From: vadimpaster@virtue.com (Vadim Pasternak)
Subject: Is this function polynomial ?
Date: 1 Jun 2001 07:39:05 -0400
Newsgroups: sci.math
Summary: Functions on products which restrict to polynomials on lines

Suppose  f:Q^2 -> Q (Q is the rationals) is a function of two
variables x and y such that f(x,y) is a polynomial in y for each
fixed x, and a polynomial in x for each fixed y . 
Is f(x,y) a polynomial in both x and y ?
i.e f(x,y) = p(x,y) where p is polynomial in 2 variables ? 

Thanks
==============================================================================

From: Douglas Zare <zare@math.columbia.edu>
Subject: Re: Is this function polynomial ?
Date: Fri, 01 Jun 2001 23:00:04 -0400
Newsgroups: sci.math

Azmi Tamid wrote:

> [...]
> Douglas can you please share the construction with the rest of us that
> do not know what is the experts site ?

Well, since someone else asked, for any countable subset of the
complex numbers S, there are functions on SxS which are not
polynomials of two variables but whose restriction to any {a}xS or
Sx{b} is a polynomial. Here is a construction: Enumerate S:
S={s_0,s_1,s_2,...}. Choose values for the diagonal {(s_i,s_i)}. Let
the function be constant on {s_0}xS and Sx{s_0}. Let the function be
linear on {s_1}xS and Sx{s_1} (interpolating through the values at s_0
and s_1), at most quadratic on {s_2}xS and Sx{s_2} (using the values
at s_0, s_1, and s_2) etc.

For some choices of the values on the diagonal, the result will be a
polynomial, but this will not be the case if the choice is
f(s_i,s_i)=i or (-1)^i for any enumeration of the rationals. One could
also choose (s_i,s_i) at the ith step to force the polynomial function
on {s_i}xS to have degree exactly i.

Douglas Zare

[reformatted --djr]