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