The next lecture is moved from its usual Friday time slot to Wednesday (tomorrow), November 26, 10:00. This is one-time move.

## Tuesday, November 25, 2014

### Lecture 7 (Nov. 24)

## 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 .

**Main Theorem**.

.

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 .

**References**.

- 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.

## Thursday, November 20, 2014

### Lecture 6 (Nov. 21, 2014)

## Zeros of solutions of linear equations

Nontrivial (i.e., not identically zero) solutions of linear ordinary differential equations obviously possess certain properties concerning their roots (points where these solutions vanish). The simplest, in a sense paradigmal property, is the following.

**Prototheorem**. *Let be a nontrivial solution of a sufficiently regular linear ordinary differential equation of order . Then cannot have a root of multiplicity greater or equal than *.

Here by regularity we mean the condition that the operator has coefficients smooth enough to guarantee that any solution near any point in the domain of its definition is uniquely determined by the initial conditions .

Indeed, if has a root of multiplicity , that is, all first derivatives of at vanish, then by virtue of the equation and hence the by the uniqueness must be identically zero.

In particular, solutions of first order equation are nonvanishing, solutions of any second order equation may have only simple roots etc.

## What about quantitative versions?

**Theorem ** (de la Vallee Poussin, 1929). *Assume that the coefficients of the LODE*

*are explicitly bounded, .*

*Assume that the bounds are small relative to the length of the interval, i.e.,*

*Then any nontrivial solution of the equation has no more than isolated roots on .*

## Novikov’s counterexample

What about linear systems of the first order?

Consider the system with and the norm explicitly bounded on . Consider all possible linear combinations . Can one expect a uniform upper bound for the number of roots of all combinations?

Let be a polynomial having many zeros on . Consider the -system of the form

The first equation defines a nonvanishing function , the second equation – its derivative which vanishes at all roots of .

By replacing by one can achieve an arbitrarily small sup-norm of the coefficients of this system on the segment (or even any open complex neighborhood of this real segment). Thus no matter how small are the coefficients, the second component will have the specified number of isolated roots.

## Complexification

What about complex valued versions? There is no Rolle theorem for them.

I will describe three possible replacements, Kim’s theorem (1963), nearest in the spirit, and two versions of the argument principle.

**Theorem** (W. Kim)

*Assume that an analytic LODE*

*is defined in a convex compact subset of diameter and the condition (*) holds. Then this equation is disconjugate in : any solution has at most isolated roots.*

This result follows from the interpolation inequality of the following type: if is a function holomorphic in and has isolated roots there, then (the maximum modulus norm is assumed).

Consider the equation on the real interval but with complex-valued coefficients (and solutions). Solutions will be then real parameterized curves which only exceptionally rarely have roots. Instead of counting roots, one can measure their rotation around the origin , which is defined as for any continuous choice of the argument.

**Theorem**. Assume that

Then rotation of any nontrivial solution is explicitly bounded: .

If an analytic LODE with explicitly bounded coefficients is defined, say, on a triangle , then application of this result to the sides of the triangle yields an explicit upper bound for the number of isolated roots of analytic solutions inside the triangle.

**Reference**

S. Yakovenko, On functions and curves defined by differential equations, §2.

## Tuesday, November 18, 2014

### Lecture 5 (Nov. 17)

## Fuchsian equivalence and Fuchsian classification

**Definitions**

A (formal, genuine) Fuchsian operator of order is a (formal, resp., converging) series of the form with the coefficients from the ring of polynomials in the variable with .

The polynomial is called the Euler part of .

Two Fuchsian operators are -equivalent (formally or analytically), if there exist two Fuchsian operators such that and the Euler parts of are mutually prime in .

Unlike the Weyl algebra , the collection of Fuchsian operators is not a subalgebra, although it is “multiplicatively” (compositionally) closed.

The Fuchsian equivalence is indeed reflexive, transitive and symmetric. The first two properties are obvious, to prove the last one an additional effort is required. Indeed, for two Fuchsian operators of order with mutually prime Euler parts, one can construct two operators with holomorphic coefficients so that the identity holds, but in the leading terms of may well degenerate, thus violating the Fuchsian condition. However, one can always find such pair of operators of order greater by 1, which will still be Fuchsian. The rest is easy.

