For cactus, Pascal, and Pappus configurations, the matroid ideal is generated, up to radical, by circuit, Grassmann-Cayley, and liftability polynomials.
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Algebraic Geometry of Cactus, Pascal, and Pappus Matroids
For cactus, Pascal, and Pappus configurations, the matroid ideal is generated, up to radical, by circuit, Grassmann-Cayley, and liftability polynomials.