9 October 2004, College Station, Texas.

Article for Autumn 2004 issue of the Emissary, the newsletter of MSRI.

This article appeared in the Emissary, the newsletter of the MSRI, and has been translated into several languages, including norsk (By Lars Olden email), svenska, Romanian (by Alexandra Seremina of azoft), Polish (by Olga Babenko), Indonesian (by Jordan Silaen @ ChameleonJohn.com), Macedonian (by Jim Anastasovski), Portuguese by Artur Weber and Adelina Domingos, Gujarati (by Ravi Sahni), and in Urdu (by Samuel Badree).

Everyone knows that two points determine a line, and many people who
have studied geometry know that five points on the plane determine a
conic. In general, if you have *m* random points in the plane and you
want to pass a rational curve of degree *d* through all of them,
there may be no solution to this interpolation problem (if *m* is
too big), or an infinite number of solutions (if *m* is too small),
or a finite number of solutions (if *m* is just right). It turns
out that ``*m* just right'' means *m*=3*d*-1
(*m*=2 for lines and *m*=5 for conics).

A harder question is, if *m*=3*d*-1, *how many*
rational curves of degree
*d* interpolate the points? Let's call this number
*N _{d}*, so that

The research themes in the MSRI Winter 2004 semester on Topological
Aspects of Real Algebraic Geometry
included enumerative real algebraic geometry, tropical geometry,
real plane curves, and applications of real algebraic geometry.
All are woven together in the unfolding story of this interpolation
problem, a prototypical problem of *enumerative
geometry*, which is the art of counting geometric figures determined by
given incidence conditions. Here is another problem: how many lines
in space meet four given lines?
To answer this, note that three lines lie on a unique doubly-ruled
hyperboloid.

Enumerative geometry works best over the *complex* numbers, as
the number of real figures depends rather subtly on the configuration of
the figures giving the incidence conditions.
For example, the fourth line may meet the hyperboloid in two real
points, or in two complex conjugate points, and so there are either
two or no real lines meeting all four.
Based on many examples, we have come to expect that any enumerative
problem may have all of its solutions be real [So].

Another such problem is the 12 rational curves interpolating 8 points in the plane. Most mathematicians are familiar with the nodal (rational) cubic shown on the left below. There is another type of real rational cubic, shown on the right.

Since there are at most 12 such curves,
*N*()
+*N*() \leq 12,
and so there are 8, 10, or 12 real rational cubics interpolating 8
real points in the plane, depending upon the number
(0, 1, or 2) of cubics with an isolated point.
Thus there will be 12 real rational cubics interpolating any 8 of
the 9 points of intersection of the two cubics below.

Welschinger [W], who was an MSRI postdoc last Winter, developed
this example into a theory.
In general, the singularities of a real rational plane curve *C* are nodes
or isolated points.
The parity of the number of nodes is its *sign* *s*(*C*), which is
either 1 or -1.
Given 3*d*-1 real points in the plane,
Welschinger considered the absolute value of the quantity

This was a breakthrough, as *W _{d}* was
(almost) the first truly non-trivial invariant in enumerative real
algebraic geometry.
Note that

Mikhalkin, who was an organizer of the semester, provided the key to
computing *W _{d}* using tropical algebraic
geometry [Mi].
This is the geometry of the tropical semiring,
where the operations of max and + on real numbers
replace the usual operations of + and multiplication.
A tropical polynomial is a piecewise linear function of the form

Mikhalkin showed that there are only finitely many rational tropical curves
of degree *d* interpolating 3*d*-1 generic points.
While the number of such curves does depend upon the choice of
points, Mikhalkin attached positive multiplicities to each
tropical curve so that the weighted sum does not, and is in fact
equal to *N _{d}*.
He also reduced these multiplicities and the enumeration of
tropical curves to the combinatorics of lattice
paths within a triangle of side length

Mikhalkin used a correspondence involving the map
Log :(**C**^{*})^{2}
--> **R**^{2}
defined by (*x*,*y*)|-->(log|*x*|,log|*y*|),
and a certain `large complex limit' of the complex structure
on (**C**^{*})^{2}.
Under this large complex limit, rational curves of degree *d*
interpolating 3*d*-1 points in
(**C**^{*})^{2}
deform to `complex
tropical curves', whose images under Log are ordinary
tropical curves interpolating the images of the points.
The multiplicity of a tropical curve *T* is the number of
complex tropical curves which project to *T*.

What about real curves?
Following this correspondence,
Mikhalkin attached a real multiplicity to each tropical curve and
showed that if the tropical curves interpolating a given 3*d*-1
points have total real multiplicity *N*, then there are
3*d*-1 real
points which are interpolated by *N* real rational curves of
degree *d*.
This real multiplicity is again expressed in terms of
lattice paths.

What about Welschinger's invariant? In the same way, Mikhalkin attached a signed weight to each tropical curve (a tropical version of Welschinger's sign) and showed that the corresponding weighted sum equals Welschinger's invariant. As before, this tropical signed weight may be expressed in terms of lattice paths.

During the semester at MSRI, Itenberg,
Kharlamov, and Shustin [IKS] used Mikhalkin's results to
estimate Welschinger's invariant.
They showed that *W _{d}*\geq

There are two other instances of this phenomenon of lower bounds,
the first of which predates Welschinger's work.
Suppose that *d* is even and let *W*(*s*) be a real polynomial of
degree *k*(*d*-*k*+1).
Then Eremenko and
Gabrielov [EG] showed that there exist real
polynomials *f*_{1}(*s*),...,
*f _{k}*(

For example, this story was recounted over beer one evening
at the MSRI Workshop
on Geometric Modeling and Real Algebraic Geometry in April 2004.
A participant, Schicho, realized that the result
*W*_{3}=8 for cubics explained why a method he had
developed always seemed to work.
This was an algorithm to compute an approximate parametrization of an
arc of a curve, via a real rational cubic interpolating 8 points on
the arc.
It remained to find conditions that guaranteed the existence of a
solution which is close to the arc.
This was just solved by Fiedler-Le Touzé, an MSRI postdoc who had
studied cubics (not necessarily rational) interpolating 8 points to
help classify real plane curves of degree 9.

We gratefully thank our editor, Silvio Levy and the MSRI members whose work we describe.

Supported by the National Science Foundation grants CAREER DMS-0134860 and DMS-9810361 (funding the MSRI), and the Clay Mathematical Institute.

Last modified: Mon Apr 23 15:33:30 CEST 2018