Splitting Algebras

[soehms]

Splitting algebras have been considered by Dan Laksov, Anders Thorup, Torsten Ekedahl and others in order to study intersection theory of Grassmann and other flag schemes. Similarily as splitting fields they can be considered as extensions of rings containing all the roots of a given monic polynomial over that ring under the assumption that its Galois group is the symmetric group of order equal to the polynomial's degree.

Thus they can be used as a tool to express elements of a ring generated by n indeterminates in terms of symmetric functions in these indeterminates.

This ticket realizes splitting algebras by a recursive quotient ring construction splitting off some linear factor of the polynomial in each recursive step. Accordingly it will be inherited from class PolynomialQuotientRing.

CC: @tscrim

Component: commutative algebra

Keywords: splitting polynomial roots

Author: Sebastian Oehms

Branch/Commit: e52eece

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/29716

[soehms]

Branch: u/soehms/splitting_algebra

[soehms]

New commits:

f1bb41e

29716 initial version

[soehms]

Commit: f1bb41e

[soehms]

Author: Sebastian Oehms

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

ee46039	Merge branch 'u/soehms/splitting_algebra' of git://trac.sagemath.org/sage into splitting_algebra_29716
2f0c99c	29716 rebasing and pyflakes hints

[sagetrac-git]

Changed commit from f1bb41e to 2f0c99c

[sagetrac-git]

Changed commit from 2f0c99c to 5ea214f

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

8c2a4cc	Merge branch 'u/soehms/splitting_algebra' of git://trac.sagemath.org/sage into splitting_algebra_29716
5ea214f	29716: rebase to 9.2.beta0 and some style fixes

[tscrim]

Changed branch from u/soehms/splitting_algebra to public/algebras/splitting_algebra-29716

[tscrim]

comment:5

I made it through the ticket, and mostly I changed some minor things. However, there is one more serious issue that I don't know how to fix:

sage: from sage.algebras.splitting_algebra import SplittingAlgebra
....: L.<u, v, w> = LaurentPolynomialRing(ZZ); x = polygen(L)
....: S.<X, Y> = SplittingAlgebra(x^3 - u*x^2 + v*x - w)
....: (X, Y)
....: 
(Y, X)

The names of the generators do not match, but I don't know how much confusion this will cause to users or how much this even makes sense. Could you clarify about what you expect to happen here?

---

New commits:

c3fe0da

Some mild PEP8 stuff and other formatting stuff for splitting algebras.

[tscrim]

Changed commit from 5ea214f to c3fe0da

[tscrim]

Reviewer: Travis Scrimshaw

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

0ec9ab1	Merge branch 'public/algebras/splitting_algebra-29716' of git://trac.sagemath.org/sage into splitting_algebra_29716
e52eece	29716: correct _first_ngens

[sagetrac-git]

Changed commit from c3fe0da to e52eece

[soehms]

comment:7

Replying to @tscrim:

I made it through the ticket, and mostly I changed some minor things.

Thanks for all these PEP8 corrections! I'm suprised that the two missing supression of the warning didn't cause a doctest failure.

However, there is one more serious issue that I don't know how to fix:

sage: from sage.algebras.splitting_algebra import SplittingAlgebra
....: L.<u, v, w> = LaurentPolynomialRing(ZZ); x = polygen(L)
....: S.<X, Y> = SplittingAlgebra(x^3 - u*x^2 + v*x - w)
....: (X, Y)
....: 
(Y, X)

The names of the generators do not match, but I don't know how much confusion this will cause to users or how much this even makes sense. Could you clarify about what you expect to happen here?

I didn't realize the exchange of the indeterminates and this is definitely not intended. The reason for this is that I provided the _first_ngens by means of the recursive construction. But, since X is adjoined first and than Y this lead to the wrong order.

I dropped the method gens_recursive since it is needed nowhere else and replaced its invocation by the usage of the first n splitting_roots which are in the correct order.

I omit a separate test for this bug, since it is implicit in this change

-            (X + 1)*Y
+            X*Y + X

[tscrim]

comment:8

Thank you. LGTM now.

[vbraun]

Changed branch from public/algebras/splitting_algebra-29716 to e52eece