Algebraic matroids and representability

Choose elements ae of a field extension K/k, indexed by a finite ground set E. A subset is independent if no nonzero polynomial over k in its indexed coordinates vanishes. Its rank is the transcendence degree of the field generated by those coordinates over k. The exchange law for transcendence bases gives the matroid augmentation axiom; downward closure is immediate from the definition. A coordinate algebraic over k is a loop, while a minimal algebraically dependent subset is a circuit.

Equivalently, start with a prime ideal P in a polynomial ring over k and take the coordinate classes in the fraction field of the integral domain k[x]/P. A subset is independent exactly when eliminating all other variables leaves zero ideal in its coordinate subring. Primality matters: an arbitrary reducible algebraic set can combine incompatible componentwise dependencies and need not furnish the intended field-based matroid. The matroid retains supports of dependencies, but it forgets the coefficients and degrees of their defining polynomials.

Every vector configuration over k gives an algebraic representation: replace a vector by the corresponding linear form in independent indeterminates. Linear independence of those forms is equivalent to algebraic independence. The converse needs care. In characteristic zero, coordinate differentials in the K-vector space of Kähler differentials recover the algebraic matroid over a finitely generated extension field K of k. This yields a linear representation over K, not automatically over the original k.

If k is additionally algebraically closed of characteristic zero, a suitable generic specialization gives a linear representation over k itself. We state this theorem using the corrected Rosen–Sidman–Theran formulation. A Jacobian of defining relations gives linear relations among coordinate differentials; its columns generally encode the dual matroid, not the original one. To represent the original matroid, use the differential classes, or a matrix whose kernel is the span of those relation vectors. A parametrization Jacobian can instead directly furnish their coordinates.

A nonzero symbolic minor can vanish at a special numerical point. Therefore a generic representation may preserve the matroid only on a suitable nonempty open set where required denominators and witness minors remain nonzero. Exact elimination or symbolic rank can distinguish that theorem from numerical rank under rounding. In applications, the distinction separates structural identifiability of generic measurements from behavior at singular configurations or degenerate data.

Positive characteristic creates a deeper obstruction: the differential of a pth power is zero. A transcendental coordinate can therefore acquire a zero differential even though it is not an algebraic loop. This failure is not cured by tighter numerical tolerance. Algebraic matroids in positive characteristic need not be linearly representable over any field. The lesson does not classify those examples; its purpose is to make explicit why a characteristic-zero differential argument cannot simply be reused after reducing coefficients modulo a prime.