmetric" should be reserved
to the ideals and Gröbner bases of that ring.
Here some thoughts about a unification:
Both in Mike's and my implementation, an element p of the ring R
essentially has an attribute _p, which is a 'usual' polynomial: p._p gives
the value of p.
In Mike's implementation, the ring R has and attribute P (a polynomial
ring) so that p._p is guaranteed to be coercible into P, for any `p \in
R`. This allows for usual arithmetic.
In my implementation, it is possible that coercion does not provide
arithmetic for p._p and q._p, for p and q in R. ''Only'' if it does not,
then i explicitly construct a common parent for p._p and q._p. So, Mike's
arithmetic is a special case of mine.
Do we agree on the following technical details of a potential unification?
* Compared with SymmetricPolynomialRing_class,
InfinitePolynomialRing_class has one additional attribute `P`, and the
arithmetic of InfinitePolynomial is just a special case of the arithmetic
of SymmetricPolynomial. So, the init method might have an optional
parameter `implementation`, and if this is 'dense' then the ring is
initialized together with an attribute `P`. Otherwise, it is without `P`.
* In the methods of InfinitePolynomial and SymmetricIdeal, we could test
whether the underlying infinite ring has an attribute P. If it has (dense
implementation), the methods choose your algorithm, otherwise they choose
mine.
In the end, we would have a dense and a sparse implementation, and the
computation of symmetric Gröbner bases would be possible in ''both''
cases.
Do you think it works? How should the work be organised? Here? #5566? New
ticket?
Cheers
Simon
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/5453#comment:29>
Sage <http://sagemath.org/>
Sage - Open Source Mathematical Software: Building the Car Instead of
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