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Add notes on finding rational points

Vectornaut 2024-02-05 05:00:09 +00:00
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Engine-prototype.md

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## Rational points
One strategy for exploring a positive-dimensional solution variety is to enumerate rational points. When the user drags the display, we move numerically along the solution variety, snapping to the nearest known rational point when the drag ends.
* [Sage](https://doc.sagemath.org/html/en/reference/schemes/sage/schemes/affine/affine_rational_point.html) and [Macaulay2](https://macaulay2.com/doc/Macaulay2-1.22/share/doc/Macaulay2/RationalPoints2/html/_rational__Points.html) can enumerate rational points over various fields. For $\mathbb{Q}$, however, they turn out to use mostly brute force.
* Let's try the classic strategy of taking [rational-slope slices](https://www.quantamagazine.org/how-simple-math-reveals-rational-points-on-curves-20210722/) through a known point. In our case, the known point is the totally degenerate solution where all the points coincide and all the spheres coincide.
* The OP of [this question](https://mathoverflow.net/q/255567) suggests this strategy for finding rational points on a sphere.
## Basis optimization
### Best order