error in height difference bound

[bhutz]

For a morphism of projective space there is an explicitly computable constant that bounds the difference between the height and canonical height (f.height_difference_bound()). This function is returning incorrect values. It looks like the code is trying to take into account scaling through by the resultant and ending up with an incorrect lower bound.

P.<x,y>=ProjectiveSpace(QQ,1)
f=DynamicalSystem([5*x^2 + 3*x*y , y^2 + 3*x^2])
C=f.height_difference_bound()
for Q in P.points_of_bounded_height(bound=5):
    if (f(Q).global_height() - 2*Q.global_height()).abs() > C:
        print("bad:",Q)

CC: @pfili @EnderWannabe

Component: dynamics

Author: Ben Hutz

Branch/Commit: 9778fff

Reviewer: Frédéric Chapoton

Issue created by migration from https://trac.sagemath.org/ticket/32687

[bhutz]

comment:1

I'm happy to fix this as soon as I get a chance to spend a little time on it (hopefully in the next week or two).

[bhutz]

Branch: u/bhutz/32687

[bhutz]

Commit: 9778fff

[bhutz]

Author: Ben Hutz

[bhutz]

comment:3

I don't really see why this was trying to clear the denominators before computing the heights of the coefficients polys. I removed all that and am now getting the expected values.

I also noticed prec was missing on the heights of the coefficients, so I added that and used it in the new example to.

Lastly, I had a sorting issue in an unrelated doctest that I fixed.

---

New commits:

9778fff

32687: fix lower bound in height difference in projective_ds

[mkoeppe]

comment:4

Stalled in needs_review or needs_info; likely won't make it into Sage 9.5.

[fchapoton]

Reviewer: Frédéric Chapoton

[fchapoton]

comment:5

ok, setting to positive (not a mathematical review, though)

[vbraun]

Changed branch from u/bhutz/32687 to 9778fff