Sylvester matrix for polynomials

[sagetrac-carlohamalainen]

Small patch to add Sylvester matrix calculation for univariate and multivariate polynomials.

I think that my patch is a bit more general (and has doctests) compared to didier deshommes' patch here, which seems to have never been merged:

http://sage.math.washington.edu/home/dfdeshom/custom/patches/sylveste...

Component: commutative algebra

Keywords: Sylvester matrix

Author: Carlo Hamalainen, Luis Felipe Tabera Alonso

Reviewer: David Loeffler

Merged: sage-4.6.2.alpha3

Issue created by migration from https://trac.sagemath.org/ticket/7458

[lftabera]

Attachment: sylvester.patch.gz

[lftabera]

comment:1

Attachment: trac-7458-sylvester-rebase-4.6.patch.gz

This is a very basic feature that has to be in Sage.

I have rebased Carlo patch to 4.6 but have not touched the code.

I have some concerns that makes me mark the patch as needs work:

• The univariate case should accept the same syntax as the multivariate case. In the univariate case, the preferred call is f.sylvester_matrix(g), but I do not want Sage to throw an error if I wrote f.sylvester_matrix(g, x)

• Corner cases must be well documented.

sage: K.<x>=QQ[]
sage: K(1).sylvester_matrix(K(1))
[]

In particular, I am not sure how to deal with the sylvester matrix of 0 and constant or 0 and 0
Curretly it throws an error. My opinion is that this is not defined but should throw a more meaningful error.

Maple for instance return the empty matrix. So in maple:

Determinant(Sylvester_Matrix) != Resultant

In this corner cases.

I will try to check what other CAS do to get a wider picture.

• An example explicitly relating Sylvester matrix and resultant should be added.

[lftabera]

Changed keywords from none to Sylvester matrix

[lftabera]

Changed author from carlohamalainen to Carlo Hamalainen

[lftabera]

comment:2

• I have added more documentation and doctest.

• The variable argument is now optional in both univariate and multivariate. If it is not used, the first variable of the polynomial ring is used.

• I have added coercion to be able to compute the Sylvester matrix of polynomials in different rings, for ex. ZZ[x] and QQ[x]

• solved an issue for the dimension of the matrix of the sylvester matrix of (x**n, 0)

It is not ready for review because the sylvester matrix of (0,0) is not implemented.

[lftabera]

comment:3

Attachment: trac-7458-sylvester-improvements.patch.gz

Finally, if one of the polynomials is zero, the code raises a ValueError.

[lftabera]

Changed author from Carlo Hamalainen to Carlo Hamalainen, Luis Felipe Tabera Alonso

[lftabera]

comment:4

Apply trac-7458-sylvester-rebase-4.6.patch, trac-7458-sylvester-improvements.patch

[loefflerd]

comment:5

Very nice. I am impressed with the thoroughness of the testing of corner cases here. All doctests in sage/rings pass, and the reference manual builds OK.

[loefflerd]

Reviewer: David Loeffler

[jdemeyer]

Merged: sage-4.6.2.alpha3