#16739: is_weak_popov
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Reporter: ketzu | Owner:
Type: PLEASE CHANGE | Status: new
Priority: minor | Milestone: sage-6.4
Component: linear algebra | Resolution:
Keywords: matrix | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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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.
> 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. 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.
--
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Ticket URL: <http://trac.sagemath.org/ticket/16739#comment:3>
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