Simple exponential estimate for the number of real zeros of complete abelian integrals

Novikov, Dmitri, and Yakovenko, Sergei. "Simple exponential estimate for the number of real zeros of complete abelian integrals." Annales de l'institut Fourier 45.4 (1995): 897-927. <http://eudml.org/doc/75150>.

@article{Novikov1995,
abstract = {We show that for a generic polynomial $H=H(x,y)$ and an arbitrary differential 1-form $\omega =P(x,y)\,dx+Q(x,y)\,dy$ with polynomial coefficients of degree $\le d$, the number of ovals of the foliation $H=\{\rm const\}$, which yield the zero value of the complete Abelian integral $I(t)=\oint _\{H=t\}\omega $, grows at most as $\exp O_H(d)$ as $d\rightarrow \infty $, where $O_H(d)$ depends only on $H$. The main result of the paper is derived from the following more general theorem on bounds for isolated zeros occurring in polynomial envelopes of linear differential equations. Let $f_1(t),\dots ,f_n(t)$, $t\in K\Subset \{\Bbb R\}$, be a fundamental system of real solutions to a linear ordinary differential equation $Lu=0$ with rational coefficients and without singularities on the interval $K$. If the differential operator $L$ is irreducible, then any real function representable in the form $\sum \limits _\{j,k=1\}^n p_\{jk\}(t)\,f_j^\{(k-1)\}(t)$ with polynomial coefficients $p_\{jk\}\in \{\Bbb C\}[t]$ of degree less or equal to $d$, may have at most $\exp O_\{L,K\}(d)$ real isolated zeros on $K$ as $d\rightarrow \infty $.},
author = {Novikov, Dmitri, Yakovenko, Sergei},
journal = {Annales de l'institut Fourier},
keywords = {Abelian integrals; irreducible equations; Fuchsian singularities; polynomial envelopes; number of ovals; foliation},
language = {eng},
number = {4},
pages = {897-927},
publisher = {Association des Annales de l'Institut Fourier},
title = {Simple exponential estimate for the number of real zeros of complete abelian integrals},
url = {http://eudml.org/doc/75150},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Novikov, Dmitri
AU - Yakovenko, Sergei
TI - Simple exponential estimate for the number of real zeros of complete abelian integrals
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 4
SP - 897
EP - 927
AB - We show that for a generic polynomial $H=H(x,y)$ and an arbitrary differential 1-form $\omega =P(x,y)\,dx+Q(x,y)\,dy$ with polynomial coefficients of degree $\le d$, the number of ovals of the foliation $H={\rm const}$, which yield the zero value of the complete Abelian integral $I(t)=\oint _{H=t}\omega $, grows at most as $\exp O_H(d)$ as $d\rightarrow \infty $, where $O_H(d)$ depends only on $H$. The main result of the paper is derived from the following more general theorem on bounds for isolated zeros occurring in polynomial envelopes of linear differential equations. Let $f_1(t),\dots ,f_n(t)$, $t\in K\Subset {\Bbb R}$, be a fundamental system of real solutions to a linear ordinary differential equation $Lu=0$ with rational coefficients and without singularities on the interval $K$. If the differential operator $L$ is irreducible, then any real function representable in the form $\sum \limits _{j,k=1}^n p_{jk}(t)\,f_j^{(k-1)}(t)$ with polynomial coefficients $p_{jk}\in {\Bbb C}[t]$ of degree less or equal to $d$, may have at most $\exp O_{L,K}(d)$ real isolated zeros on $K$ as $d\rightarrow \infty $.
LA - eng
KW - Abelian integrals; irreducible equations; Fuchsian singularities; polynomial envelopes; number of ovals; foliation
UR - http://eudml.org/doc/75150
ER -