The following results can be proved by more or less direct computation in the algebra :

- A Fuchsian operator with a nonresonant Euler part is -equivalent to its Euler part.
- Any Fuchsian operator is -equivalent to a polynomial operator from .
- Any Fuchsian operator is -equivalent to a polynomial operator of the form with being polynomials without free terms, , which is Liouville integrable.
- A Fuchsian operator has trivial (identical) monodromy if and only if it is -equivalent to an Euler operator with pairwise different integer roots. The corresponding equation has an apparent singularity (all solutions are analytic) if and only if all these roots are pairwise different nonnegative integers.

## Wednesday, November 12, 2014

### Lecture 4 (Nov. 14, 2014)

## Algebraic theory of linear ordinary differential operators

- Differential field of meromorphic germs of functions of one variable + derivation produce noncommutative polynomials : a polynomial acts on in a natural way.
- The equation only exceptionally rarely has a solution in , but one can always construct a differential extension of which will contain solutions of this equation.
- Analytically solutions of the equation form a tuple of functions analytic and multivalued in a punctured neighborhood of the origin. The multivaluedness is very special: the linear span remains the same after the analytic continuation, hence there exists a matrix such that .
- Instead of , any other derivation can be used, in particular, the Euler derivation .
**Example.**Equations with constant coefficients have the form with constant coefficients . Such an operator can always be factorized into commuting factors, with . A fundamental system of solutions consists of*quasipolynomials*, . In a similar way the*Euler operator*has the form and its solutions are functions , (look at the model equation ).**Weyl equivalence of of two operators.**Two operators of the same order are called Weyl equivalent, if there exist an operator which maps any solution of the equation to a solution of the equation isomorphically (i.e., no solution is mapped to zero).

The above definition means that the composition vanishes on all solutions of , hence must be divisible by : for some .Note that the operator represented by each side of the above equality, is a non-commutative analog of the least common multiple of mutually prime polynomials : it is divisible by both and .**Theorem.**T*he Weyl equivalence is indeed an equivalence relationship: it is reflexive, symmetric and transitive.*

The only thing that needs to be proved is the symmetry. Since are mutually prime, there exist two operators such that , hence . This identity means that is simultaneously divisible by and by (immediately). Hence is divisible by their least common multiple : there exists an operator such that . But since the algebra is without zero divisors, the right factor can be cancelled, implying , which means that maps solutions of into those of .- Different flavors of Weyl equivalence: regular (nonsingular) requiring be nonsingular or arbitrary.
**Theorem**.*Any nonsingular operator with holomorphic coefficients , is regular Weyl equivalent to the operator*.

This result is analogous to the rectification theorem reducing any nonsingular system to .**Theorem.***Any Fuchsian operator is Weyl equivalent to an Euler operator*.

This is similar to the meromorphic classification of tame systems. The conjugacy may be non-Fuchsian.- Missing part: a genuine analog of holomorphic classification of Fuchsian systems.

## Poincare-Dulac-Fuchs classification of Fuchsian operators

Instead of representing operators as non-commutative polynomials in or in , one can represent them as non-commutative (formal) Taylor series of the form with the coefficients from the *commutative* algebra of univariate polynomials, but not commuting with the “main variable” .

Such an operator is Fuchsian of order , if and only if for all , and . The polynomial is the “eulerization” of , and the series can be considered as a noncommutative perturbation of the Euler operator .

**Definition.** The operator is non-resonant, if no two roots of differ by a nonzero integer, .

**Theorem.** *A non-resonant Fuchsian operator is Weyl equivalent to its Euler part with the conjugacy being a Fuchsian operator, , .*

## In search of the general theory (to be continued)

References.

The classical paper by Ø. Ore (1932) in which the theory of non-commutative polynomials was established, and the draft of the paper by Shira Tanny and S.Y., based on Shira’s M.Sc. thesis (Weizmann Institute of Science, 2014).

## Monday, November 10, 2014

### Lecture 3 (Nov. 10, 2014).

## Local theory of Fuchsian systems (cont.)

- Resonant normal form.

**Definition.**A meromorphic Fuchsian singularity , , is in the (Poincare-Dulac) normal form, if for all , the identities hold. **Theorem.**Any Fuchsian system is holomorphically gauge equivalent to a system in the normal form.- Integrability of the normal form: let (in fact, the sum is finite). Then the solution is given by the (non-commutative) product . The monodromy is the (commutative) product, .

References: [IY], section 16.

## Linear high order homogeneous differential equations

- Differential operators as noncommutative polynomials in the variable with coefficients in a differential field of meromorphic germs at the origin.
- Composition and factorization.
- Reduction of a linear equation to a system of linear first order equations and back. Singular and nonsingular equations.
- Euler derivation and Fuchsian equations (“nonsingular with respect to “).
- Division with remainder, greatest common divisor of two operators, divisibility and common solutions of two equations.
**Sauvage theorem.**Tame equations are Fuchsian.

