Some Hessian curves with many real components

Frank Sottile
Adriana Ortiz
    The hessian of a plane curve f(x,y) of degree d is the polynomial
Hf   :=   fxxfyy - fxy2 ,
which has degree 2d-4. A natural question to ask is, what could be the topology of real hessian curves? If d is at least 4 then the set of hessian curves is a proper subvariety of the set of all plane curves of degree 2d-4, so there is some content to this question. Moreover, standard tools in real algebraic geometry do not seem to apply. We do not know of any restrictions on real hessian curves, besides restrictions that hold for all real curves of degree 2d-4. Likewise, constructions of curves with controlled topology do not necessarily give hessian curves.
    A natural question would be if hessian curves can be M-curves, that is, if a hessian curve acheived the Harnack bound for the number of ovals. Since its degree is 2d-4, the Harnack bound for hessian curves is (2d-5)(d-3)+1 ovals. As these Hessian curves are affine, one may also ask about the Harnack bound for affine curves, which is (2d-5)(d-3) and 2d-4 unbounded components.
    Prior to this investigation, the best constructions for hessians of curves of degree d gave hessians with (d-1)(d-2)/2 ovals, due to one of us (Ortiz) [O1, O2]. Here is a table with these numbers, for small values of d.
degree of fd  3 45 67
degree of hessian2d-4 2 4 6 8 10 
Harnack bound for hessian(2d-5)(d-3)+1 1411  22 37
Affine Harnack bound  (2d-5)(d-3)+d  27 15  2743
Ovals of Ortiz hessians(d-1)(d-2)/2 136 1015

    We investigated hessians of quartic and quintic polynomials, experimentally. This involved generating millions of random polynomial, computing their hessians, filtering those whose hessian could not have many ovals or components, and then examining the remainder. In all, we looked at 150 million quartics, 40 million quintics, 40 million sextics. These were sone on several machines at Texas A&M University from November 2004 through March 2005. In March to April 2006 we ran a separate computation to look at sextics whose hessian curve was compact and had many ovals. This investigated over 208 million sextics and used 360 gigaHertz-days of computation to find a single example of a sextic whose hesssian was compact with 11 ovals. Each computation on a machine was controlled by a bash shell script. (Here is an example.) This shell script called a MAPLE script, Hessian.maple, setting the random number generator _seed to the loop variable, saving potentially interesting polynomials for further examination. Here are some of the more interesting Hessian curves that we found. Maple scripts for drawing them yourself are linked to the maple leaves: .
Hessian of quartic with 4 ovals
Hessian of quartic with 2 ovals
and 4 unbounded components
Hessian of quintic with 7 ovals
Another Hessian of a quintic with 7 ovals
Hessian of quintic with 6 ovals
and 4 unbounded components
Hessian of quintic with 6 ovals
and 4 unbounded components
(close up)
Hessian of quintic with 7 ovals and 2 unbounded components
Hessian of quintic with 8 ovals
Hessian of sextic with 10 ovals
and 4 unbounded components
Hessian of sextic with 11 ovals
and 2 unbounded components
Hessian of sextic with 11 ovals
Hessian of sextic with 11 ovals
(close up)
       
[O1] A.  Ortiz-Rodríguez, On the special parabolic points and the topology of the parabolic curve of certain smooth surfaces in R3, C. R. Math. Acad. Sci. Paris 334 (2002), no. 6, 473--478.
[O2] A.  Ortiz-Rodríguez, Quelques aspects sur la géométrie des surfaces algébriques réelles, Bull. Sci. math. 127 (2003), 149--177.
Work of Sottile supported by the National Science Foundation under CAREER Grant DMS-0134860.
Work of Ortiz supported by DGAPA and CONACyT.
Modified since: 18 May 2006 by Frank Sottile