M-curves

Much of this follows Dr. Korchagin's paper [5].

1 Real Projective Plane

Lets look at three definitions of the projective plane P 2 . Let x , y , and z &Space; be the coordinates in the real affine topological space 3 &Space; with origin O . Define the equivalence relation in 3 \ O &Space; as follows: the points a = x y z &Space; and b &Space; are equivalent iff there exists t , t 0 , such that b = t x t y t z . So all points on a line passing through the origin O &Space; are equivalent.

Definition 1 The quotient space 3 \ O / ~ &Space; is called the projective plane P 2 .

Definition 2 The set of the lines passing through the origin O &Space; in 3 &Space; is called the projective plane P 2 .

Definition 3 The quotient space 𝕊 2 / a , - a &Space; which is generated by the equivalence relation a ~ - a &Space; is called a projective plane P 2 .

Note: 𝕊 2 = { ( x , y , z ) 3 | x 2 + y 2 + z 2 = 1 } .

Below is a picture of the projective plane P 2 &Space; (called the sphere with cross-cap model). The second picture is a dissection of the same projective plane.

Projective plane: cut sphere with crosscap

Another way to visualize the projective plane is to imagine the disk with boundary and identify the diametrically opposite points of the boundary (this is the Poincaré disk model).

The projective plane is non-orientable (there is no difference between left and right):

The coordinates of the projective plane are given by x y z &Space; which is the ratio of x, y, z. So the point x y z t x t y t z , where t 0 . We call x y z &Space; the homogeneous coordinates of P 2 &Space; . The projective plane with axes model is shown below. It also shows the 3 different origins and the quadrants.

2 Real Algebraic Plane Curves (Algebraic Curve)

A real algebraic plane curve of degree d &Space; is a real polynomial f x y &Space; of degree d , considered up to constant factors. Any polynomial can be converted into a homogeneous polynomial. This is done by the following steps:

We are given a polynomial f x y &Space; of degree d .
We let x = X Z &Space; and y = Y Z &Space; . Plugging these into the polynomial we get f X Z Y Z .
Multiply through by Z d &Space; to clear out the denominators.
This gives us a homogeneous polynomial F X Y Z &Space; of degree d .

Example: Suppose we have the polynomial f x y = y x 2 .
Let x = X Z &Space; and y = Y Z &Space; .
Then f X Z Y Z = Y Z X Z 2 . Since the degree of f is 2, we have

Z 2 f X Z Y Z = Z 2 Y Z X Z 2 = Y Z X 2 . So the homogeneous polynomial is F X Y Z = Y Z X 2 .
To convert a homogeneous polynomial to an affine polynomial, just reverse the above steps.

A real homogeneous algebraic plane curve of degree d &Space; is a real homogeneous polynomial F x y z &Space; of degree d , considered up to constant factors. The set F = x y z P 2 | F = 0 &Space; is called the set of real points of the curve and F = x y z P 2 | F = 0 &Space; the set of complex points of the curve. A curve is said to be non-singular if the system

{ F x ' = 0 F y ' = 0 F z ' = 0 has no non-zero solution. If this system has a non-zero solution the curve is said to be singular and the solution is said to be a singular point of the curve.

Unless other wise noted all curves will be projective curves.

If a curve f is non-singular then each connected component of f is homeomorphic to a circle. There are two such kinds of circles. The first kind separates the plane P 2 &Space; into two pieces, one homeomorphic to a disk and the other homeomorphic to a Möbius band. This is shown below:

Projective plane being seperated by an oval.

With respect to the oval the disk is called the inner component and the Möbius band is called the outer component. The second type of connected component is called an odd branch and does not separate the projective plane. The maximum number of components that a curve of degree d can have is less than or equal to M = ( d - 1 ) ( d - 2 ) 2 + 1 . This was proved in 1876 by A. Harnack [2]. He also proved that for each degree d there exists a curve with M components. In 1938 I.G. Petrovsky [4] gave curves with the maximum number of components M the name M-curves. D. Hilbert [3] included the problem of isotopy classification of M -curves as the 16^th problem when he gave his talk at the International Mathematical Congress at Paris in 1900. The 1^st part of Hilbert's 16^th problem is ``What can the pair P 2 , F &Space; be up to homeomorphism?'', where F is an M-curve of degree d, F is its set of real points in P 2 . In other words how many different pictures are there for each degree d.

The first fact about M -curves is that they are all nonsingular. If the M-curve is of even degree, then it consists of only ovals. If it is of odd degree, then it consists of one odd branch and the rest of the components are ovals.
Suppose we have two disjoint ovals, how can they be situated in P 2 ? There are only two arrangements. The first is that each oval lies in the outer component of the other oval. The second is that one oval lies in the inner component of the other oval. This arrangement is called an injective pair.

