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The multiplicity of a zero of a restriction of a polynomial of degree q in C, n ≥ 2, to a non-singular trajectory of a polynomial vector field ξ with coefficients of degree p does not exceed [pq(p + q)] n−2 , if the polynomial does not vanish identically on this trajectory. For a system of polynomial vector fields, this implies an effective estimate on degree of nonholonomy i.e., the minimal order of brackets necessary to generate a subspace of maximal possible dimension at each point. In particular, this allows to check effectively whether a given system of polynomial vector fields is totally nonholonomic (controllable) at each point. Similar estimates are found for systems of vector fields with analytic coefficients satisfying polynomial Pfaffian equations. This allows to check effectively whether such a system is totally nonholonomic (controllable) at a given point. Introduction. The problem of an effective bound on the multiplicity of a zero of a polynomial Q in n variables restricted to a trajectory of a polynomial vector field ξ was suggested by J.-J. Risler [18] in connection with the problem of controllability of polynomial control systems in nonholonomic control theory [4, 21]. For n = 2, the solution is given in [6]. In this case, the problem is a special case of the estimates for the Pfaffian multiplicities [7]. However, for n ≥ 3, the problem cannot be reduced to the Pfaffian case. J.-J. Risler [18] found that the order at z of the restriction of Q to a trajectory of ξ through z does not exceed μ+ n− 1 where μ is the multiplicity at z of the ideal generated by the polynomial Q and its n − 1 consecutive derivatives along ξ at z. If μ is finite, it can be easily estimated through the Bezout theorem. The estimate is single exponential in n. In this paper, we give an effective estimate of the multiplicity κ of a zero of an arbitrary polynomial of degree q on a trajectory of a polynomial vector field with coefficients of degree p in C (Theorem 1): κ ≤ p2−1q2(p+ q)2 . (1) Polynomial vector fields and degree of nonholonomy Page 2 This estimate is doubly exponential in n. For a system of polynomial vector fields of degree p in C, the estimate (1) allows to define an effective upper bound on the degree of nonholonomy i.e., the minimal order N of brackets necessary to generate a subspace of maximal possible dimension d at each point (Theorem 3): N ≤ (d− 2) ( 23d−75p3 )2n−2 + d− 1, for d ≥ 3; N ≤ ( 6p )2n−2 + 1, for d = 2. (2) Obviously, N = 1 for d = 1. In particular, a system of polynomial vector fields is totally nonholonomic i.e., d = n at each point, when a finite system of polynomial equations defined by coefficients of all brackets of order not exceeding N is incompatible. The latter property can be checked (at least for polynomials with rational coefficients) with any standard algorithm for deciding consistency of a system of algebraic equations. Thus, the problem whether a system of polynomial vector fields is totally nonholonomic is algorithmically decidable (Corollary from Theorem 3). The estimates for vector fields with “Noetherian” coefficients i.e., analytic functions satisfying polynomial Pfaffian equations, can be easily deduced from the estimates (1) and (2) for polynomial vector fields (Theorems 2 and 4). In order to check whether a given system of vector fields with Noetherian coefficients is totally nonholonomic, we have to decide whether a system of transcendental equations with Noetherian functions is compatible. Usually this cannot be done algorithmically. However, if we want to decide whether our system is controllable at a given point where its coefficients are computable, the problem is again reduced to compatibility of a system of algebraic equations, which is algorithmically decidable (Corollary from Theorem 4). Totally nonholonomic systems play important role in control theory, where they are called controllable [4, 21]. The estimate (2) allows to check effectively controllability of a given polynomial control system (see [18] for a discussion). Systems of this kind appear also in the theory of hypoelliptic partial differential equations where the total nonholonomy is known as the “Hörmander’s condition,” and the degree of nonholonomy is involved in the a priori subelliptic estimates. This connection was first established by Hörmander [12] (see [5, 11, 16, 19, and 13, Ch. 22] for further developments). In the theory of probabilities, Hörmander’s condition appears in connection with the diffusion processes along trajectories of vector fields (Stratonovich diffusion). See [15] for discussion and further references. Polynomial vector fields and degree of nonholonomy Page 3 In differential geometry, totally nonholonomic systems appear as anisotropic CarnotCarathéodory [9, 10] or sub-Riemannian [20] metrics. The degree of nonholonomy determines how far the metric is from a Riemannian metric: for a system with degree of nonholonomy r, the Carnot-Carathéodory distance in the worst direction is proportional to 1/r-th power of the Riemannian distance. In the theory of several complex variables, the degree of nonholonomy is closely connected to the ‘type’ condition for boundaries of non-strictly pseudoconvex domains introduced by Kohn [14], and studied in [2, 3]. See [1] for the latest development in this area. Polynomials on trajectories of polynomial vector fields. Lemma 1. Let Q(x) be a germ of an analytic function at 0 ∈ C, such that Q|{x1=...