White’s conjecture for arbitrary matroids

Prove White’s conjecture for every matroid: establish that the toric ideal of the matroid base configuration is generated by quadratic binomials, and, in the strong form, by binomials arising from symmetric basis exchanges.

Background

For a matroid M, the paper defines the toric ideal I_M as the kernel of the monomial map sending each variable indexed by a basis to the product of the variables indexed by the elements of that basis. The weak form of White’s conjecture asserts that I_M is generated by quadrics, while the strong form specifies quadratic generators corresponding to symmetric basis exchanges.

The paper notes that these conjectures are known for several special classes of matroids, including graphic, sparse paving, and strongly base orderable matroids, but remain unresolved in general. The main theorem establishes preservation of the strong conjecture under deleting a single basis when the resulting collection is again the set of bases of a matroid, but does not resolve the conjecture for all matroids.