Re: divisors of zero

In article <cjnelson9-852E3E.15440530042006@xxxxxxxxxxxxxxxx>,
Clifford Nelson <cjnelson9@xxxxxxxxxxx> wrote:
http://groups.google.com/group/geometry.research/browse_frm/thread/23704a
ecd086c689/e365d3da1264633d?lnk=st&q=4D+field&rnum=12#e365d3da1264633d

If you can get the URL above to work you can see that Robin Chapman and
John Rickard found divisors of zero for B_5 numbers in about a day. I
don't know how they found them. How would you find them with
Mathematica?

I don't know what they did, either. But here's what I might do in Maple.
This is the formula for (a,b,c,d)*(e,f,g,h), quoted in Chapman's article:

L:= [1/5*((a + b + c + d)*e - d*f - c*g - a*(-e - f - g - h) -
b*h), 1/5*(-a*e + (a + b + c + d)*f - d*g -
b*(-e - f - g - h) - c*h),
1/5*(-b*e - a*f + (a + b + c + d)*g -
c*(-e - f - g - h) - d*h),
1/5*(-c*e - b*f - a*g - d*(-e - f - g - h) +
(a + b + c + d)*h)];

Since the components are all homogeneous of degree 2, we might try for
a=e=1. And to cut down the number of variables some more, try c=d and
g=h (OK, I cheated and noticed that Chapman's and Rickard's solutions
had that).

solve(eval(L, {a=1,e=1,c=d,g=h}));

One of the results Maple gives is

{h = 1/2*RootOf(_Z^2+_Z-1,label = _L1),
f = -1/2-1/2*RootOf(_Z^2+_Z-1,label = _L1),
d = -1/2-1/2*RootOf(_Z^2+_Z-1,label = _L1),
b = 1/2*RootOf(_Z^2+_Z-1,label = _L1)}

Applying allvalues to this gives one solution as

h = -1/4+1/4*5^(1/2), b = -1/4+1/4*5^(1/2), f = -1/4-1/4*5^(1/2),
d = -1/4-1/4*5^(1/2)

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.