#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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Changes (by ketzu):
* keywords: => matrix
* priority: major => minor
* component: PLEASE CHANGE => linear algebra
Old description:
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:2>
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