Existence of canonical form for cubic and quartic form?

Question

I am a post graduate student who is currently studying some optimization for quadratic form. From the class lecture, I know that we can always turn any quadratic functions into theirs corresponding canonical form using some changes of variable.

For example: $f\left( {x,y} \right) = x \times y$ can be rewritten as $f\left( {X,Y} \right) = \frac{1}{4}{X^2} - \frac{1}{4}{Y^2}$ through the following change of variable $\left\{ {\begin{array}{*{20}{c}} {X = x + y}\\ {Y = x - y} \end{array}} \right.$.

Out of pure curiosity, my question is:

Does "canonical cubic form" and "canonical quartic form" for function of the form $f\left( {x,y,z} \right) = xyz$ and $g\left( {x,y,z,t} \right) = xyzt$ respectively even existed ?

If these canonical form existed then what would be a systematic way to find them ?

Thank you for your enthusiasm !

Are you working over $\Bbb C$? $\Bbb R$? In either case, not all quadratic forms can be written in the form $X^2 - Y^2$ for some choice of basis, e.g., the zero quadratic form. One can likewise ask about normal forms for symmetric trilinear forms on $\Bbb R^2$, i.e., in $2$ variables. — Travis Willse, Apr 05 '24 at 20:17
@TravisWillse sorry for not being clear enough, I am working with the real number since this is an engineering problem. But if complex number does reveal some interesting insight, I am willing to take a look at it too. — Tuong Nguyen Minh, Apr 06 '24 at 12:20
There's always the polynomial analogue of the Waring Problems and e.g. that the form defining a cubic surface can be written as as sum of five cubes of linear forms. — Jan-Magnus Økland, Apr 06 '24 at 14:16
@TuongNguyenMinh It sounds like you're essentially asking for a cubic/quartic version of Sylvester's Law of Inertia. Is that correct? — Travis Willse, Apr 07 '24 at 01:33
Wow I have just check it. Yes I am not sure if the cubic or quartic version even exist and if it exist then how to find it ? https://math.stackexchange.com/questions/1817906/sylvesters-law-of-inertia — Tuong Nguyen Minh, Apr 07 '24 at 07:24
Maybe see Laws of inertia in higher degree binary forms by Bruce Reznick Proc. Amer. Math. Soc. 138 (2010), 815-826 — Jan-Magnus Økland, Apr 07 '24 at 11:01
That article is really interesting but it seems to be true for even degree — Tuong Nguyen Minh, Apr 07 '24 at 13:51
Try searching for the polynomial Waring problem or rank. Your question is about monomials, increasing the degree but also the dimension. The geometry of the homogeneous form $xy$ gives two points, $0$ and $\infty,$ on the projective line, $xyz$ the degree three curve of three “coordinate” lines in the projective plane and $xyzt$ the degree four surface of the four coordinate planes in projective space. A quadric surface given by a general degree four form in four variables is an example of a K3-surface. The general cubic surface in space is rational, and in this context especially interesting — Jan-Magnus Økland, Apr 07 '24 at 17:30
Oh, I was referring to another comment of mine; uniquely decomposable as a sum of five cubes of linear forms. — Jan-Magnus Økland, Apr 08 '24 at 02:35

score 3 · Accepted Answer · answered Apr 07 '24 at 21:32

3

$2\cdot 2! xy=(x+y)^2-(x-y)^2$

$2^2\cdot 3! xyz=(x+y+z)^3-(x+y-z)^3-(x-y+z)^3+(x-y-z)^3 $

$2^3\cdot 4! xyzt=(x+y+z+t)^4-(x+y+z-t)^4-(x+y-z+t)^4-(x-y+z+t)^4 +(x+y-z-t)^4+(x-y+z-t)^4+(x-y-z+t)^4-(x-y-z-t)^4$

$\vdots$

answered Apr 07 '24 at 21:32

Jan-Magnus Økland

6,019

2

I learned this from page one of The Waring problem for polynomials Geometry and applications the Master’s thesis (online version) by Leo Kayser. – Jan-Magnus Økland Apr 08 '24 at 07:37
Oh thank you so much, I have already upvoted your answer ! – Tuong Nguyen Minh Apr 14 '24 at 10:22

Existence of canonical form for cubic and quartic form?

1 Answers1