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From: hanrahan398 on 4 Jun 2010 11:12
Does f(x,y) have to be a polynomial function of x and y, if for
constant x it is a polynomial in y and for constant y it is a
polynomial in x? I haven't been able to prove this.

Michael
From: hanrahan398 on 4 Jun 2010 11:35
On Jun 4, 4:12 pm, hanrahan...(a)yahoo.co.uk wrote:
> Does f(x,y) have to be a polynomial function of x and y, if for
> constant x it is a polynomial in y and for constant y it is a
> polynomial in x? I haven't been able to prove this.

I should have specified: x and y are real, so is f(x,y), and the
function is a polynomial in y for EVERY constant x, and also in x for
EVERY constant y.

Michael
From: Stephen Montgomery-Smith on 5 Jun 2010 01:20
hanrahan398(a)yahoo.co.uk wrote:
> On Jun 4, 4:12 pm, hanrahan...(a)yahoo.co.uk wrote:
>> Does f(x,y) have to be a polynomial function of x and y, if for
>> constant x it is a polynomial in y and for constant y it is a
>> polynomial in x? I haven't been able to prove this.
>
> I should have specified: x and y are real, so is f(x,y), and the
> function is a polynomial in y for EVERY constant x, and also in x for
> EVERY constant y.
>
> Michael

I solved this problem many years ago. It isn't totally straightforward.
You have to use the fact that the reals are uncountable.
From: William Elliot on 5 Jun 2010 04:22

On Sat, 5 Jun 2010, Stephen Montgomery-Smith wrote:
> hanrahan398(a)yahoo.co.uk wrote:
>> On Jun 4, 4:12 pm, hanrahan...(a)yahoo.co.uk wrote:

>>> Does f(x,y) have to be a polynomial function of x and y, if for
>>> constant x it is a polynomial in y and for constant y it is a
>>> polynomial in x? I haven't been able to prove this.
>>
>> I should have specified: x and y are real, so is f(x,y), and the
>> function is a polynomial in y for EVERY constant x, and also in x for
>> EVERY constant y.
>
> I solved this problem many years ago. It isn't totally straightforward.
> You have to use the fact that the reals are uncountable.
>
Does f(x,y) as described above have the form
sum(j=1,..n) pk(x).qk(y)
where, for k = 1,.. n,
pk(x) in R[x], qk(y) in R[y].


From: hanrahan398 on 5 Jun 2010 09:30
On Jun 5, 6:20 am, Stephen Montgomery-Smith
<step...(a)math.missouri.edu> wrote:
> hanrahan...(a)yahoo.co.uk wrote:
> > On Jun 4, 4:12 pm, hanrahan...(a)yahoo.co.uk wrote:
> >> Does f(x,y) have to be a polynomial function of x and y, if for
> >> constant x it is a polynomial in y and for constant y it is a
> >> polynomial in x? I haven't been able to prove this.
>
> > I should have specified: x and y are real, so is f(x,y), and the
> > function is a polynomial in y for EVERY constant x, and also in x for
> > EVERY constant y.
>
> > Michael
>
> I solved this problem many years ago.  It isn't totally straightforward..
>   You have to use the fact that the reals are uncountable.

Hi Stephen. Thanks for this. I haven't yet worked out how the
uncountability of the reals can be used, but do you remember whether
the answer is yes or no to whether f(x,y) must be a polynomial?

Michael
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