# 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.

## 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)

## Monday, October 22, 2007

### “Auxiliary Lesson” #1 שעור עזר (Oct. 25, 2007)

Analytic ODEs in real and complex domain: similarities and differences.

1. Background on holomorphic functions. Weierstrass compactness principle.
2. (Ordinary) Differential Equations and their solutions.
3. Contracting mapping principle (recall).
4. Picard integral operator and its contractivity.
5. Existence/uniqueness theorem.
6. Example: Matrix exponent and its computation.
7. Holomorphic vector fields and their trajectories. Equivalence of vector fields.
8. Flow box theorem and rectification theorem for nonsingular vector fields.

Attached is Section 1. It will be available on these pages for a limited time and is password-protected from printing 😦 … I must obey  the requirements of the Publisher.

###### Disclaimer. In full compliance with the strike rules (were it still be underway), this meeting is defined as a research/orientation seminar on a novel teaching technology. 🙂

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