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Imlpicit function theorem for polynomial systems with vanishing Jacobian and it's applications to flexible structures
by
Victor Alexandrov
Sobolev Institute of Mathematics
Introduction. Let F: Rn×Rm --> Rl; t, t0 in Rn; X, X0 in Rm and let F(t0, X0)=0. The standard implicit function theorem gives conditions implying existence of an implicit function X=X(t) in a neighborhood of the point (t0, X0) which is defined by the equation F(t, X)=0, i. e., for which F(t, X(t)) \equiv 0. The most important condition is that the operator F'X(t0, X0) is invertible.
As far as is known to the author, no versions of the implicit function theorem are known in the case when F'X(t0, X0) is not invertible. We are going to fill in this gap partially in the present talk.
Our investigations are motivated by studying flexible polyhedra and frameworks. Mappings F occurring in that field are polynomial mappings which do not depend on the parameter t. We'll focus our attention on this particular case.
A typical system of polynomial equations to which on can apply our arguments
is as follows:
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Parameter t is not involved to system (1), the point X0=(5, 5, 7) satisfies system (1), and the determinant of the Jacobi matrix of system (1) vanishes at X0. Nevertheless, it will follow from our results that X0 is not an isolated solution to system (1), but it belong to a continuous family of solutions X=X(t), X(0)=X0, which can be referred to as an explicit function defined by system (1) and by point X0=(5, 5, 7). [In this example, the family defines nothing but a straight line generator of the one-sheet hyperboloid which is defined by system (1), of course.]
The main result.
Let F: Rn --> Rl be a polynomial mapping.
A polynomial X(t)=\sumk=0n tkXk is said to be an n-th order
approximate solution to the system F(X)=0 at the point X0 if
F(X(t))=0 mod tn.
We construct positive integers N
and L, a linear mapping C: RN --> RL and a bilinear mapping B: RN×RN --> RL
such that a polynomial X(t)=\sumk=0n tkXk is a n-approximate solution to
the system F(X)=0 if and only if the vectors X0, X1, ... , Xn
satisfy the equation
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Our main result is as follows: Let X0+-tX1+-... +-tnXn be an n-th order approximate solution to the system F(X)=0 and let Ln denotes the linear span of the vectors X1, ... , Xn. If B(Xi, Xj)+-B(Xj, Xi) in CLn for all 1 <= i, j <= n then there exists an analytic family of solutions Y(t)=\sumn=0\infty tnYn to the system F(X)=0 such that Yj=Xj for all 0 <= j <= n.
Applications to flexible frameworks. Let F=(X, E) be a framework in Rd, i. e., let X={ x1, ... , xv}, xj in Rd, and E subset { 1, ... , v}×{ 1, ... , v}. F is said to be flexible if there exists an analytic family F(t) of frameworks, F(t)=(X(t), E), such that
(i) F(0)=F;
(ii) || xi(t)-xj(t)||=const for all (i, j) in E;
(iii) there exists (i0, j0) such that || xi(t)-xj(t)|| is not a constant.
In other words, studying flexible frameworks we are looking for a ``non-trivial''
family of solutions to the following system of polynomial equations
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Applying our main result, one can prove that if a framework has a non-trivial infinitesimal bending (=approximate solution to (2)) of sufficiently high order and if this bending is ``regular'' then the framework is flexible. In the case of flexible polyhedra, exact formulations may be found in the author's article Sufficient conditions for the extensibility of an N-th order flex of polyhedra, Beitr. Algebra Geom. 39, No.2 (1998) 367-378.
Date received: June 12, 1999
Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacy-39.