#5566: [with patch, needs review] Symmetric Groebner bases and Infinitely
Generated Polynomial Rings
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Reporter: SimonKing | Owner: SimonKing
Type: enhancement | Status: new
Priority: major | Milestone: sage-4.0
Component: commutative algebra | Keywords: Symmetric Ideal
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Comment(by SimonKing):
The new patch (still self-contained) should apply to sage-3.4.1.rc3
Let me address your comments:
* I built the documentation, and it looks OK to me.
* Now, I have separate dense and sparse implementations for
{{{InfinitePolynomial}}}s.
* I changed {{{sco}}} into {{{symmetric_cancellation_order}}} and
{{{AddGenerator}}} into {{{add_generator}}}.
'''__New Feature__'''
* I added a {{{__getattr__}}} method to {{{InfinitePolynomial}}}s that
forwards to the underlying finite polynomials. Consequence: Any attribute
of the underlying finite polynomial is directly available to the Infinite
Polynomial, unless that method is overwritten. In that way, I
automatically have latex typeset for the Infinite Polynomials.
* A propos latex: I also added latex methods for Infinite Polynomial
Rings and Symmetric Ideals.
Latex looks like this:
{{{
sage: X.<x,y> = InfinitePolynomialRing(QQ)
sage: latex(x)
x_{\ast}
}}}
I think this makes sense. After all, {{{x}}} generates a series of
variables, indexed by natural numbers, and in this situation it is common
to use a star as index. Similarly:
{{{
sage: latex(X)
\mathbf{Q}[x_{\ast}, y_{\ast}]
sage: latex(x[2]*y[0]+3*x[4]+1)
y_{0} x_{2} + 3 x_{4} + 1
sage: latex(X*(x[1]*y[2]))
\left(y_{2} x_{1}\right)\mathbf{Q}[x_{\ast},
y_{\ast}][\mathfrak{S}_{\infty}]
}}}
Perhaps the last line requires some explanation.
1. G=S_infinity shall denote the symmetric group over the natural
numbers. I hope this is understandable.
2. R=Q[x_*,y_*] is the infinite polynomial ring.
3. R[G] is the group ring of G with coefficients in R.
4. A symmetric ideal is nothing but a R[G] module. Since G acts from the
right, we thus have the above notation.
Rather huge patch, sorry...
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/5566#comment:33>
Sage <http://sagemath.org/>
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