#1956: implement multivariate power series arithmetic
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Reporter: was
| Owner: pernici
Type: enhancement
| Status: needs_review
Priority: major
| Milestone: sage-4.6.2
Component: commutative algebra
| Keywords: multivariate power series
Author: Niles Johnson
| Upstream: N/A
Reviewer: Martin Albrecht, Simon King
| Merged:
Work_issues: multivariate series on 1 generator should remain different from a
univariate series |
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Comment(by pernici):
Please disregard my previous message :(
In this ticket series with total degree truncation are considered.
I would like to ask how the case of series with partial degree truncation
works.
Consider for example the product of two series in variables `x,y`
with precisions `prec_x,prec_y`; if they start with a constant term one
can
just truncate with `O(x^prec_x) , O(y^prec_y)`.
I do not see how it works if there are no constant terms; consider for
example
two series with precision 3 in both variables; add a variable `b` to track
the monomials which should be eliminated
{{{
sage: R.<x,y,b> = QQ[]
sage: p1 = x + y + x^2 + x*y + y^2 + b*(x^3 + y^3)
sage: p2 = x^2 + x*y + y^2 + b*(x^3 + y^3)
sage: p3 = p1*p2
sage: p3.coefficient({b:0})
x^4 + 2*x^3*y + 3*x^2*y^2 + 2*x*y^3 + y^4 + x^3 + 2*x^2*y + 2*x*y^2 + y^3
sage: p3.coefficient({b:1})
2*x^5 + 2*x^4*y + 2*x^3*y^2 + 2*x^2*y^3 + 2*x*y^4 + 2*y^5 + x^4 + x^3*y +
x*y^3 + y^4
}}}
`x^4, x^3*y, x*y^3, y^4` appear in `p3.coefficient({b:1})`
so they should be disregarded; the product of the series should be
{{{
3*x^2*y^2 + x^3 + 2*x^2*y + 2*x*y^2 + y^3
}}}
but I do not see how to get this simply with truncations;
neglecting `3*x^2*y^2` one would get the total degree truncation
{{{
sage: R.<x,y> = QQ[[]]
sage: p1 = x + y + x^2 + x*y + y^2 + R.O(3)
sage: p2 = x^2 + x*y + y^2 + R.O(3)
sage: p1*p2
x^3 + 2*x^2*y + 2*x*y^2 + y^3 + O(x, y)^4
}}}
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/1956#comment:77>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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