Incidence geometry in the projective plane via almost-principal minors of symmetric matrices

Understand

We present an encoding of a polynomial system into vanishing and non-vanishing constraints on almost-principal minors of a symmetric, principally regular matrix, such that the solvability of the system over some field is equivalent to the satisfiability of the constraints over that field.

• This implies two complexity results about Gaussian conditional independence structures.
• First, all real algebraic numbers are necessary to construct inhabitants of non-empty Gaussian statistical models defined by conditional independence and dependence constraints.
• This gives a negative answer to a question of Petr \v{S}ime\v{c}ek.