### A Condition for a Polynomial Map to be Invertible.

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We apply the well-known homotopy continuation method to address the motion planning problem (MPP) for smooth driftless control-affine systems. The homotopy continuation method is a Newton-type procedure to effectively determine functions only defined implicitly. That approach requires first to characterize the singularities of a surjective map and next to prove global existence for the solution of an ordinary differential equation, the Wazewski equation. In the context of the MPP, the aforementioned...

The global convergence of a direct method for determining turning (limit) points of a parameter-dependent mapping is analysed. It is assumed that the relevant extended system has a singular root for a special parameter value. The singular root is clasified as a $bifurcationsingularity$ (i.e., as a $degenerate$ turning point). Then, the Theorz for Imperfect Bifurcation offers a particular scenario for the split of the singular root into a finite number of regular roots (turning points) due to a given parameter imperfection. The relationship...

In this paper we introduce the notion of "$p$-dimensional rate of convergence" which generalizes the notion of rate of convergence introduced by V. Pták. Using this notion we give a generalization of the Induction Theorem of V. Pták, which may constitute a basis for the study of the iterative procedures of the form ${X}_{n+1}=F({x}_{n-p+1},{X}_{n-p+2},...,{x}_{n})$, $n=0,1,2,...$. As an illustration we apply these results to the study of the convergence of the secant method, obtaining sharp estimates for the errors at each step of the iterative procedure.

A general existence and uniqueness result of Picard-Lindelöf type is proved for ordinary differential equations in Fréchet spaces as an application of a generalized Nash-Moser implicit function theorem. Many examples show that the assumptions of the main result are natural. Applications are given for the Fréchet spaces ${C}^{\infty}\left(K\right)$, $S\left({\mathbb{R}}^{N}\right)$, $B\left(\mathbb{R}{R}^{N}\right)$, ${D}_{{L}_{1}}\left({\mathbb{R}}^{N}\right)$, for Köthe sequence spaces, and for the general class of subbinomic Fréchet algebras.