# Analgebraic Geometry

It so happened that at the beginning of 2016 I gave a talk on the conference “Geometric aspects of modern dynamics” in Porto, delivered a minicourse at Journées Louis Antoine in Rennes and wrote an expository paper for the European Mathematical Society Newsletter, all devoted to the same subject. The subject, provisionally dubbed as “Analgebraic geometry”, deals with algebraic-like properties (especially from the point of view of intersection theory) of real and complex analytic varieties defined by ordinary and Pfaffian differential equations with polynomial right hand sides. Thus

analgebraic = un-algebraic + analytic + algebraic (background) + weak algebraicity-like properties.

It turns out that this analgebraic geometry has very intimate connections with classical problems like Hilbert 16th problem, properties of periods of algebraic varieties, analytic number theory and arithmetic geometry.

For more details see the presentation prepared for the minicourse (or the shorter version of the talk) and the draft of the paper.

Any remarks and comments will be highly appreciated.

## Noetherian chains

Computation of the (local intersectional) degree of a phase curve of a polynomial vector field, produced in Lecture 11, is based on the length of the ascending chain of polynomial ideals generated by consecutive derivations.

Let $D:\mathbb C[x_1,\dots,x_n]\to\mathbb C[x_1,\dots,x_n]$ be the Lie derivation of the algebra of polynomials along the vector field $v$. It increases the degrees by at most $d-1$. Let $p_0\in\mathbb C[x]$ be a seed polynomial of degree $\delta\in\mathbb N$ and consider the ascending chain of ideals

$I_0\subseteq I_1\subseteq I_2\subseteq\cdots\subseteq I_k\subseteq\cdots \subseteq\mathbb C[x],\qquad I_k=\left,$

where $p_k=Dp_{k-1},\ k=1,2,\dots$.  By Noetherianity, this chain must eventually stabilize at some step: $I_N=I_{N+1}=\cdots$. In addition to this chain of ideals, one can consider the associated descending chain of algebraic varieties

$\mathbb C^n\supseteq X_0\supseteq X_1\supseteq\cdots\supseteq X_k\supseteq\cdots, \qquad X_k=\{p_0(x)=\cdots=p_k(x)=0\}.$

This chain also stabilizes  no later than on the $N$th step, but may stabilize earlier.  The following properties of these chains can be verified by elementary arguments.

1. The chain of ideals is strictly ascending: If $I_N=I_{N+1}$, then all subsequent ideals in the chain coincide.
2. The chain of varieties may be nonstrictly ascending: e.g., $n=1,\ p_0(x)=x^m,\ D=\frac{\mathrm d}{\mathrm dx}$.
3. The length of the descending chain measures the maximal nontrivial order of contact between the trajectories of $v$ and the hypersurface $X_0=\{p_0=0\}$.

In general, the length of a strictly ascending chain of polynomial ideals generated by the sequence of polynomials of degrees not exceeding an explicit (growing) function of $k$, can be bounded by an algorithmically computable function. However, even in the simplest case where $\deg p_k\le \delta+k(d-1)$ (as above), this function turns out to be the Ackermann generalized exponential, a recursive but not primitively recursive function of $n,d,\delta\in\mathbb N$ which grows faster than any elementary (or primitive recursive) function. It is the algebraic origin of the sequence of polynomials, which allows to establish better results.

Example. Assume that $A:\mathbb C[x]\to\mathbb C[x]$ is an endomorphism of the ring of the polynomials, and instead of the iterations $p_k=Dp_{k-1}$ of the Lie derivation, we consider the sequence $p_k=Ap_{k-1}$. Then analogous chains can be constructed, yet their properties will be slightly different (in a sense, better). In particular, the chain of varieties becomes strictly descending and its length can be relatively simply bounded by simple function of $n,d,\delta$. If the growth rate of $\deg p_k$ is linear (as above), the bound will be double exponential in $n$. However,  in general the growth rate of iterates $A^k p_0$ is exponential, which leads to the bound given by a tower function (iterated exponent) of height $n=\dim x$.

