A Bound for the Degree of Nonholonomy in the Plane

Let Σ = (V1,…, Vs) be a system made with vector fields V1,…,Vs in Rn whose coordinates are polynomials of degree ⩽ d. To such a system is associated the control system x = ∑u1V1. It is proven that in the case n = 2, the degree of nonholonomy of such a system is bounded by a function θ(2, d) ⩽6d2 − 2d + 2.