=xn−1=0} 6≡ 0. (3) Let Q = ∂Q/∂xn, and let Ii be the ideal in C{x} generated by Q, . . . , Q. Suppose that Zi = V (Ii) has dimension d, and let μi be the multiplicity of Ii at a generic point z of any d-dimensional irreducible component of Zi, i.e. the rank at z of Oz(C)/Ii over Oz(Zi). Then dimZi+μi < d. Proof. Let z be a generic point of a d-dimensional component of Zi. Due to (3) the the direction of the xn axis is not tangent to Zi at z. Consider a surface S of codimension d, transversal to Zi at z, containing the line L parallel to the xn axis through z. Let ν be the order of Q|L at z. We have μi = dimCOz(S)/ ( Ii ⊗Oz(S) ) . At the same time, all the generators of Ii, when restricted to L, belong to (t ν−i) where t is a local parameter on L at z. Hence the epimorphism of rings Oz(S)→ C{t} (the restriction from S to L) induces an epimorphism of rings Oz(S)/(Ii ⊗Oz(S))→ C{t}/(tν−i). As the dimension over C of the second ring is equal to ν − i, we have ν − i ≤ μi. This means that the point z cannot belong to Zi+μi , q.e.d. Theorem 1. Let ξ(y) = ∑ j Pj(y)∂/∂yj be a vector field in C , n ≥ 2, with polynomial in y = (y1, . . . , yn) coefficients Pj of degree not exceeding p ≥ 1. Let γ be a non-singular trajectory of ξ, i.e., ξ 6= 0 at each point of γ. For a polynomial Q(y) of degree q, either Q|γ ≡ 0 or the order of a zero of Q|γ at each point of γ does not exceed p −1q2 n−2 (p+ q) n−2 . (4) Proof. Let y0 be a point of γ where (4) is violated. Let us choose the coordinates x in C near y0 so that ξ = ∂/∂xn, and use the notations of the lemma 1 in these coordinates. Polynomial vector fields and degree of nonholonomy Page 4 Then Q is the i-th derivative of Q along ξ. Suppose that Q|γ 6≡ 0. Then Q 6≡ 0, hence dimZ0 = n − 1. We can suppose also that Q is irreducible, because the estimate for Q1Q2 follows from the estimates for Q1 and Q2. In this case, the multiplicity of Q at a generic point of Z0 is equal 1. From the lemma 1 we have dimZ1 = n − 2. To estimate the multiplicity μ1, we note that Q (1) = ∑ j Pj∂Q/∂yj is a polynomial in y of degree not exceeding q + p − 1. From the Bezout theorem applied to Q and Q restricted to a 2-dimensional linear (in the coordinates y) subspace transversal to Z1 in its generic point, μ1 ≤ q(q + p − 1) ≤ q(p + q) − 1. From the lemma 1 we have dimZq(p+q) < n − 2. Let us choose ν ≤ q(p + q) so that dimZν = n − 3. For a generic linear combination G of Q, . . . , Q, the functions Q, Q, and G define a complete intersection Y of dimension n−3. The function G is a polynomial in y of the degree not exceeding q+(p−1)q(p+q). Let L be a linear (in the coordinates y) 3-dimensional subspace transversal to Y in its generic point z, and let I be the ideal generated inOz(L) by the restrictions of Q, Q, and G to L. From the Bezout theorem dimCOz(L)/I does not exceed q(p+q−1)[q+(p−1)q(p+q)] ≤ pq(p+ q) − q(p+ q). Hence μν does not exceed pq(p+ q) − q(p+ q). From the lemma 1 we have dimZpq2(p+q)2 < n− 3. The same argument inductively shows that dimZν < n− k, for 2 ≤ k ≤ n and ν ≥ p −1q2 k−2 (p+ q) k−2 . (5) In particular, dimZν < 0, i.e. Zν is empty, for ν ≥ p −1q2 n−2 (p+ q) n−2 , q.e.d. Definition 1. (Khovanskii, unpublished; see Tougeron [22].) A Noetherian chain of order m and degree α is a system f = (f1(y), . . . , fm(y)) of germs of analytic functions at the origin 0 of a complex or real n-dimensional space, satisfying Pfaffian equations ∂fi ∂yj = gij ( y, f1(y), . . . , fm(y) ) , for i = 1, . . . ,m and j = 1, . . . , n, (6) where gij are polynomials in y and f of degree not exceeding α ≥ 1. A function φ(y) = P ( y, f1(y), . . . , fm(y) ) , where P is a polynomial in y and f of degree not exceeding p, is called a Noetherian function of degree p, with the Noetherian chain f1, . . . , fm. For the real case, all the functions and polynomials in this definition are supposed to be real. Theorem 1 can be extended from polynomial to Noetherian functions as follows: Theorem 2. Let f = (f1, . . . , fm) be a Noetherian chain of order m and degree α, and let ξ = ∑ j φj(y)∂/∂yj be a vector field with the coefficients φj Noetherian of degree p, with the Noetherian chain f . Let ψ be a Noetherian function of degree q, with the Polynomial vector fields and degree of nonholonomy Page 5 Noetherian chain f . Suppose that ξ(0) 6= 0 and that ψ does not vanish identically on the trajectory γ of ξ through 0. Then the multiplicity of the zero of ψ|γ at 0 does not exceed (p+ α) −1q2 m+n−2 (p+ α+ q) m+n−2 . Proof. The statement follows from Theorem 1 after

Multiplicities of zeroes of polynomials on trajectories of polynomial vector fields and bounds on degree of nonholonomy