The easiest way to estimate the length of varieties generated by consecutive derivations is based on the explicit Nullstellensatz. By this  theorem, for any polynomial $q\in\mathbb C[x]$ which vanishes on the variety $X\subseteq\mathbb C^n$ which is the zero locus of an ideal $I\subseteq\mathbb C[x]$ there exist a finite power $\rho$ such that $q^\rho\in X$. The number $\rho$ can be explicitly bounded from above (J. Kollar, 1988): if $I$ is generated by polynomials of degree no greater than $m$, then $\rho\leqslant m^n$. Having this bound, for each irreducible component of the variety $X_k$ which does not belong to the stable limit, one can predict, how many steps in can survive before being eliminated.  The resulting upper bound will be double exponential in $n$.

However, a better, more realistic and simple exponential in $n$ upper bound can be achieved by completely different argument.

Example. Assume that $n=2$ and we look at an isolated contact between a (nonsingular) trajectory of a vector field $v$ and an algebraic curve $X_0=X$ of degree $\delta$ at a point $a\in\mathbb C^2$. Consider the local analytic chart in which $v$ is parallel to the $y$-axis and the point $a$ is at the origin. If the curve $X$ has tangency of order $\mu$ with the vertical axis, then its projection on the $x$-axis is locally a ramified covering of order $\mu$. Consider a small bidisk neighborhood of the origin and apply a small analytic perturbation to $X$. The multiple tangency point will be scattered into several points of simple (quadratic) tangency, while the topological covering property will persist. Denote by $\nu$ the number of obtained simple tangencies: at each tangency exactly two leaves of the covering “collide”. Thus the total number of leaves $\mu$ cannot be greater than $2\nu$. The problem thus becomes to estimate $\nu$. However, the set of points of quadratic contact is algebraic: it is defined by the equations $X_1=\{p_0=0, p_1=0\}$ of degrees $\delta$ and $\delta+d-1$, so by the Bezout theorem the number of points does not exceed the product of these two numbers.

To generalize this argument for the multidimensional settings, one has to modify the topological part of the argument dealing with “exactly two leaves of the covering collide”. Instead of just one set $X_1$, one has to consider the sets $X_1,\dots, X_{n-1}$ (recall that $n$ is the dimension of the ambient space), and instead of counting points, one should consider their Euler characteristics. The corresponding combinatorics can be elegantly expressed by the “integration over the Euler characteristic” discovered by O. Viro (1988), while the bounds for the Euler characteristic of algebraic varieties can be bounded by virtue of the J. Milnor’s result (1964).

The result, due to A. Gabrielov and A. Khovanskii, is simple exponential (in $n$) bound, was achieved in 1998. However, for some problems in the analytic number theory (algebraic independence of transcendental numbers) it is important to have a more precise estimate of the maximal tangency order for $d, n$ fixed, but $\delta$ variable and growing to infinity. The most recent achievements in this direction are due to G. Binyamini [4], see below.

Besides, a different (and considerably more difficult) problem arises in the singular context, when one tries to estimate the order of contact of an algebraic hypersurface with a separatrix of a polynomial vector field, an invariant analytic curve (usually non-smooth) which contains a singular point of the vector field $v$.  Here again the most recent breakthroughs are due to Binyamini [5].

References (in addition to those mentioned earlier).

1. A. Gabrielov, A. KhovanksiiMultiplicity of a Noetherian intersection.  Geometry of differential equations, 119–130,
Amer. Math. Soc. Transl. Ser. 2, 186, Amer. Math. Soc., Providence, RI, 1998.
2. O. ViroSome integral calculus based on Euler characteristic. Topology and geometry—Rohlin Seminar, 127–138,
Lecture Notes in Math., 1346, Springer, Berlin, 1988.
3. J. Milnor, On the Betti numbers of real varieties.  Proc. Amer. Math. Soc. 15 1964 275–280.
4. G. Binyamini, Multiplicity Estimates: a Morse-theoretic approach, arXiv:1406.1858 (2014).
5. G. Binyamini, Multiplicity estimates, analytic cycles and Newton polytopes, arXiv:1407.1183

## Monday, December 29, 2014

### Lecture 11 (Mon, Dec 15, 2014)

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 4:43
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## Trajectories of polynomial vector fields

Definition
Let $\gamma:(-r,r)\to\mathbb R^n$ be a finite piece of a real analytic curve. Its intersection complexity (or intersection degree) is the maximal number of isolated intersections of $\gamma$ with real affine hyperplanes,

$\displaystyle \deg\gamma=\max_{\varPi\subset\mathbb R^n}\#\{t\in(-r,r):\gamma(t)\in\varPi\}<+\infty.$

The goal of this lecture is to explain the following result (D. Novikov, S.Y., 1999) which claims that a sufficiently small piece of a nonsingular trajectory of a polynomial vector field has a finite intersection degree bounded in terms of the dimension and the degree of the field.

