Symmetric Bessmertny\u{i} Realizations and Field Extension Problems in Characteristic 2 - A Differential Algebra Approach

We present a short, purely algebraic proof of the Symmetric Bessmertny\u{i} Realization Theorem in the characteristic $2$ case recently proved in [EOW26]. Symmetric Bessmertny\u{i} realizations are Schur complements of affine linear symmetric matrix pencils, and they arise naturally as state-space representations in linear systems theory. In contrast with the algorithmic approach in [EOW26], we use differential algebra: by defining formal partial derivatives on multivariate rational functions over fields of positive characteristic and considering their corresponding field of constants, we obtain scalar criteria for symmetric and homogeneous symmetric realizability in characteristic $2$, effectively reducing the matrix-valued problem to its diagonal entries. As a consequence, we prove a new theorem on the field extension problem for symmetric and homogeneous symmetric Bessmertny\u{i} realizations. Finally, in the scalar case, we identify realizable rational functions with vector spaces over appropriate fields of constants and quantify the abundance of counterexamples in characteristic $2$.