10 1. OVERVIEW

that W4 = 240 and W5 = 18264. Solomon [136] has also found an intersection-

theoretic interpretation for these invariants.

These ideas have also found an application. Gahleitner, J¨ uttler, and Schi-

cho [58] proposed a method to compute an approximate parameterization of a

plane curve using rational cubics. Later, Fiedler-Le Touz´ e [49] used the result of

Kharlamov (that W3 = 8), and an analysis of pencils of plane cubics to prove that

this method works.

While the story of this interpolation problem is fairly well-known, it was not

the first instance of lower bounds in enumerative real algebraic geometry. In their

investigation of the Shapiro Conjecture, Eremenko and Gabrielov found a similar

invariant σm,p which gives a lower bound on the number of real points in the inverse

image

Wr−1(Φ)

under the Wronski map of a real polynomial Φ ∈

RPmp).

Assume

that p ≤ m. If m + p is odd, set σm,p to be

(1.10)

1!2! · · · (m−1)!(p−1)!(p−2)! · · · (p−m+1)!(

mp

)!

(p−m+2)!(p−m+4)! · · · (p+m−2)!

(

p−m+1

2

)

!

(

p−m+3

2

)2

! · · ·

(

p+m−1

2

)

!

.

If m + p is even, then set σm,p = 0. If p m, then set σm,p := σp,m.

Theorem 1.14 (Eremenko-Gabrielov [45]). If a polynomial Φ(t) ∈

RPmp

of

degree mp is a regular value of the Wronski map, then there are at least σm,p

real

m-dimensional subspaces of polynomials of degree m+p−1 with Wronskian Φ.

Remark 1.15. The number of complex points in

Wr−1(Φ)

is #m,p (1.5). It

is instructive to compare these numbers. We show them for m+p = 11 and m =

2,..., 5.

m 2 3 4 5

σm,p 14 110 286 286

#m,p 4862 23371634 13672405890 396499770810

We also have σ7,6 ≈ 3.4 ·

104

and #7,6 ≈ 9.5 ·

1018.

Despite this disparity in their

magnitudes, the asymptotic ratio of log(σm,p)/ log(#m,p) appears to be close to

1/2. We display this ratio in the table below, for different values of m and p.

log(σm,p)

log(#m,p)

m

2

m+p−1

10

2

m+p−1

10

3

m+p−1

10

4

m+p−1

10

5

m+p−1

10

m

+

p

−

1

100 0.47388 0.45419 0.43414 0.41585 0.39920 0.38840

1000 0.49627 0.47677 0.46358 0.45185 0.44144 0.43510

10000 0.49951 0.48468 0.47510 0.46660 0.45909 0.45459

100000 0.49994 0.48860 0.48111 0.47445 0.46860 0.46511

1000000 0.49999 0.49092 0.48479 0.47932 0.47453 0.47168

10000000 0.50000 0.49246 0.48726 0.48263 0.47857 0.47616

Thus, the lower bound on the number of real points in a fiber of the Wronski map

appears asymptotic to the square root of the number of complex solutions.

It is interesting to compare this to the the result of Shub and Smale [132]

that the expected number of real solutions to a system of n Gaussian random

polynomials in n variables of degrees d1,...,dn is

√

d1 · · · dn, which is the square

root of the number of complex solutions to such a system of polynomials. Thus

1

2

is the ratio of the logarithm of the expected number of complex solutions to the

logarithm of the expected number of real solutions.