More specifically, we consider the polynomial vector field associated with the system of polynomial differential equations

$\dot x_i=v_i(x_1,\dots,x_n),\qquad i=1,\dots,n,\quad v_i\in\mathbb R[x_1,\dots,x_n],\ \deg v_i\leqslant d$.

Denote its integral trajectory passing through an arbitrary point $a\in\mathbb R^n$ by $\gamma_a$.

Theorem
For any $a\in\mathbb R^n$ a sufficiently small piece of $\gamma_a$ has the intersection degree not exceeding $2^{2^{O(n^3 d)}}$.

References.

1. D. Novikov and S. Yakovenko, Trajectories of polynomial vector fields and ascending chains of polynomial ideals, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 2, 563–609, MR1697373 (2001h:32054)
2. S. Yakovenko, Quantitative theory of ordinary dierential equations and the tangential Hilbert 16th problem, 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)

## Oscillatory behavior of Fuchsian equations

### Semilocal theory

Consider a holomorphic linear equation in the unit disk $0<|t|\le 1$, having a unique Fuchsian singularity at the origin $t=0$. Such an equation can be always reduced to the form $Lu=0,\ L=\epsilon^n+a_1(t)\epsilon^{n-1}+\cdots+a_n(t)$, with holomorphic bounded coefficients $a_1,\dots,a_n\in\mathscr O(D)$, $D=\{|t|\leqslant 1\}$, $|a_k(t)|\leqslant A$.

The previous results imply that one can produce an explicit upper bound for the variation of argument of any nontrivial solution $u$ of the equation $Lu=0$ along the boundary of the unit disk $\partial D$: $\left.\mathrm{Var\,Arg\,}u(t)\right|_{t=1}^{t=\mathrm e^{2\pi \mathrm i}}\leqslant V_L=C\cdot n(A+1)$ for some universal constant $C$.

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 $u$ in $D$. Unfortunately, solutions are usually ramified and the argument principle does not work. Denote by $\mathbf M$ the monodromy operator along the boundary.

##### Definition

The Fuchsian point is called quasiunipotent, if all eigenvalues $\mu_1,\dots,\mu_n$ of the matrix $\mathbf M$ have modulus one, $|\mu_k|=1$.

##### Theorem 1

The number of isolated roots of any solution of the equation $Lu=0$ in the Riemann domain $\Pi=\{0<|t|\leqslant 1,\ |\mathrm{Arg\,}t\le 2\pi\}$ having real coefficients $a_k(\mathbb R)\subseteq\mathbb R,\ k=1,\dots,n$ and a single quasiunipotent singularity at the origin does not exceed $(2n+1)(2V_L+1)$, where $V_L=Cn(A+1)$ is the parameter bounding the magnitude of coefficients of $L$.

The proof is based on a version of the flavor of the Rolle theorem for the “difference operators” $\mathbf P_\mu=\mu^{-1}\mathbf M-\mu\mathbf M^{-1}$ for any unit $\mu$ such that $\mu^{-1}=\bar\mu$:

$\#\{t\in\Pi:\ u(t)=0\}\leqslant \#\{t\in\Pi:\ \bigl(\mathbf P_\mu u\bigr)(t)=0\}+2V_L.$

A version of the Cayley-Hamilton theorem asserts that the (commutative) composition $\mathbf P=\prod_{\mu}\mathbf P_\mu$ over all eigenvalues of the monodromy operator (counted with their multiplicities) vanishes on all solutions of the real Fuchsian equation.

### Global theory

A linear ordinary differential equatuib with rational coefficients from $\Bbbk=\mathbb C(t)$ can always be transformed to the form

$Lu=0,\qquad p_0(t)\partial^n+p_1(t)\partial^{n-1}+\cdots+p_n(t),\qquad p_0,\dots,p_n\in\mathbb C[t].\qquad (*)$

It may depend on additional parameters $\lambda=(\lambda_1,\dots,\lambda_r)\in\mathbb C^r$: if this dependence is rational, then we may assume that the coefficients of the operator are polynomials from $\mathbb C[t,\lambda]$. The new feature then will be appearance of singular perturbations: for some values of the parameters $\lambda=\lambda_*$ the leading coefficient $p_0(~\cdot~,\lambda_*)$ may vanish identically in $t$, 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 $\tau=1/t$ allows to investigate the nature of singularity at the infinite point $t=\infty\in\mathbb C P^1$. The equation is called Fuchsian, if it is Fuchsian at each its singular point on the Riemann sphere $\mathbb C P^1=\mathbb C\cup\{\infty\}$.

