A Bound for a Typical Differential Dimension of Systems of Linear Differential Equations

We prove upper and lower bounds for the leading coefficient of Kolchin dimension polynomial systems partial linear differential equations in case codimension two, quadratic with respect to orders system. A notion typical plays an important role algebra, some its estimations were proved by J. Ritt E. Kolchin; they also advanced several conjectures that later refuted. Our bound generalizes analog Bézout theorem one indeterminate. It is better than estimation D. Grigoriev.