Geometric form of non-oscillation theorems
Solutions of linear systems can be highly oscillating relatively to hyperplanes . However, there exists a class of system for which one can produce such bounds.
Let be a smooth parametrized curve. Its osculating frame is the tuple of vector functions (velocity), (acceleration), till . Generically these vectors are linear independent for all except isolated points. The differential equations defining the curve up to a rigid motion have a “companion form”,
Note that this is a vector ODE with scalar coefficients, i.e., a tuple of identical scalar ODEs. Besides, it may exhibit singularities: if the osculating frame degenerates (which may well happen at isolated points of the curve), the coefficients of this equation exhibit a pole at the corresponding moments of time .
However, the osculating frame is not a natural object: it depends on the parametrization. The invariant notion is the osculating flag, the flag of subspaces spanned (in ) by the vectors . The flag can be naturally parametrized by the orthogonalization procedure applied to the osculating frame: by construction, this means that we consider the -tuple of orthonormal vectors with the property that
This new frame satisfies the Frenet equations: their structure follows from the invariance of the flag and the orthogonality of the frame.
The functions are called Frenet curvatures: they are nonnegative except for the last one (hypertorsion) which has sign and may change it at isolated hyperinflection points.
Definitions. (Absolute) integral curvatures of a smooth (say, real analytic) curve , parametrized by the arclength , are the quantities , , and (the last quality, equal to the number of hyperinflection points up to the constant , is called integral hyperinflection).
Let be a smooth curve avoiding the origin in the space. Its absolute rotation around the origin is defined as the length of its spherical projection on the unit sphere, . The absolute rotation around any other point is defined by translating this point to the origin.
If is a -dimensional affine subspace disjoint from and the orthogonal projection on the orthogonal complement , the absolute rotation of $\latex \Gamma$ around is the absolute rotation of the curve around the point .
The absolute rotation of around an affine hyperplane is defined as .
Formally the 0-sphere is not connected, but it is convenient to make it into the metric space with two “antipodal” points at the distance , similarly to higher dimensional unit spheres with antipodal points always distanced at .
Denote by the supremum , where the supremum is taken over all affine subspaces of dimension $k$ in .
The proof of this theorem is based on a combination of arguments from integral geometry and the Frobenius formula for a differential operator vanishing on given, say, real analytic functions . Denote by the Wronski determinant of the first functions , adding for convenience . These Wronskians are real analytic, and assuming that does not vanish identically, we can construct the linear th order differential operator
One can instantly see that this operator is monic (composition of monic operators of order 1) and by induction prove that it vanishes on all functions .
The straightforward application of the Rolle theorem guarantees that if all the Wronskians are nonvanishing on , then the operator is disconjugate and no linear combination of functions can have more than isolated root.
In the case where the Wronskians are allowed to have isolated roots, numbering if counted with multiplicity, then the maximal number of zeros that a linear combination as above may exhibit, is bounded by .
- A. Khovanskii, S. Yakovenko, Generalized Rolle theorem in and . Contains detailed description of the so called Voorhoeve index, the total variation of argument of an analytic function on the boundary of its domain and why this serves as a substitute for the Rolle theorem over the complex numbers. As a corollary, rather sharp bounds for the number of complex roots of quasipolynomials , in complex domains are obtained.
- D. Novikov, S. Yakovenko, Integral curvatures, oscillation and rotation of smooth curves around affine subspaces. Contains the proof of the Main theorem cited below, with a slightly worse weights attached to the integral curvatures.
- D. Nadler, S. Yakovenko, Oscillation and boundary curvature of holomorphic curves in . A complex analytic version of the Main theorem with improved estimates.