[GAP Forum] Automorphism groups of homogeneous polynomials

[Alexander Hulpke]

Dear Graham Gerrard, dear Gap-forum,

On Oct 6, 2009, at 10/6/09 12:26, Graham Gerrard wrote:

> Any homogeneous polynomial in n variables has an automorphism group  
> which
> preserves the polynomial.
>
> How do I tackle generation of such a group using GAP?

If your n is small (say up to 12 or probably 15), you could compute  
the automorphism group immediately as stabilizer.

In your example (you mailed privately):
r:=PolynomialRing(Rationals,8);
inds:=IndeterminatesOfPolynomialRing(r);
a:=inds[1];
b:=inds[2];
c:=inds[3];
d:=inds[4];
e:=inds[5];
f:=inds[6];
g:=inds[7];
h:=inds[8];

s:=SymmetricGroup(Length(inds));
pol:= a*a*a*a+b*b*b*b+c*c*c*c+d*d*d*d+e*e*e*e+f*f*f*f+g*g*g*g+h*h*h*h
        
-2 
*a 
*a 
*b 
*b 
-2 
*a 
*a 
*c 
*c 
-2 
*a 
*a 
*d 
*d 
-2 
*c 
*c 
*f 
*f 
-2 
*d 
*d 
*f 
*f 
-2 
*e 
*e 
*f*f-2*b*b*g*g-2*d*d*g*g-2*e*e*g*g-2*b*b*h*h-2*c*c*h*h-2*e*e*h*h       
-8*a*b*e*f-8*a*c*e*g-8*a*d*e*h      +2*b*b*c*c+2*b*b*d*d+2*c*c*d*d 
+2*a*a*e*e+2*b*b*e*e+2*c*c*e*e+2*d*d*e*e+2*a*a*f*f+2*b*b*f*f+2*a*a*g*g 
+2*c*c*g*g+2*f*f*g*g+2*a*a*h*h+2*d*d*h*h+2*f*f*h*h+2*g*g*h*h
       +8*b*c*f*g+8*b*d*f*h+8*c*d*g*h;

gap> u:=Stabilizer(s,pol,OnIndeterminates);
Group([ (1,5)(2,6)(3,7)(4,8), (1,5)(2,6)(3,8)(4,7), (1,5)(2,7)(3,6) 
(4,8) ])
gap> Size(u);
12

However this is a naive orbit/stabilizer computation whose cost will  
grow with n! and you probably won't be able to go much further. If you  
can find a partition condition for the indeterminate which would let  
you work in a wreath product instead, this would let you go further.

In general,translating to a graph problem, and using nauty (via the  
GRAPE package) is the way to go.

Best,

    Alexander Hulpke







-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke