Multiplier spectra for projective morphisms

[sagetrac-gjorgenson]

Implement a function to compute the n multiplier spectra, the set of multipliers of periodic points of formal period n, of a projective morphism. This currently only makes sense for morphisms over projective space of dimension 1.

Depends on #18409

CC: @bhutz

Component: algebraic geometry

Author: Grayson Jorgenson

Branch/Commit: 6112c29

Reviewer: Ben Hutz

Issue created by migration from https://trac.sagemath.org/ticket/18443

[sagetrac-gjorgenson]

Branch: u/gjorgenson/ticket/18443

[sagetrac-git]

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

e50b845

18443: added default values to documentation

[sagetrac-git]

Commit: e50b845

[bhutz]

Reviewer: Ben Hutz

[bhutz]

comment:4

A few things here, the big one is that the multiplier spectrum is defined with multiplicities. So if a perioidic point has multiplicity > 1 its multiplier should be repeated. Such as example should be added to the doc tests.

Other minor issues:

• a positive integer, the period.

• numberfield -> number field

• n = int(n)

• change_ring now works for QQbar, so this is not needed

if not PS.base_ring() is QQbar:
 emb = PS.base_ring().embeddings(QQbar)[0]
 f = self.change_ring(QQbar,embedding=emb) # to be able to find all periodic points
else:
 f = self

• f.multiplier(P,n)[0][0] -> f.multiplier(P,n)[0,0]

• multipliers.sort() -> not needed

• sigma_invariants - 2nd senctence should say all periodic points. (same for INPUT)

• from sage.combinat.sf.sf import SymmetricFunctions -> move to top

• make len(multipliers) a constant in for loop

[sagetrac-git]

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

cc8b735

18443: Implemented changes from review.

[sagetrac-git]

Changed commit from e50b845 to cc8b735

[sagetrac-git]

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

ab5c82e	18443: Merged with latest beta
3333f6a	18443: implemented last change from reivew

[sagetrac-git]

Changed commit from cc8b735 to 3333f6a

[bhutz]

comment:8

A couple things in multiplier_spectrum

• you should check that the domain is projective space to avoid subschemes

• K = f._number_field_from_algebraics() should be outside the if

• if [1,0] is a point of multplicity you will miss this (You also only check if it is fixed not n-periodic). You could instead see if the poly F is of the form y^m*G(x,y) for some poly G, then m is the multplicity of [1,0].

• it doesn't seem like your conversion to QQbar is working. Trying your number field example, when the map is actually defined over a number field

sage: set_verbose(None)
sage: z = QQ['z'].0
sage: K.<w> = NumberField(z^4 - 4*z^2 + 1)
sage: P.<x,y> = ProjectiveSpace(K,1)
sage: H = End(P)
sage: f = H([x^2 - w/4*y^2,y^2])
sage: f.multiplier_spectra(2,False)

Not sure exactly what you should do here. One option is to get change_ring for polynomials to deal with QQbar, by passing in and/or creating an embedding. If you look in schemes.generic.morphism.SchemeMorphism_polynomial.change_ring() it gives a simple way to construct a hom of polynomials given a hom of the base rings. It does appear that the change_ring of the parent is working, so you probably just need to create the hom between the poly rings and apply it to the element to be changed.

[sagetrac-git]

Changed commit from 3333f6a to 71d1123

[sagetrac-git]

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

71d1123

18443: implemented changes from second review

[sagetrac-gjorgenson]

comment:10

For the formal==True case I used the technique in the morphism change_ring function to create the homomorphism between polynomial rings to use on F and this seems to have addressed the issues so far. For the formal==False case, the conversion already seemed to be working, but the _number_field_from_algebraics version of the map wasn't needed for finding roots.

I found another problem though:

P.<x,y> = ProjectiveSpace(QQ,1)
H = End(P)
f = H([x^2 - 3/4*y^2,y^2])
f.multiplier_spectra(2)

gives [1,1] back which isn't correct.

I didn’t catch this before, but multiplier_spectra should only return [1] here, because there is only one two cycle. The collapsing causes this two cycle to turn into a root of multiplicity 2 of the dynatomic polynomial, so the algorithm I use to pick representatives from cycles doesn’t work.

I will implement a fix attempt for this next, but I wanted to push up the current commit before doing so.

[bhutz]

comment:11

For x^2-3/4, what you have are no 2-cycles. There are 2 formal 2-periodic points, but they are a doubled fixed point. So, I'm not sure [1,1] is incorrect in this case. Why do you think this is wrong?

[sagetrac-gjorgenson]

comment:12

I thought getting [1,1] back was messing up the resulting elementary symmetric polynomial. For the other x^2 + c, the formal 2-multiplier spectra consists of one multiplier and so there is only one elementary symmetric polynomial, sigma_{1}^{(2)} = sigma_2 + 4. For x^2 - 3/4, if we use [1,1] we have an extra multiplier, and sigma_{1}^{(2)} = 1+1 = 2 when we expect sigma_{1}^{(2)} = sigma_2 + 4 = 1.

I was thinking that it might be better to treat the doubled fixed point as a single 2-cycle so that we only get one multiplier. Would that work?

[bhutz]

comment:13

hmm...yes I see your point. I think having the correct number of multipliers so that the elementary symmetric polynomials are continuous as they are supposed to be would be the correct approach.

[sagetrac-git]

Changed commit from 71d1123 to 3aca932

[sagetrac-git]

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

3aca932

18443: attempt to adapt algorithm for selecting cycle representatives

[sagetrac-gjorgenson]

comment:15

I modified the selection loop slightly so now there's no break when iterating points to find their cycles. I think this will address the issue of extra representative points for collapsed cycles, and shouldn't change anything for maps with no collapsing.

[bhutz]

comment:16

Just a couple very minor things here. You should actually convert n, not just check the conversion. ie. use n=int(n)

Also, I'd rather not have you accessing the ._polys list directly. Instead you can use G[0], G[1], etc.

Finally, I'd like to see an example with multiplicities.

I see you directly implemented the QQbar change ring for the dynatomic polynomial. I still think the .change_ring() should be fixed as well, but that can be done in a different ticket.

[sagetrac-git]

Changed commit from 3aca932 to 6112c29

[sagetrac-git]

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

6112c29

18443: minor changes and new example

[sagetrac-gjorgenson]

comment:18

I added an example using x^2 - 7/4 which has a collapsed 3-cycle. I left the x^2 -3/4 example in to illustrate that the doubled fixed point is treated as one 2-cycle, but I can remove it if it seems unnecessary.

[vbraun]

Changed branch from u/gjorgenson/ticket/18443 to 6112c29