#16739: is_weak_popov
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Reporter: ketzu | Owner:
Type: enhancement | Status: new
Priority: minor | Milestone: sage-6.4
Component: linear algebra | Resolution:
Keywords: matrix weak- | Merged in:
popov-form #16742 | Reviewers:
Authors: | Work issues:
Report Upstream: N/A | Commit:
Branch: | 37898695c0dc92f7e9cd1d0a99fe63c854e18253
u/ketzu/is_weak_popov | Stopgaps:
Dependencies: |
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Description changed by ketzu:
Old description:
> The target of this ticket is to provide/add a function is_weak_popov to
> the matrix interface. The function should return true if the matrix it is
> called on is in weak popov form. This ticket is independent from but
> connected to !#16742.
>
> Short description of weak popov form: Let R be an ordered Ring and A^mxn^
> a matrix over R. The leading position of a row is called the position i
> in [1,m) such that the order of A[i,_] is maximal within the row. If
> there are multiple entries with the maximum order, the highest i is the
> leading position (the furthest to the right in the matrix). A is in weak
> popov form if all leading positions are different (zero lines ignored).
>
> The function will implement this only for polynomial rings, the order
> function is the degree of the polynomial. Example:
>
> [x^2^+1, x]
>
> [x, x+1]
>
> is in weak popov form: Row 1 has the degrees 2 and 1, the leading
> position is for i=0, row 2 has two times degree 1 so the higher i is
> chosen with i=1.
>
> [x^2^+1, x]
>
> [x,0]
>
> is NOT in weak popov form, row 1 has now degrees 1 and -1, so the leading
> position is i=0 as in row 1.
New description:
The target of this ticket is to provide/add a function is_weak_popov to
the matrix interface. The function should return true if the matrix it is
called on is in weak popov form. This ticket is independent from but
connected to [http://trac.sagemath.org/ticket/16742 #16742].
Short description of weak popov form: Let R be an ordered Ring and A^mxn^
a matrix over R. The leading position of a row is called the position i
in [1,m) such that the order of A[i,_] is maximal within the row. If
there are multiple entries with the maximum order, the highest i is the
leading position (the furthest to the right in the matrix). A is in weak
popov form if all leading positions are different (zero lines ignored).
The function will implement this only for polynomial rings, the order
function is the degree of the polynomial. Example:
[x^2^+1, x]
[x, x+1]
is in weak popov form: Row 1 has the degrees 2 and 1, the leading position
is for i=0, row 2 has two times degree 1 so the higher i is chosen with
i=1.
[x^2^+1, x]
[x,0]
is NOT in weak popov form, row 1 has now degrees 1 and -1, so the leading
position is i=0 as in row 1.
--
--
Ticket URL: <http://trac.sagemath.org/ticket/16739#comment:11>
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