Assume that infinity is non-singular (this can always be achieved by a Mobius transformation of the independent variable $t$). Then a Fuchsian equation with the singular locus $Z=\{z_1,\dots,z_m\}\subset\mathbb C$ can always be transformed to the form $Mu=0$, where $M$ is the operator

$M= E^n+q_1(t)E^{n-1}+\cdots+q_{n-1}(t)E+q_n(t),$

$E=E_Z=(t-z_1)\cdots(t-z_m)\partial$

(nonsingularity at infinity implies certain bounds on the degrees of the polynomials $p_k\in\mathbb C[t]$). 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 $\{z_1, \dots,z_m\}$.

#### Definition.

The slope of this operator (*) is defined as the maximum

$\displaystyle\angle L=\max_{k=1,\dots,n}\frac{\|p_k\|}{\|p_0\|}$

where the norm of a polynomial $p(t)=\sum_0^r c_j t^j\in\mathbb C[t]$ is the sum $\|p\|=\sum_j |c_j|$.

Simple inequalities:

1. Any polynomial $p(t)$ of known degree $d=\deg p$ and norm $M=\|p\|$ admits an explicit upper bound for $|p(t)|$ on any disk $\{|t|\leqslant R\}$: $|p(t)| \leqslant MR^d$ for $R>1$.
2. A polynomial of unit norm $\|p\|=1$ admits a lower bound for $|p(t)|$ for points distant from its zero locus $Z=\{t:\ p(t)=0\}$. More precisely,

$\displaystyle |p(t)|\geqslant 2^{-O(d)}\left(\frac rR\right)^d,\qquad r=\mathrm{dist }(t,Z),\quad R=|t|>1.$

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 $n=\mathrm{ord }L,\ d=\max\deg p_k$ and $B=\angle L$. 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 $p_0$ is dangerously small.

## Tuesday, November 25, 2014

### Lecture 7 (Nov. 24)

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 11:50
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## Geometric form of non-oscillation theorems

Solutions of linear systems $\dot x(t)=A(t)x(t), \ x\in\mathbb R^n,\ t\in[0,\ell]$ can be highly oscillating relatively to hyperplanes $(p,x)=0, \ p\in\mathbb R^{n*}\smallsetminus 0$. However, there exists a class of system for which one can produce such bounds.

Let $\Gamma:t\mapsto x(t)$ be a smooth parametrized curve. Its osculating frame is the tuple of vector functions $v_1(t)=\dot x(t)$ (velocity), $v_2(t)=\dot v_1(t)$ (acceleration), till $v_n(t)=\dot v_{n-1}(t)$. Generically these vectors are linear independent for all $t$ except isolated points. The differential equations defining the curve up to a rigid motion have a “companion form”,

$\dot v_k=v_{k+1},\quad k=1,\dots,n-1,\qquad \dot v_n=\sum_{i=1}^n\alpha_i(t)v_i,\quad \alpha_i\in\mathbb R.$

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 $t\in[0,\ell]$.

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 $T_x\mathbb R^n\simeq\mathbb R^n$) by the vectors $\mathbb R_1 v_1\subset \mathbb Rv_1+\mathbb Rv_2\subset\cdots$. The flag can be naturally parametrized by the orthogonalization procedure applied to the osculating frame: by construction, this means that we consider the $n$-tuple of orthonormal vectors $e_1(t),\dots,e_n(t)$ with the property that

$\mathrm{Span\ }(v_1,\dots,v_k)= \mathrm{Span\ }(e_1,\dots,e_k),\qquad \forall k=1,\dots, n-1.$

This new frame satisfies the Frenet equations: their structure follows from the invariance of the flag and the orthogonality of the frame.

