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From: quasi on 10 Aug 2010 16:59

Prove or disprove:

If f in Q[x,y,z] is a symmetric polynomial such that

f = g^3 + h^3

for some g,h in Q[x,y,z], then g and h are symmetric.

quasi
From: Robert Israel on 10 Aug 2010 19:35
quasi <quasi(a)null.set> writes:

>
> Prove or disprove:
>
> If f in Q[x,y,z] is a symmetric polynomial such that
>
> f = g^3 + h^3
>
> for some g,h in Q[x,y,z], then g and h are symmetric.
>

As in my recent posting:
g = (a x + b y + c z) (a y + b z + c x) (a z + b x + c y)
h = (a x + b z + c y) (a y + b x + c z) (a z + b y + c x)

where a,b,c are distinct.

g and h are not symmetric, but are interchanged by odd permutations of x,y,z.
--
Robert Israel israel(a)math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
From: Gerry Myerson on 11 Aug 2010 01:41
In article <rbisrael.20100810232803$4794(a)news.acm.uiuc.edu>,
Robert Israel <israel(a)math.MyUniversitysInitials.ca> wrote:

> quasi <quasi(a)null.set> writes:
>
> >
> > Prove or disprove:
> >
> > If f in Q[x,y,z] is a symmetric polynomial such that
> >
> > f = g^3 + h^3
> >
> > for some g,h in Q[x,y,z], then g and h are symmetric.
> >
>
> As in my recent posting:
> g = (a x + b y + c z) (a y + b z + c x) (a z + b x + c y)
> h = (a x + b z + c y) (a y + b x + c z) (a z + b y + c x)
>
> where a,b,c are distinct.
>
> g and h are not symmetric, but are interchanged by odd permutations of x,y,z.

Perhaps a simpler counterexample is g(x, y, z) = x, h(x, y, z) = -x.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: quasi on 11 Aug 2010 03:02
On Tue, 10 Aug 2010 18:35:49 -0500, Robert Israel
<israel(a)math.MyUniversitysInitials.ca> wrote:

>quasi <quasi(a)null.set> writes:
>
>>
>> Prove or disprove:
>>
>> If f in Q[x,y,z] is a symmetric polynomial such that
>>
>> f = g^3 + h^3
>>
>> for some g,h in Q[x,y,z], then g and h are symmetric.
>>
>
>As in my recent posting:
> g = (a x + b y + c z) (a y + b z + c x) (a z + b x + c y)
> h = (a x + b z + c y) (a y + b x + c z) (a z + b y + c x)
>
>where a,b,c are distinct.
>
>g and h are not symmetric, but are interchanged by odd permutations of x,y,z.

Yes, very nice.

Thanks, Robert.

I'll have to rethink the underlying idea.

quasi
From: quasi on 11 Aug 2010 03:10
On Wed, 11 Aug 2010 15:41:30 +1000, Gerry Myerson
<gerry(a)maths.mq.edi.ai.i2u4email> wrote:

>In article <rbisrael.20100810232803$4794(a)news.acm.uiuc.edu>,
> Robert Israel <israel(a)math.MyUniversitysInitials.ca> wrote:
>
>> quasi <quasi(a)null.set> writes:
>>
>> >
>> > Prove or disprove:
>> >
>> > If f in Q[x,y,z] is a symmetric polynomial such that
>> >
>> > f = g^3 + h^3
>> >
>> > for some g,h in Q[x,y,z], then g and h are symmetric.
>> >
>>
>> As in my recent posting:
>> g = (a x + b y + c z) (a y + b z + c x) (a z + b x + c y)
>> h = (a x + b z + c y) (a y + b x + c z) (a z + b y + c x)
>>
>> where a,b,c are distinct.
>>
>> g and h are not symmetric, but are interchanged by odd permutations of x,y,z.
>
>Perhaps a simpler counterexample is g(x, y, z) = x, h(x, y, z) = -x.

Yipes!

That's _too_ simple!

Evidently, odd powers had no chance!

quasi
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