From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math.num-analysis
Subject: Re: Simple form for cubic polynomial over 2-space
Date: 8 Sep 1995 19:40:11 GMT

In article <42n5l6$drt@sunsystem5.informatik.tu-muenchen.de>,
Carsten Steger <stegerc@Informatik.TU-Muenchen.DE> wrote:
>It is well known that a quadratic function over the 2-space of reals
>    f(x,y) = a*x^2 + b*x*y + c*y^2
>can be transformed by a rotation into a "simple" canonical form
>    f(x',y') = d*x'^2 + e*y'^2.
>
>My question is: Is there a similar rotation that transforms a cubic
>function
>    f(x,y) = a*x^3 + b*x^2*y + c*x*y^2 + d*y^3
>into some "simple" form? If so, how does this simple form look like?
>How can the transformation be determined?

By insisting upon rotations you leave only very small equivalence classes,
so that the cubics will still fall into very many equivalence classes.
(The rotations form a one-parameter group of transformations, still
leaving a 3-parameter family of "simple" functions.)

I might suggest viewing the transformations geometrically. By homogeneity,
your  f  is determined by its values along the unit circle in the plane.
On this compact set, f must attain a minimum at least once. You can rotate
coordinates so that the location of the minimum is at (-1,0). Then it
turns out the coefficient  b  will be zero.

I played a bit with maple as follows. The circle can be parameterized as
the set ((1-t^2)/(1+t^2), (2t)/(1+t^2)). Substitute into  f  to get a
function  g(t)  on the real line, which extends to the point at infinity.
Compute  g'  to find the minima and maxima of  g; it is by setting
the limit of g' (as t -> oo) to zero that we get the condition that  b=0.
Then  t=0  (i.e. the point  (1,0)) is also a critical point for  g. There
remains a quartic to determine the location of any other minima or maxima, 
but this is really a quadratic equation in  u=t-1/t, so that if there is
a critical point at  t,  there is also one at  -1/t  (that is, if there is
one at (x,y), there is another at  (-x,-y).) So you can parameterize the
equivalence classes of cubics by pairs of lines through the plane, together
with an overall scale factor (e.g. the value of  f  at  (-1,0) ). (This
would have to be interpreted suitably to cover the cubics which have only
one pair of real critical points on the circle.)

Note that in the quadratic case you have more than one function of the
"simple" type in each equivalence class (e.g. x^2-y^2 and -x^2+y^2).
For cubics you have either a unique representation (in the case  f  has
only one minimum on the circle) or three (in case  f  has three minima).
With  b=0, these two cases are distinguished by the sign of  9d^2-12ca+8c^2.

Your question suggests others in which the cubics are not homogeneous;
the number of variables in increased; the degree of the equation is increased;
additional equations are imposed; or the equivalence classes of functions
are enlarged. This provides many perspectives on the problem, most of
them incorporated into classical parts of algebraic geometry.

dave