arXiv subjects
Publications and source records attributed to Zbigniew Szafraniec.
In this paper we study the stable set of the gradient flow associated with a critical point of an analytic function. In particular we present simple topological conditions which imply that this set contains an infinite family of trajectories, or has a non-empty interior.
There are presented sufficient conditions for existence of an infinite family of trajectories of an analytic gradient flow which converge to a critical point.
For a polynomial mapping from S^{n-k} to the Stiefel manifold \widetilde{V}_k(\R^{n}), where n-k is even, there is presented an effective method of expressing the corresponding element of the homotopy group π_{n-k}\widetilde{V}_k(\R^{n})\simeq\Z in terms of signatures of quadratic forms. There is also given a method of computing the intersection number for a polynomial immersion from S^m to \R^{2m}.
Let f_t , where t is close to zero, be an analytic family of plane-to-plane mappings. There are presented effective methods of computing the number of cusps of f_t emanating from the origin and having positive/negative cusp degree.
There is shown how to compute the sum of indices of critical points of rank two for polynomial selmaps of R^4
Let M be an oriented 3-manifold. For a generic f \in C^ \infty(M,R^3), there is a discrete set of swallowtail critical points. In that case, at any swallowtail point p there exists a well-oriented coordinate system centered at p, and a coordinate system centered at f(p), such that locally f has the form f_\pm(x,y,z)=(\pm xy+x^2 z+x^4,y,z), so one may associate with p a sign I(f,p)\in \{\pm 1\}. A geometric definition of the sign associated with a swallowtail was recently introduced by Goryunov. We shall show how to compute the number of swallowtail points having the positive/negative sign, in the case where f : R^n \rightarrow R^n is a polynomial mapping, in terms of signatures of quadratic forms.
Let f be a generic polynomial mapping mapping from the plane to the plane. There are constructed quadratic forms whose signatures determine the number of positive and negative cusps of f.
There is presented a method for computing the number of branches of a real analytic curve germ from $R^n$ to $R^m$, where m is greater or equal to n, having a singular point at the origin, and the number of half--branches of the set of double points of an analytic germ from $R^2$ to $R^3$.
There is presented an algorithm for computing the topological degree for a large class of polynomial mappings. As an application there is given an effective algebraic formula for the intersection number of a polynomial immersion M --> R^2m, where M is an m-dimensional algebraic manifold.
Let L be an algebraic set and let g : R^(n+1) \times L --> R^(2n) (n is even) be a polynomial mapping such that for each l in L there is r(l)>0 such that the mapping g_l = g(.,l) restricted to the sphere S^n(r) is an immersion for every 0<r<(l), so that the intersection number I(g_l|S^n(r)) is defined. Then the function which maps l in L to I(g_l|S^n(r)) is algebraically constructible.
We show that on real algebraic sets algebraically constructible functions coincide with the finite sums of signs of polynomials. Then we give some applications.