References: [IY], Section 19.

### Lecture 2 (Nov 7, 2014)

## Local theory of Fuchsian singular points

- Monodromy and holonomy.
- Growth of multivalued solutions.
- Tame singularities.
- Principal example: the Euler system , . Solution:

, monodromy , . - Fuchsian condition.
- Gauge classification of linear systems, .
- Meromorphic gauge classification of tame (regular) systems.
- Holomorphic gauge classification of Fuchsian singularities: ,

, , . - Resonances (integer differences between eigenvalues of .
- Holomorphic Eulerization of non-resonant Fuchsian singularities.

Reference: [IY], section 16.

## Monday, November 3, 2014

### Lecture 1 (Nov. 3, 2014)

The first lecture was introductory, containing the motivation for the forthcoming subjects.

The world of (real or complex) algebraic sets is tame: any question on the topological complexity admits an algorithmic solution and explicitly bounded answer.In particular, any algebraic set which consists of finitely many isolated points, admits an explicit bound for the number of these points by the product of degrees of equations defining this set (Bézout theorem). All the way around, equations involving nonalgebraic solutions to even simplest algebraic differential equations (sine/cosine), may define infinite sets (integer numbers). We will try to find out how the algebraic universe can be enlarged to include transcendental objects which still admit explicit bounds on their complexity.

It turns out that periods, integrals of rational forms over algebraic cycles, do possess such constructive finiteness, although this is far from easy to see. This finiteness is characteristic for solutions of rational Pfaffian systems with moderate singularities and special monodromy group.

## Part 1: General linear systems.

A linear system locally lives on a cylinder, the product of a (complex) linear space and an open base . If is an -matrix of holomorphic 1-forms on the base $U$, then a linear system defined by this matrix 1-form, is a matrix differential equation , whose solution is a holomorphically invertible matrix function , . If the base is one-dimensional, then with a holomorphic matrix function , and the linear system takes the familiar shape [IY, sect. 15]

A necessary and sufficient condition for a local existence of solution is vanishing of the curvature, which amounts to the matrix identity (the right hand side is the matrix 2-form with the entries , ). See [NY, sect. 1].

Solution of a linear system is defined modulo a right multplicative constant matrix factor: for any , and any other solution has such form. Using this observation, any piecewise curve $latex\gamma$ on the base can be covered by small neighborhoods with local solutions in these neighborhoods, which agree on the pairwise intersections . If this was not the case for the initial choice of local solutions, this can be always achieved by suitably twisting them (replacing by so that on the intersections). This explains how solutions can be continued analytically along any simple curve, yet after continuation along a closed path the solution may acquire a non-trivial monodromy factor.

### Analytic theory of linear differential equations: a new course

Carissimi!

This post is to announce the midi-course (about 24 hours) that will be given in November-December in Universita di Pisa. The weekly timetable is as follows,

Lunedi 11-13 Aula 1

Venerdi 9-11 Aula 1.

The course will be based (among other) on several principal sources, all available online. Here are the links:

- Yu. Ilyashenko, S. Yakovenko, Lectures on analytic differential equations, MR2085816 (2005f:34255). Mainly Chapter III and Section 26.
- D. Novikov, S. Yakovenko, Lectures on meromorphic flat connections, In: Normal forms, bifurcations and finiteness problems in differential equations, 387–430, NATO Sci. Ser. II Math. Phys. Chem., 137, Kluwer Acad. Publ., Dordrecht, 2004 (Preprint math.CA/0212334).
- Yakovenko, S. Quantitative theory of ordinary differential equations and tangential Hilbert 16
^{th}problem, Preprint math.DS/0104140 (2001).*On finiteness in differential equations and Diophantine geometry*, CRM Monogr. Ser., vol.**24**, Amer. Math. Soc., Providence, RI, 2005, pp. 41–109, MR2180125 (2006g:34062)

More specialized references will be added in the appropriate posts.

Feel free to leave your questions and comments

Arrivederci!

## Saturday, September 6, 2014

### Polynomials, matrices and differential equations

The transcendental elementary functions are traditionally introduced in the high school/(pre)calculus courses in the way that obscures their inter-relations and properties. The text below is an extended version of the one-hour lecture delivered on Ulpana de-Shalit for undergraduate students. It purports to introduce these functions as solutions to linear differential equations in the way similar to radicals being introduced as solutions to algebraic equations.

Polynomials, matrices and differential equations, 11 pages (pdf).