**Oleg Viro's home page, texts for students**
#
introductions to topology of real algebraic varieties

## What is topology of real algebraic
varieties?

This is not an easy question. The difficulty is that
a reasonable answer must rely on some background, which varies.
Indeed, the answer depends on the answer to the question:

"**What is variety?**"

**1.** For a simple-minded answer at the high school level, please,
read

*
Easy reading on topology of real
plane algebraic curves* by V. Kharlamov and O. Viro.

This is an elementary introduction, which requires
preliminary knowledge neither in topology, nor algebraic geometry, but
can give the first impression of the subject. It contains almost no
proofs, and should be considered a sort of pure advertisement.

This was written as an informal introduction to a book
*Topological properties of real plane algebraic
curves*
by V. M. Kharlamov, V. A. Rokhlin and O. Y. Viro. The book has not
been completed.

**2.** Another introduction to the same subject is given in the
introductory part of my book
*Real algebraic varieties with
prescribed topology,*.
It requires less topology, at least at the beginning. The book is
unfinished.

**3.** A tremendous role in shaping of the subject was played by
Hilbert's sixteenth problem. Take a look at transparencies of my talk
*
The 16th Hilbert problem, a story of mystery, mistakes and
solution*,
screen version.

There is also a complete text of Hilbert's talk
at the second International Congress of Mathematicians in 1900 with all
23 problems.

**7.** Soon after the sixteenth Hilbert problem was formulated,
a former Hilbert's student
Virginia Ragsdale
published a paper
*On the arrangement of the real branches of plane algebraic
curves*, Amer. J. Math. **28** (1906), 377-404,
in which she analyzed the results by Harnack and Hilbert and posed
several conjectures. Some of them were proved, the others disproved. A
story about this is presented in the paper *
Patchworking algebraic curves disproves the Ragsdale conjecture,* by
I. Itenberg, O. Ya. Viro, Math. Intelligencer **18** (1996), no. 4,
19-28.