Sergei Yakovenko's blog: on Math and Teaching

Sunday, December 14, 2014

Lecture 10 (Fri, Dec 12, 2014)

Uniform bounds for parametric Fuchsian families

The previous lectures indicate how zeros of solutions can be counted for linear differential equations on the Riemann sphere. For an equation of the form

u^{(n)} u+a_1(t)u^{(n-1)}u+\cdots+a_{n-1}(t)u'+a_n(t)u=0 ,\quad a_1,\dots, a_n\in\mathbb C(t)\qquad(*)

one has to assume that:

  1. The equation has only Fuchsian singularities at the poles of the coefficients a_1,\dots,a_n;
  2. The monodromy of each singular point is quasiunipotent (i.e., all eigenvalues of the corresponding operator are on the unit circle);
  3. The slope of the differential equation is known.

The slope is a badly formed and poorly computable number that characterizes the relative strength of the non-principal coefficients of the equation. It is defined as follows:

  1. For a given affine chart t\in\mathbb C on \mathbb P^1, multiply the equation (*) by the common denominator of the fractions for a_k(t), reducing the corresponding operator to the form b_0(t)\partial^n+b_1(t)\partial^{n-1}\cdots+b_n(t) with b_0,\dots,b_n\in \mathbb C[t];
  2. Define the affine slope as the \max_{k=1,\dots,n}\frac{\|b_k\|}{\|b_0\|}, where the norm of a polynomial b(t)=\sum_j \beta_j t^j is the sum \sum_j |\beta_j|;
  3. Define the conformal slope of an equation (*) as the supremum of the affine slopes of the corresponding operators over all affine charts on \mathbb P^1.
  4. Claim. If the equation (*) is Fuchsian, then the conformal slope is finite.

The rationale behind the notion of the conformal slope of an equation is simple: it is assumed to be the sole parameter which allows to place an upper bound for the variation of arguments along “simple arcs” (say, circular arcs and line segments) which are away from the singular locus \varSigma of the equation (*).

The dual notion is the conformal diameter of the singular locus. This is another badly computable but still controllable way to subdivide points of the singular locus into confluent groups that stay away from each other. The formal definition involves the sum of relative lengths of circular slits.

The claim (that is proved by similar arguments as the precious claim on boundedness of the conformal slope) is that a finite set points of the Riemann sphere \mathbb P^1 has conformal diameter bounded. Moreover, if \varSigma\subseteq\mathbb P^m is an algebraic divisor of degree d in the m-dimensional projective space, then the conformal diameter of any finite intersection
\varSigma_\ell=\ell\cap\varSigma for any 1-dimensional line \ell\subseteq\mathbb P^m is explicitly bounded in terms of m,d.

Together these results allow to prove the following general result.

Theorem (G. Binyamini, D. Novikov, S.Y.)

Consider a Pfaffian n\times n-system \mathrm dX=\Omega X on the projective space \mathbb P^m with the rational matrix 1-form of degree d. Assume that:

  1. The system is integrable, \mathrm d\Omega=\Omega\land\Omega;
  2. The system is regular, i.e., its solution matrix X(t) grows at worst polynomially when t tends to the polar locus
    \varSigma of the system;
  3. The monodromy of the system along any small loop around \varSigma is quasiunipotent.

Then the number of solutions of any solution is bounded in any triangle T\subseteq\ell free from points of $late \varSigma$.

If in addition the system is defined over \mathbb Q and has bitlength complexity $c$, then this number is explicitly bounded by a double exponential of the form 2^{c^{P(n,m,d)}}, where P(n,m,d) is an explicit polynomial of degree \leqslant 60 in these variables.

Remark. The quasiunipotence condition can be verified only for small loops around the principal (smooth) strata of \varSigma by the Kashiwara theorem.

Reference

G. Binyamini, D. Novikov, and S. Yakovenko, On the number of zeros of Abelian integrals: A constructive solution of the infinitesimal Hilbert sixteenth problem, Inventiones Mathematicae 181 (2010), no. 2, 227-289, available here.

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