Hyperbolyc polycycles and appearing multiple limit cycles

Consider a 2-dimensional oriented manifold $M$ and a smooth vector field $v_0$ on $M$.
Definition 1. A directed graph $\gamma$ imbedded to $M$ is called a hyperbolic polycycle of a vector field if it satisfies the following properties:

• its vertices are hyperbolic saddles of the vector field;
  
• its edges are separatrix connections; the time determines the direction;
  
• the graph $\gamma$ is Eulerian (there exists a path that visits each edge exactly once).
  

If a polycycle has a monodromy map from a transversal section to itself then the polycycle called monodromic.
Let $\gamma$ be a hyperbolic polycycle formed by $n$ separatrix connections. After a small perturbation of the polycycle $\gamma$ some limit cycles appear. The characteristic number of a saddle is the modulo of the ratio of its eigenvalues, the negative one is in the numerator. A limit cycle is of multiplicity $m$ if after any its generic perturbation it splits into not greater than $m$ hyperbolic limit cycles.
The main result is formulated in the following two theorems.
Theorem 1. For any positive integer $n$ there exists a non-trivial polynomial $\mathcal{L}_n(\lambda_1, \ldots, \lambda_n)$ such that the following statement holds. Let $v_0$ be a vector field with a hyperbolic polycycle $\gamma$ and the characteristic numbers $\lambda_1, \ldots, \lambda_n$ of the saddles satisfy the inequation

\begin{align} \mathcal{L}_n(\lambda_1, \ldots, \lambda_n) \neq 0. \label{eq:L_n_ineq} \end{align}

Then for any $C^\infty$-smooth finite-parameter family the multiplicity of any appearing limit cycle is not greater than $n$.
The following theorem is opposite to the previous one.
Theorem 2. Let $\gamma$ be a monodromic hyperbolyc polycycle formed by $n$ saddles and $n$ separatrix connections. Denote by $\lambda_1,\ldots,\lambda_n$ the characteristic numbers of the saddles, suppose $\lambda_1\ldots\lambda_n = 1$. Let $V$ be a generic $C^\infty$-family perturbing the polycycle $\gamma$. Then $n+1$-multiple limit cycle appears ($n+1$ limit cycles appear) in the family $V$.
These two theorems connect to a polynomial system that discribes the behavior of the perturbed polycycle. Hence, their proofs use the theory of the commutative algebra.