Opened 4 years ago

Closed 4 years ago

# implement rational preperiodic points for polynomials over number fields

Reported by: Owned by: bhutz bhutz minor sage-6.7 algebraic geometry Ben Hutz Grayson Jorgenson N/A f4bd855 (Commits) f4bd8554f573f5557fcad138fb34ecf58c859b40

### Description

Through the use of Weil restriction we can reduce the problem of rational points over number fields to QQ-rational points. Combining the Weil restriction functionality with the rational preperiodic point functionality, we can extend the latter to number fields.

### comment:1 Changed 4 years ago by bhutz

• Branch set to u/bhutz/ticket/18279
• Created changed from 04/22/15 14:22:22 to 04/22/15 14:22:22
• Modified changed from 04/22/15 14:22:22 to 04/22/15 14:22:22

### comment:2 Changed 4 years ago by bhutz

• Commit set to ff79600f7aa59a301e6bb6ca560b6616de46039f
• Status changed from new to needs_review
• Summary changed from implemented rational preperiodic points for polynomials over number fields to implement rational preperiodic points for polynomials over number fields

New commits:

 ​ff79600 `18279: rational preperiodic for number fields`

### comment:3 Changed 4 years ago by bhutz

• Authors set to Ben Hutz

### comment:4 Changed 4 years ago by gjorgenson

• Status changed from needs_review to needs_work
• the point at infinity will always be a totally ramified fixed point so no check is needed
• Maps over QQbar seem get incorrect output:
```P.<x,y> = ProjectiveSpace(QQbar,1)
H = End(P)
f = H([x^2-y^2,y^2])
print f.rational_preperiodic_points()
```

returns None

but

```P.<x,y> = ProjectiveSpace(QQbar,1)
H = End(P)
f = H([x^2-y^2,y^2])
f = f._number_field_from_algebraics()
print f.rational_preperiodic_points()
```

returns [(1 : 1), (1 : 0), (-1 : 1), (0 : 1)]

### comment:5 Changed 4 years ago by git

• Commit changed from ff79600f7aa59a301e6bb6ca560b6616de46039f to 55312ac9603c6d67cc7c2a1157fd4c1ca1764016

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

 ​55312ac `18279: fix issues from review`

### comment:6 Changed 4 years ago by bhutz

• Status changed from needs_work to needs_review

fixed issues. I also noticed that the 'inverted' polynomial case, i.e., where 0 is totally ramified and fixed was not actually working since the inverse of the weil restriction was not producing the correct points. I fixed this as well.

### comment:7 Changed 4 years ago by gjorgenson

Everything looks good, but there is just a small typo (I think it was there before the commits though)

-in line 3110, Determined should be Determines

Also, it looks like the first lines of the documentation for `rational_preperiodic_graph` are the same as those of `rational_preperiodic_points`, but the function returns the graph of the preperiodic points

### comment:8 Changed 4 years ago by git

• Commit changed from 55312ac9603c6d67cc7c2a1157fd4c1ca1764016 to f4bd8554f573f5557fcad138fb34ecf58c859b40

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

 ​f4bd855 `18279: fix typos in docs`

### comment:9 Changed 4 years ago by gjorgenson

• Reviewers set to Grayson Jorgenson
• Status changed from needs_review to positive_review

### comment:10 Changed 4 years ago by vbraun

• Branch changed from u/bhutz/ticket/18279 to f4bd8554f573f5557fcad138fb34ecf58c859b40
• Resolution set to fixed
• Status changed from positive_review to closed
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