feat(math): add plane algebraic curve operations (#1877)

[morluto]

Summary

Implements the domain with 3 atomic operations for plane algebraic curve theory using SymPy exact symbolic computation.

Closes #1877

Operations

Operation	Description
algebraic_geometry.affine_plane_curve.check	Check polynomial defines a valid affine plane curve
algebraic_geometry.plane_curve.projective_closure.compute	Homogenize an affine curve to obtain projective closure
algebraic_geometry.projective_curve.affine_chart.compute	Dehomogenize a projective curve at a chart variable

Design

• SymPy backend: All operations use SymPy for exact symbolic polynomial computation.
• Homogenization: The projective closure is computed by homogenizing with a new variable z, ensuring the result is a homogeneous polynomial of the same total degree.
• Dehomogenization: The affine chart is extracted by setting the chart variable to 1, recovering the original affine curve from its projective closure.

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[morluto]

Deep review summary

Verdict: REQUEST CHANGES — the wire boundary does not require polynomials in the declared ring, the homogenizing variable can collide, and arbitrary nonhomogeneous expressions are accepted as projective curves.

1. affine_plane_curve.check accepts objects that are not affine plane curves

The request uses raw sympify and only tests poly != 0. Consequently it can accept undeclared symbols, rational/transcendental expressions such as 1/x or sin(x), and the constant polynomial 1, whose zero set is empty rather than a plane curve. It also does not require the two variable names to be distinct or valid.

Parse one canonical polynomial in QQ[x,y], require exactly two unique declared variables, no foreign symbols, and positive total degree. Reuse the repository's polynomial value/parser instead of introducing another string boundary.

2. Homogenization can produce a malformed variable list

The new homogeneous variable is hard-coded as z. If the affine variables are ("x","z"), the output declares ("x","z","z"), and the existing z coordinate is conflated with the homogenizing coordinate. Either require a caller-supplied fresh variable or deterministically generate one that is not already present.

For a genuine polynomial and a fresh variable, the monomial homogenization formula itself is correct. Bind the result by checking:

• every monomial has the same total degree;
• setting the new variable to 1 recovers the original polynomial exactly;
• the output variable list is unique and canonical.

3. Affine-chart extraction does not require a projective curve

Any expression in three names is accepted; homogeneity is never checked. Setting one variable to 1 is a valid chart operation only after the input is bound as a homogeneous polynomial defining a projective plane curve. Add that invariant, validate chart_variable at the request boundary, and ensure the resulting two-variable polynomial has no foreign symbols.

4. String length is not a work bound

Inputs such as (x+y)**1000000 are short but can make expansion/output infeasible. Bound parsed degree, monomial count, coefficient size, expression-node count, and projected output size before expanding.

The round trip dehomogenize(homogenize(f), z)=f should be the central property test, including nonhomogeneous affine inputs, variable-name collisions, constants, and foreign-symbol rejection.