For an injective pair, the oval that lies in the outer component of the other oval is called the outer oval and the other oval is called the inner oval. An oval that does not have a oval in its inner component is called an empty oval.

Suppose we have h ovals and any two ovals form an injective pair,

then these ovals are said to form a nest of depth h.

In order to describe the isotopy class of a curve we will use the coding scheme devised by Viro[6]. The odd branch is coded by ⟨ J ⟩ &Space; and an oval by ⟨ 1 ⟩ . Let ⟨A⟩ be some collection of ovals, then ⟨ 1 ⟨ A ⟩ ⟩ &Space; means that one oval surrounds the collection ⟨A⟩. If ⟨ A ⟩ &Space; and ⟨ B ⟩ &Space; are two collections of ovals such that they are disjoint and no oval of one collection surrounds an oval of the other collection, then we denote there union as ⟨ A ∐ B ⟩ . The following abbreviations are also used:

A ∐ · · · ∐ A ⏟ n &Space; times = n A
n × 1 = n &Space; represents n empty ovals.
1 ⟨ 1 ⟨ · · · ⟨ 1 ⟨ A ⟩ ⟩ · · · ⟩ ⟩ = 1 n ⟨ A ⟩ &Space; where the number of repetitions of the fragment 1 ⟨ &Space; is n.

Examples: Example of Viro's coding scheme.

⟨ J ∐ 4 ∐ 1 ⟨ 2 ⟩ ∐ 1 ⟨ 1 ∐ 1 ⟨ 2 ⟩ ⟩ ∐ 1 3 ⟨ 2 ⟩ ⟩

Let us look at some theorems concerning algebraic curves.
Euler's identity Any homogeneous polynomial f x 1 x 2 . . . x n &Space; of degree r satisfies the identity

r f = x 1 f x 1 + x 2 f x 2 + . . . + x n f x n One of the most useful basic theorems concerning algebraic curves is the following:
Bézout's theorem If F , G &Space; are projective algebraic curves of degrees d 1 , d 2 &Space; and the set F G &Space; is finite and counted properly, then # F G = d 1 d 2 .
This theorem is used to prohibit curves.
Example: Does there exist an eighth degree curve having the code ⟨ 1 ⟨ 1 ⟨ 1 ⟨ 1 ⟨ 1 ⟩ ⟩ ⟩ ⟩ ⟩ ?
By Harnack's theorem we know that the maximum number of components is M = 8 1 8 2 2 1 = 22 . So it can certainly have 5 components.
Now let F be the algebraic curve with the code ⟨ 1 ⟨ 1 ⟨ 1 ⟨ 1 ⟨ 1 ⟩ ⟩ ⟩ ⟩ ⟩ . Let G be a line (curve of degree one) and let them be such that they intersect as shown below:

We can see that # F G = 10 , but by Bézout's theorem # F G = 8 . So this curve can not exist.

As of 1997 the problem had been solved for degree's d < 8 . For d < 6 &Space; by A. Harnack [2], d = 6 &Space; by D. A. Gudkov [1], for d = 7 &Space; by O. Ya. Viro [6].

more stuff on d=8

A1: Catalog of M-curves

This a catalog of M-curves.

Degree 1
Isotopy class:
⟨ J ⟩

Degree 2
Isotopy class:
⟨ 1 ⟩

Degree 3
Isotopy class:
⟨ J ∐ 1 ⟩

Degree 4
Isotopy class:
⟨ 4 ⟩

Degree 5
Isotopy class:
⟨ J ∐ 6 ⟩

Degree 6
Isotopy class:
⟨ 9 ∐ 1 ⟨ 1 ⟩ ⟩
⟨ 5 ∐ 1 ⟨ 5 ⟩ ⟩
⟨ 1 ∐ 1 ⟨ 9 ⟩ ⟩

Degree 7
Isotopy class:
⟨ J ∐ 1 ∐ 1 ⟨ 13 ⟩ ⟩
⟨ J ∐ 2 ∐ 1 ⟨ 12 ⟩ ⟩
⟨ J ∐ 3 ∐ 1 ⟨ 11 ⟩ ⟩
⟨ J ∐ 4 ∐ 1 ⟨ 10 ⟩ ⟩
⟨ J ∐ 5 ∐ 1 ⟨ 9 ⟩ ⟩
⟨ J ∐ 6 ∐ 1 ⟨ 8 ⟩ ⟩
⟨ J ∐ 7 ∐ 1 ⟨ 7 ⟩ ⟩
⟨ J ∐ 8 ∐ 1 ⟨ 6 ⟩ ⟩
⟨ J ∐ 9 ∐ 1 ⟨ 5 ⟩ ⟩
⟨ J ∐ 10 ∐ 1 ⟨ 4 ⟩ ⟩
⟨ J ∐ 11 ∐ 1 ⟨ 3 ⟩ ⟩
⟨ J ∐ 12 ∐ 1 ⟨ 2 ⟩ ⟩
⟨ J ∐ 13 ∐ 1 ⟨ 1 ⟩ ⟩
⟨ J ∐ 15 ⟩

