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 for answering me. Your argument is correct, but 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