$\dot e_k(t)=\varkappa_{k-1}(t)e_{k-1}(t)+\varkappa_{k}(t)e_{k+1}(t),\qquad \varkappa_0\equiv\varkappa_{n}\equiv0.$

The functions $\varkappa_1(t),\dots,\varkappa_{n-1}(t)$ 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 $\Gamma:[0,\ell]\to\mathbb R^n$, parametrized by the arclength $t$, are the quantities $K_j=\int_0^\ell|\varkappa_j(t)|\,\mathrm dt$, $j=1,\dots,n-1$, and $K_n=\pi\#\{t:\ \varkappa_{n-1}(t)=0\}$ (the last quality, equal to the number of hyperinflection points up to the constant $\pi$, is called integral hyperinflection).

Let $\Gamma:[0,\ell]\to\mathbb R^n\smallsetminus\{0\}$ be a smooth curve avoiding the origin in the space. Its absolute rotation around the origin $\Omega(G,0)$ is defined as the length of its spherical projection on the unit sphere, $x\mapsto \frac x{\|x\|}$.  The absolute rotation $\Omega(\Gamma, a)$ around any other point $a\notin\Gamma$ is defined by translating this point to the origin.

If $L\subset\mathbb R^n$ is a $k$-dimensional affine subspace disjoint from $\Gamma$ and $P_L:\mathbb R^n\to L^\perp$ the orthogonal projection on the orthogonal complement $L^\perp$, the absolute rotation $\Omega(\Gamma, L)$ of $\latex \Gamma$ around $L$ is the absolute rotation of the curve $P_L\circ\Gamma$ around the point $P_L(L)\in L^\perp\simeq \mathbb R^{n-k}$.

The absolute rotation of $\Gamma$ around an affine hyperplane $L$ is defined as $\pi\cdot \#(\Gamma\cap L)$.

Formally the 0-sphere $\mathbb S^0=\{\pm 1\}\subset\mathbb R^1$ is not connected, but it is convenient to make it into the metric space with two “antipodal” points at the distance $\pi$, similarly to higher dimensional unit spheres with antipodal points always distanced at $\pi$.

Denote by $\Omega_k(\Gamma)$ the supremum $\sup_{\dim L=k}\Omega(\Gamma,L)$, where the supremum is taken over all affine subspaces $L$ of dimension $k$ in $\mathbb R^n$.

Main Theorem.

$\Omega_k(\Gamma)\leqslant n + 4\bigl(K_1(\Gamma)+\cdots+K_{k+1}(\Gamma)\bigr) \qquad \forall k=0,\dots,n-1$.

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 $f_1(t),\dots,f_n(t)$. Denote by $W_k(t)$ the Wronski determinant of the first $k$ functions $f_1,\dots,f_k$, adding for convenience $W_0\equiv 1,\ W_1\equiv f_1$. These Wronskians are real analytic, and assuming that $W_n$ does not vanish identically, we can construct the linear $n$th order differential operator

$\displaystyle \frac{W_n}{W_{n-1}}\,\partial\,\frac{W_{n-1}}{W_{n}}\cdot\frac{W_{n-1}}{W_{n-2}}\,\partial\,\frac{W_{n-2}}{W_{n-1}}\,\cdots\, \frac{W_2}{W_1}\,\partial\,\frac{W_1}{W_2}\cdot\frac{W_1}{W_0}\,\partial\,\frac{W_0}{W_1}.$

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 $f_1,\dots, f_n$.

The straightforward application of the Rolle theorem guarantees that if all the Wronskians are nonvanishing on $[0,\ell]$, then the operator is disconjugate and no linear combination of functions $\sum c_i f_i(t)$ can have more than $n-1$ isolated root.

In the case where the Wronskians $W_k(t)$ are allowed to have isolated roots, numbering $\nu_k$ if counted with multiplicity, then the maximal number of zeros that a linear combination as above may exhibit, is bounded by $(n-1)+4\sum_{k=1}^n \nu_k$.

References.

1. A. Khovanskii, S. Yakovenko, Generalized Rolle theorem in $\mathbb R^n$ and $\mathbb C$. 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 $\sum_k p_k(z)\mathrm e^{\lambda_k z}$, $\lambda_k\in\mathbb C,\ p_k\in\mathbb C[z]$ in complex domains are obtained.
2. 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.
3. D. Nadler, S. Yakovenko, Oscillation and boundary curvature of holomorphic curves in $\mathbb C^n$. A complex analytic version of the Main theorem with improved estimates.

