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Traversally Generic & Versal Vector Flows: Semi-Algebraic Models of Tangency to the Boundary

Let $X$ be a compact smooth manifold with boundary. In this article, we study the spaces $\mathcal V^\dagger(X)$ and $\mathcal V^\ddagger(X)$ of so called boundary generic and traversally generic vector fields on $X$ and the place they occupy in the space $\mathcal V(X)$ of all fields (see Theorems \ref{th3.4} and Theorem \ref{th3.5}). The definitions of boundary generic and traversally generic vector fields $v$ are inspired by some classical notions from the singularity theory of smooth Bordman maps \cite{Bo}. Like in that theory (cf. \cite{Morin}), we establish local versal algebraic models for the way a sheaf of $v$-trajectories interacts with the boundary $\d X$. For fields from the space $\mathcal V^\ddagger(X)$, the finite list of such models depends only on $\dim(X)$; as a result, it is universal for all equidimensional manifolds. In specially adjusted coordinates, the boundary and the $v$-flow acquire descriptions in terms of universal deformations of real polynomials whose degrees do not exceed $2\cdot \dim(X)$.