Maury Barbato
Posts:
792
From:
University Federico II of Naples
Registered:
3/15/05


Re: Polynomials' Zeros
Posted:
Apr 26, 2013 1:07 PM


I wrote:
> 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
I have found a way to construct countable I and J. Let {x_1,x_2,?} be a countable subset of K such that for every x_i we have p(x_i,y)=0 for finitely many y in K (this subset exists for property (1)). Analogously, let {y_1,y_2,?} be a countable subset of K such that for every y_i p(x,y_i)=0 for finitely many x in K.
Let Z_{+}={1,2,3,?} be the set of positive integers. For every positive integer n, define the (finite) set
S_n={ m in Z_{+}  p(x_n,y_m)=0}.
Analogously, let T_n be the set
T_n={m in Z_{+}  p(x_m,y_n)=0}.
We construct I={x_{i_1}, x_{i_2},?} and J={y_{j_1}, y_{j_2},?} by induction in the following way. Define i_1=1. Suppose that we have defined i_1,?,i_k, and j_1,?,j_{k1}. Then we define
j_k = min Z_{+} \ ({j_1,?,j_{k1}} \/ T_{i_1} \/ T_{i_2} \/ ?\/ T_{i_k}),
and
i_{k+1}= min Z_{+} \({i_1,?,i_k} \/ S_{j_1} \/ S_{j_2} \/ ? \/ S_{j_k}).
It is immediate from construction that I and J have the desired property. I think the same construction can be used to show that we can find I and J with I=J=K by using transfinite induction insted of ordinary induction. But I don't know the theory of ordinal numbers so well, so I am not sure. What do you think? Thank you very very much for your attention. My Best Regards, Maurizio Barbato

