qqideal

Exact ideals over QQ: python-flint for arithmetic, msolve for Gröbner bases, verdicts that refuse to guess. Emptiness, dimension, and 0-dimensional degree from grevlex leading ideals. No solver-mode I/O.

Install

pip install qqideal

That pulls msolveio and python-flint from PyPI. You also need a system msolve 0.10.x binary on PATH.

Usage

from qqideal import Ideal, Kind, Ring, ideal_verdict

R = Ring("x", "y")
I = Ideal(["x^2-1", "y-x"], ring=R)

verdict = ideal_verdict(I)
print(verdict.kind)       # Kind.NONEMPTY
print(verdict.dim)        # 0
print(verdict.degree)     # 2
print(verdict.certainty)  # Certainty.PROVEN

if verdict.kind is Kind.NONEMPTY:   # never `if verdict:` -- that raises
    print(I.groebner())             # (Poly('x - y', ...), Poly('y^2 - 1', ...))

Verdict.__bool__ raises TypeError. TIMEOUT and ERROR are not answers, and a truthiness test would silently fold them into one of the two that are.

opens= saturates before the test, so you can ask about the complement of a hypersurface:

ideal_verdict(["x*y", "x"], ring=R, opens=["x"]).kind   # Kind.EMPTY
Ideal(["x^2"], ring=R).radical_member("x").kind         # Kind.EMPTY: x is in the radical

Certainty

A unit ideal over Q from msolve -g is Certainty.MODULAR, not PROVEN: msolve 0.10.1 returns after its first modular prime and still prints characteristic 0. A nonempty result over Q uses a lifted -g 2 basis and is Certainty.PROVEN, as is any result over a prime field.

Saturation

I.saturate(f) is Rabinowitsch: it returns I + (u*f - 1) in the ring extended by one slack variable. Its variety is V(I) \ V(f), so emptiness, dimension, and degree are those of I : f^∞ -- but its generators are not that ideal written back in R, which would need an elimination order msolve's Gröbner mode does not offer. colon is an alias, and in v0.1 the colon is the saturation.

Not in v0.1

Primary decomposition, positive-dimensional radicals, radical computation of any kind (membership only), solver mode / -P, Macaulay2. These raise NotImplementedError rather than returning an approximation.

License

MIT © 2026 DC Posch — https://github.com/dcposch/qqideal