Implemented an algorithm to find a rational point on a conic by friyaz · Pull Request #19807 · sympy/sympy

friyaz · 2020-07-20T19:14:30Z

References to other Issues or PRs

#19320

Brief description of what is fixed or changed

Implemented an algorithm to find a rational point on the conic. To parametrize a monoid of degree d, we need to find a point of multiplicity d - 1. This implies for curves of degree 2, we need to determine a rational point on it. While determining a point of multiplicity >= 2 is easy using sympy's non-linsolve, a separate algorithm needs to be implemented for points of multiplicity 1 or regular points.

The regular_point was based on iterating over a set of points and checking whether they lie on the conic. This PR fixes this for conics. I have not yet implemented the algorithm for quadrics.

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vector
- Added a function to find a rational point on conic

sympy-bot · 2020-07-20T19:14:31Z

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- Added a function to find a rational point on conic (#19807 by @friyaz and @Upabjojr)

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#### References to other Issues or PRs
#19320 
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#### Brief description of what is fixed or changed
Implemented an algorithm to find a rational point on the conic. To parametrize a monoid of degree d, we need to find a point of multiplicity d - 1. This implies for curves of degree 2, we need to determine a rational point on it. While determining a point of multiplicity >= 2 is easy using sympy's non-linsolve, a separate algorithm needs to be implemented for points of multiplicity 1 or regular points. 

The `regular_point` was based on iterating over a set of points and checking whether they lie on the conic. This PR fixes this for conics. I have not yet implemented the algorithm for [quadrics](https://en.wikipedia.org/wiki/Quadric). 

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#### Release Notes

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<!-- BEGIN RELEASE NOTES -->
* vector
    * Added a function to find a rational point on conic
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Update

The release notes on the wiki have been updated.

sympy/vector/implicitregion.py

Upabjojr · 2020-07-20T20:33:31Z

sympy/vector/implicitregion.py

        if len(self.singular_points()) != 0:
            singular_points = self.singular_points()
            for spoint in singular_points:
+                syms = Tuple(*spoint).free_symbols


what about syms = set(spoint) ?

sympy/vector/implicitregion.py

sympy/vector/tests/test_implicitregion.py

friyaz · 2020-07-21T12:37:36Z

sympy/vector/implicitregion.py

+            sol = list(sol)[0]
+            syms = Tuple(*sol).free_symbols
+            rep = {s: 3 for s in syms}
+


There is a bug here. In cases like x*y = 1, the solution of diophantine equation is {(p2 + q2, -2pq, p2 - q2)}. If I substitute p = 3 and q = 3, z turns out to be zero. This leads to the result being nan. The algorithm requires the solution of diophantine equation to be non-trivial.

What about pushing the solution of the diophantine equation into the equation solver? For example, in the case {(p2 + q2, -2pq, p2 - q2)}, can it help if you get p as a function of q?

Yes, I think this should work. Solving for z != 0, then using one of its solutions.

This is how I am trying to get the solutions for z != 0

for sol in solutions: syms = Tuple(*sol).free_symbols z = sol[2] z_syms = ztemp.free_symbols if len(z_syms) == 2: print(solveset(Unequality(z, 0), next(iter(z_syms)), S.Integers))

But solveset is too slow for returning an Integer solution.

Have you thought about solving the equality and then taking the complement of the solutions? It should be equivalent to solveset(Unequality( ... )).

This should work. In the case of two variables p and q, I will iterate p over S.Integers and then solve q for z != 0.

ping @Upabjojr.

sympy/vector/implicitregion.py

sympy/vector/tests/test_implicitregion.py

sympy/vector/implicitregion.py

Co-authored-by: Naman Gera <namangera15@gmail.com>

codecov · 2020-07-28T16:19:35Z

Codecov Report

Merging #19807 into master will decrease coverage by 0.009%.
The diff coverage is 71.328%.

@@              Coverage Diff              @@
##            master    #19807       +/-   ##
=============================================
- Coverage   75.719%   75.709%   -0.010%     
=============================================
  Files          664       666        +2     
  Lines       172374    172712      +338     
  Branches     40653     40717       +64     
=============================================
+ Hits        130520    130759      +239     
- Misses       36147     36225       +78     
- Partials      5707      5728       +21

Upabjojr · 2020-07-30T09:45:45Z

sympy/vector/implicitregion.py


        - Christoph M. Hoffmann, Conversion Methods between Parametric and
-        Implicit Curves and Surfaces.
+        Implicit Curves and Surfaces, 1990. Available:


Publisher and authors are more important than the title itself. Christoph M. Hoffmann? Perdue e-Pubs?

OK, I will add it.

Upabjojr · 2020-07-30T09:47:37Z

This looks good to me. Please add the authors and publisher of all citations you've added (in the literatura, authors and publishing review company are usually more important than the title of the paper itself).

Implemented algorithm to find a rational point on a conic

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friyaz marked this pull request as draft July 20, 2020 19:15

friyaz requested a review from Upabjojr July 20, 2020 19:15

Upabjojr reviewed Jul 20, 2020

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jksuom reviewed Jul 21, 2020

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Improved docstrings and made minor corrections

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friyaz commented Jul 21, 2020

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friyaz requested a review from Upabjojr July 21, 2020 12:38

namannimmo10 reviewed Jul 21, 2020

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Upabjojr reviewed Jul 22, 2020

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Upabjojr and others added 3 commits July 26, 2020 23:03

Update sympy/vector/implicitregion.py

6542c0d

Co-authored-by: Naman Gera <namangera15@gmail.com>

regular_point: Use only non-trivial solutions of diophantine equation

ea478b4

Resolved merge conflict in implicitregion.py

895675b

friyaz marked this pull request as ready for review July 28, 2020 14:32

friyaz requested a review from Upabjojr July 28, 2020 14:33

Upabjojr reviewed Jul 30, 2020

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friyaz requested a review from Upabjojr July 30, 2020 10:07

implicitregion.py: Added publisher's name in citation

85da83b

friyaz force-pushed the reg-point branch from a989eda to 85da83b Compare July 30, 2020 10:21

Upabjojr merged commit 3cabf94 into sympy:master Jul 30, 2020

friyaz mentioned this pull request Aug 3, 2020

Allow vector_integrate to handle ImplicitRegion objects #19883

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Upabjojr commented Jul 30, 2020

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