Solver API

Polynomial system solving via Gröbner bases. The groebner Cargo feature is included in all PyPI wheels by default since 2.3.1 — no special build flag needed.

alkahest.solve(equations: list[Expr], variables: list[Expr]) → list[dict] or GroebnerBasis

Solve a system of polynomial equations symbolically.

Uses Lex Gröbner basis computation followed by triangular back-substitution. Quadratic factors are solved exactly with symbolic square roots.

Parameters:

  • equations – List of polynomial expressions (each equal to zero).

  • variables – Variables to solve for.

Returns:

  • list[dict[Expr, Expr]] — one dict per solution, mapping variable → value (symbolic, e.g. sqrt(2)/2).

  • [] (empty list) — if the system is inconsistent.

  • GroebnerBasis — for parametric (infinite) solution sets.

Example:

pool = ExprPool()
x = pool.symbol("x")
y = pool.symbol("y")

# Linear system
sols = solve([x + y - pool.integer(1), x - y], [x, y])
# → [{x: 1/2, y: 1/2}]

# Circle ∩ line (irrational)
sols = solve([x**2 + y**2 - pool.integer(1), y - x], [x, y])
# → [{x: sqrt(2)/2, y: sqrt(2)/2},
#    {x: -sqrt(2)/2, y: -sqrt(2)/2}]

Note

Solutions are symbolic by default. Evaluate numerically with eval_expr() when needed:

from alkahest import eval_expr
for sol in sols:
    for var, val in sol.items():
        print(f"{var} ≈ {eval_expr(val, {}):.6f}")

Pass numeric=True to return float values directly instead of symbolic Expr solutions.

GroebnerBasis

class alkahest.GroebnerBasis

A Gröbner basis for a polynomial ideal.

classmethod compute(polys: list[Expr], vars: list[Expr], order: str = 'GRevLex') → GroebnerBasis

Compute a Gröbner basis using the F4 algorithm.

Parameters:

order – Monomial order — "Lex", "GrLex", or "GRevLex". Use "Lex" for elimination; "GRevLex" is generally fastest.

Computes the basis using the F4 algorithm with product-criterion pruning.

reduce(expr: Expr) → Expr

Reduce expr modulo the ideal.

contains(expr: Expr) → bool

Test ideal membership.

eliminate(vars: list[Expr]) → GroebnerBasis

Compute the elimination ideal by removing generators that involve the specified variables.

Useful for implicitization of parametric curves and surfaces.

class alkahest.GbPoly

A polynomial element of a Gröbner basis computation, with rational coefficients represented as FLINT rationals.