Maury Barbato
Posts:
791
From:
University Federico II of Naples
Registered:
3/15/05


Re: Polynomials' Zeros
Posted:
Apr 25, 2013 8:52 AM


Butch Malahide:
> On Apr 23, 1:42 pm, Butch Malahide > <fred.gal...@gmail.com> wrote: > > On Apr 23, 12:25 pm, dullr...@sprynet.com wrote: > > > > > > > > > > > > > On Tue, 23 Apr 2013 09:52:05 0700 (PDT), > sputaro...@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. > > > [. . .] So p = 0. > > > > I think it's a little bit more complicated. How do > you get from the > > negation of the statement > > > > "p(x,y) = 0 for all (x,y) in IxJ, where I and J are > infinite" > > > > to the OP's statement > > > > "p(x,y) =/= 0 for all (x,y) in IxJ, where I and J > are infinite"? > > > > I think something like the following will work. I > believe the argument > > you gave actually proves the stronger result: > > > > [(for infinitely many x) (for infinitely many y) > p(x,y) = 0] implies p > > = 0. > > > > That is, if there exist an infinite set I and > infinite sets J_x (x in > > I) such that p(x,y) = 0 whenever x in I and y in > J_x, then p = 0. > > > > In other words, if p =/= 0, then (writing "a.e." > for "all but finitely > > many"): > > > > (1) (for a.e. x) (for a.e. y) p(x,y) =/= 0. > > > > Also, interchanging the roles of x and y, > > > > (2) (for a.e. y) (for a.e. x) p(x,y) =/= 0. > > > > From (1) and (2) we can construct infinite sets I > and J such that > > p(x,y) =/= 0 whenever x in I and y in J. > > Moreover, if K = aleph_{nu}, we can make I = J > = aleph_{nu}.
Dear Butch, as you said, prof. Ullrich's argument actually proves that (1) and (2) are true, but I don't see how to use these two statements in order to prove that there exist two sets I and J with the same cardinality of K, such that p(x,y) =/= = for every x in I and y in J. Could you give some hint, please?
Thank you very very much for your invaluable help. I couldn't have really realized alone how to use prof. Ullirch's argument to give an answer to my question. My Best Regards, Maurizio Barbato
A person who never made a mistake never tried anything new. A. Einstein

