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Topic: Polynomials' Zeros
Replies: 7   Last Post: Apr 26, 2013 1:07 PM

 Messages: [ Previous | Next ]
 Maury Barbato Posts: 792 From: University Federico II of Naples Registered: 3/15/05
Re: Polynomials' Zeros
Posted: Apr 23, 2013 1:58 PM

David C. Ullrich wrote:

> On Tue, 23 Apr 2013 09:52:05 -0700 (PDT),
> sputarospo@alice.it wrote:
>

> >Hello,
> >I have the following question.
> >Let P(X_1,...,X_n) be a non zero polynomial in

> X_1,...,X_n with coefficients in an infinite field K.
>

> >Does there exist n infinite subsets J_1,...,J_n of K
> such that
> >P(x_1,...,x_n) =/= 0
> >for every (x_1,...,x_n) in J_1 x J_2 x...x J_n?
> >The answer is urely yes if K is a subfield of C:

> just consider a neighborhood
> >of a point in which the poynomial has a non zero
> value.
> >But I don't know the answer for the general case.
> >What do you think about?

>
> Induction on n.
>
> Consider the case n = 2 to make things easier to
> type.
> Say p(x,y) = 0 for all (x,y) in IxJ, where I and J
> are infinite.
>
> For each fixed y, consider the polynomial p_y(x),
> defined by
>
> p_y(x) = p(x,y).
>
> For each y in J we have p_y = 0, since p_y has
> infinitely
> many zeroes.
>
> But there are polynomials q_j so that
>
> p(x,y) = p_y(x) = sum_j q_j(y) x^j.
>
> If y is in J then all the coefficients of p_y vanish.
> This says that q_j(y) = 0 for all y in J, and hence
> q_j = 0 since J is infinite. So p = 0.
>

> >My Best Regards,
> >Maurizio Barbato

>

Dear Prof. Ullrich, first of all thank you very much
it does not answer my question.
What you prove is that if p(x,y)=0 for every
(x,y) in IxJ, with I,J infinite, then p(X,Y) is
the zero polynomial.
My original question is different. I ask if
there exists some infinite sets I,J, such that
p(x,y) has always a non zero value in IxJ.
As I said in my original post, if K is a subfield
of C, then the answer is yes. We know that
p(x,y) is non zero for some (x,y) in K^2 (this
immediately follows from the statement you proved).
Then consider a neighborhood of (x,y).
By continuity, p(x,y) is non zero in this
neighborhood. So for some real intervals
I, J, we get p(x,y) =/= 0 for every (x,y) in IxJ.
Since K /\ I and K /\ J are infinite (they contain
all the rationals in I and J respectively)
we are done.
What about a generic infinite field K?
I don't know ...
Thank you very much again for your attention.
My Best Regards,
Maurizio Barbato

Date Subject Author
4/23/13 Maury Barbato
4/23/13 David C. Ullrich
4/23/13 Maury Barbato
4/23/13 Butch Malahide
4/23/13 Butch Malahide
4/25/13 Maury Barbato
4/26/13 Maury Barbato
4/24/13 David C. Ullrich