References

[1]: Gudkov D. A.,G. A. Utk, The topology of curves of degree 6 and surfaces of defree 4
Proc. Lobachevsky Univ., 87(1969)(Russian),English transl.,Transl. AMS 112.
[2]: Harnack,A., Uber Wielfatigkeit der ebenen algebraischen Kurven
Math. Ann., 10(1876), 189-199.
[3]: Hilbert,D., Uber die rellen Zuge der ebenen algebraischen Kurven
Math. Ann., 38(1891), 155-137.
[4]: Petrovsky,I.G., On the topology of real plane algebraic curves
Ann. Math.,39:1(1938), 187-209.
[5]: Korchagin,A. B., The 1^st Part of Hilbert's 16^th Problem: History and Results
Visiting Scholar's Lectures-1997, Texas Tech Univeristy Mathematical Series NO. 19
[6]: Viro, O. Ya., The curves of degree 7, the curves of degree 8, and Ragsdale conjecture
Dokl. AN SSSR, 254:6(1980), 1305-1310(Russian); English transl. in Sovist Math. Dokl. 22(1980).

Definitions

Homeomorphism: A function f is said to be a homeomorphism if it is a bijection and f and f - 1 &Space; are both continious.

Homogeneous: A polynomial

f x 1 x 2 . . . x m = &ohgr; 1 + &ohgr; 2 + . . . + &ohgr; n m a &ohgr; x 1 &ohgr; 1 x 2 &ohgr; 2 &Space; · · · &Space; x m &ohgr; m &Space; , &Space; &Space; &Space; &Space; &ohgr; = &ohgr; 1 . . . &ohgr; m is a homogenious polynomial of degree d if &ohgr; 1 + · · · + &ohgr; m = d &Space; for each a &ohgr; 0 .
An easier definition is: the polynomial f of degree d is homogeneous iff f t x t y = t d f x y . (This can be generalized to any number of variables.)

Connected: A set X &Space; is said to be connected if there does not exist nonempty, disjoint, open sets U , V &Space; whose union is X.

Möbius Band: A Möbius band can be constructed by labeling the edges and corners of a rectangle as shown below.

Then glue the edges with arrows to each other such that the arrows point in the same direction. The corners labeled a will be glued together as will the corners labeled b. You will then get an object shaped as follows: Mobius band

This is an example of a non-orientable surface. Note that it has only one side and one edge.
add more later.

Real polynomial: A real polynomial is a polynomial whose coefficients are all real.

Proofs

We first give Bezout's theorem.

Bezout's Theorem If F , G &Space; are projective algebraic curves of degrees d 1 , d 2 &Space; and the set F G &Space; is finite and counted properly, then # F G = d 1 d 2 .

Proof:

Harnack's Theorem The maximum number of connected components a nonsingular projective curve of degree d &Space; can have is M = ( d - 1 ) ( d - 2 ) 2 + 1 .

Proof: Let f = 0 &Space; be a nonsingular curve of degree d , so

{ f x ' = 0 f y ' = 0 f z ' = 0 has no non-zero solutions.
Assume the contrary, that f &Space; has M + 1 &Space; components.

Case 1 Let d &Space; be even.
Since d &Space; is even , the only components of our curve are ovals. We want to run a curve C d 2 &Space; of degree d 2 &Space; through f . So we need N = d 2 d 1 2 &Space; points to specify C d 2 . Let us pick one point on each of the first M &Space; components. So we need to pick N M = d 3 &Space; points on the remaining oval.
We can draw f C d 2 &Space; as Now # f C d 2 = 2 M d 3 1 = d 2 2 d 2 . But by Bezout's theorem # f C d 2 = d 2 2 d . Contradiction.
Case 2 The proof is similar for the odd case.

Theorem: An M-curve has at most one odd branch.

Proof 1: Suppose that an M-curve has more than one odd brach. Then those odd branches must have a point of intersection. Hence the M-curve is singular.
Contradiction.
Therefore an M-curve has at most one odd brach.

Proof 2: This proof is more technical. An odd branch realizes a non-zero element of the group H 1 P 2 , which has non-zero self-intersection. So any two odd branches must intersect each other.