## Thursday, November 20, 2014

### Lecture 6 (Nov. 21, 2014)

Filed under: Analytic ODE course — Sergei Yakovenko @ 8:20
Tags: , ,

## 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 $u$ be a nontrivial solution of a sufficiently regular linear ordinary differential equation $Lu=0$ of order $n>0$. Then $u$ cannot have a root of multiplicity greater or equal than $n-1$.

Here by regularity we mean the condition that the operator $L=\partial^n+a_1(t)\partial^{n-1}+\cdots+a_{n-1}(t)\partial+a_n(t)$ has coefficients smooth enough to guarantee that any solution $u(t)$ near any point $a$ in the domain of its definition is uniquely determined by the initial conditions $u(a),u'(s),\dots,u^{(n-1)}(a)$.

Indeed, if $u$ has a root of multiplicity $n$, that is, all first $n-1$ derivatives of $u$ at $a$ vanish, then $u^{(n)}(a)=0$ by virtue of the equation and hence the by the uniqueness $u(t)$ must be identically zero.

In particular, solutions of first order equation $u'+a_1(t)u=0$ are nonvanishing, solutions of any second order equation $u''+a_1(t)u'+a_2(t)u=0$ may have only simple roots etc.

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

$u^{(n)}+a_1(t)u^{(n-1)}+\cdots+a_{n-1}(t)u'+a_n(t)u=0,\qquad t\in[0,\ell],\qquad (\dag)$

are explicitly bounded,  $|a_k(t)|\leqslant A_k\in\mathbb R_+,\ \forall t\in[0,\ell],\ k=1,\dots,n$.

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

$\displaystyle \sum_{k=1}^n \frac{A_k}{k!}\ell^k<1.\qquad (*)$

Then any nontrivial solution of the equation has no more than $n-1$ isolated roots on $[0,\ell]$ .

## Novikov’s counterexample

What about linear systems of the first order?

Consider the system $\dot x=A(t)x$ with $x=(x_1,\dots,x_n)\in \mathbb R^n$ and the norm $\|A(t)\|$ explicitly bounded on $[0,\ell]$. Consider all possible linear combinations $u=\sum_k c_k x_k(t),\ c\in\mathbb R^n$. Can one expect a uniform upper bound for the number of roots of all combinations?

Let $a(t)$ be a polynomial having many zeros on $[0,t]$. Consider the $2\times 2$-system of the form

$\displaystyle \dot x_1=a(t)x_1,\qquad \dot x_2=(\dot a+ a^2)x_1.$

The first equation defines a nonvanishing function $x_1(t)$, the second equation – its derivative which vanishes at all roots of $a(t)$.

By replacing $a(t)$ by $\varepsilon a(t)$ one can achieve an arbitrarily small sup-norm of the coefficients of this system on the segment $[0,\ell]$ (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

$u^{(n)}+a_1(z)u^{(n-1)}+\cdots+a_{n-1}(z)u'+a_n(z)u=0,\qquad z\in D\subseteq\mathbb C$

is defined in a convex compact subset $D$ of diameter $\ell$ and the condition (*) holds. Then this equation is disconjugate in $D$: any solution has at most $n-1$ isolated roots.

This result follows from the interpolation inequality of the following type: if $u(z)$ is a function holomorphic in $D$ and has $n$ isolated roots there, then $\|u\|_D\leqslant \frac{\ell^n}{n!}\|u^{(n)}\|$ (the maximum modulus norm is assumed).

Consider the equation $(\dag)$ on the real interval but with complex-valued coefficients (and solutions). Solutions will be then real parameterized curves $u:[0,\ell]\to\mathbb C$ which only exceptionally rarely have roots. Instead of counting roots, one can measure their rotation around the origin $0\in\mathbb C$, which is defined as $R(u)=|\mathrm{Arg}~u(\ell)-\mathrm{Arg}~u(0)|$ for any continuous choice of the argument.

Theorem. Assume that

$\displaystyle \sum_{k=1}^n \frac{A_k}{k!}\ell^k<\frac12.$

Then rotation of any nontrivial solution $u$ is explicitly bounded: $R(u)<\pi (n+1)$.

If an analytic LODE with explicitly bounded coefficients is defined, say, on a triangle $D$, 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.

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