linearSpanTropCone -- linear span of the tropical cone associated to the matching field

Suppose that $I$ is an ideal and $in_w(I)$ is a binomial initial ideal of $I$ with resepct to a weight $w$. Let $C_w$ be the cone in the Groebner fan of $I$ that contains $w$ in its relative interior. The linear span of $C_w$ can be constructed from a generating set of $in_w(I)$. Each generator $x^u - x^v$ gives a hyperplane defined by kernel of $(0 .. 0, 1_u, 0 .. 0, -1_v, 0 .. 0)$. The intersection of these hyperplanes gives the linear span of the Groebner cone.

The function linearSpanTropCone checks if the supplied matching field gives rise to a toric degeneration, which happens if and only if the initial ideal of the Pluecker ideal is toric, i.e., the ideal is generated by binomials and is prime. If it is already known that the matching field gives rise to a toric degeneration then set the option VerifyToricDegeneration to false to avoid repeating this check.

The linear span is a realisation of the algebraic matroid associated to the matching field. See the function algebraicMatroid.