Oscillatory behavior of Fuchsian equations
Consider a holomorphic linear equation in the unit disk , having a unique Fuchsian singularity at the origin . Such an equation can be always reduced to the form , with holomorphic bounded coefficients , , .
The previous results imply that one can produce an explicit upper bound for the variation of argument of any nontrivial solution of the equation along the boundary of the unit disk : for some universal constant .
If the solution itself is holomorphic (e.g., in the case of apparent singularities), such bound would imply (by virtue of the argument principle) a bound for the number of zeros of in . Unfortunately, solutions are usually ramified and the argument principle does not work. Denote by the monodromy operator along the boundary.
The Fuchsian point is called quasiunipotent, if all eigenvalues of the matrix have modulus one, .
The number of isolated roots of any solution of the equation in the Riemann domain having real coefficients and a single quasiunipotent singularity at the origin does not exceed , where is the parameter bounding the magnitude of coefficients of .
The proof is based on a version of the flavor of the Rolle theorem for the “difference operators” for any unit such that :
A version of the Cayley-Hamilton theorem asserts that the (commutative) composition over all eigenvalues of the monodromy operator (counted with their multiplicities) vanishes on all solutions of the real Fuchsian equation.
A linear ordinary differential equatuib with rational coefficients from can always be transformed to the form
It may depend on additional parameters : if this dependence is rational, then we may assume that the coefficients of the operator are polynomials from . The new feature then will be appearance of singular perturbations: for some values of the parameters the leading coefficient may vanish identically in , meaning that the order of the corresponding equation drops down to a smaller value. Such phenomenon is known to cause numerous troubles of analytic nature.
Changing the independent variable allows to investigate the nature of singularity at the infinite point . The equation is called Fuchsian, if it is Fuchsian at each its singular point on the Riemann sphere .
Assume that infinity is non-singular (this can always be achieved by a Mobius transformation of the independent variable ). Then a Fuchsian equation with the singular locus can always be transformed to the form , where is the operator
(nonsingularity at infinity implies certain bounds on the degrees of the polynomials ). However, the coefficients of this form depend in the rational way not only on the coefficients of the original equation (*), but also on the location of the points .
The slope of this operator (*) is defined as the maximum
where the norm of a polynomial is the sum .
- Any polynomial of known degree and norm admits an explicit upper bound for on any disk : for .
- A polynomial of unit norm admits a lower bound for for points distant from its zero locus . More precisely,
We expect that for an equation having only quasiunipotent Fuchsian singular points, the number of isolated roots of solutions can be explicitly bounded in terms of and . Indeed, it looks like we can cut out circular neighborhoods of all singularities and apply Theorem 1.
The trouble occurs when singularities are allowed to collide or almost collide. Then any slit separating them will necessarily pass through the area where the leading coefficient is dangerously small.