Is a bivariate function that is a polynomial function with respect to each variable necessarily a bivariate polynomial?
Let $ \mathbb{F} $ be an uncountable field. Suppose that $ f: \mathbb{F}^{2} \rightarrow \mathbb{F} $ satisfies the following two properties:
- For each $ x \in \mathbb{F} $, the function $ f(x,\cdot): y \mapsto f(x,y) $ is a polynomial function on $ \mathbb{F} $.
- For each $ y \in \mathbb{F} $, the function $ f(\cdot,y): x \mapsto f(x,y) $ is a polynomial function on $ \mathbb{F} $.
Is it necessarily true that $ f $ is a bivariate polynomial function on $ \mathbb{F}^{2} $? What if $ \mathbb{F} $ is merely countably infinite?
Solution 1:
Maybe this works for the countably infinite case. Order the rationals (or whatever countably infinite field you have) as $r_1,r_2,\dots$. Let $$f(x,y)=(x-r_1)(y-r_1)+(x-r_1)(x-r_2)(y-r_1)(y-r_2)+\cdots$$ Then if $r$ is any rational, say, $r=r_j$, then $f(r,y)$ is a polynomial of degree $j-1$ in $y$, and similarly for $f(x,r)$. But clearly $f$ is not a polynomial function --- what would be its degree?
Solution 2:
As shown is Gerry Myerson’s answer, the answer is NO when $\mathbb F$ is countably infinite.
The answer is YES when $\mathbb F$ is uncountable, however.
Sketch of proof : since there are only countably many degrees, the polynomials will share a common degree on an uncountable set. This bound on the degree allows one to use interpolation, and to retrieve the whole of $f$.
More detailed proof : Denote by $d(x)$ the degree of the univariate polynomial $f(x,.)$ for $x\in {\mathbb F}$ (recall that the degree of the zero polynomial is $-\infty$), and put $U_d=\lbrace x \in {\mathbb F} | d(x)=d\rbrace$ for $d\in \lbrace -\infty \rbrace \cup {\mathbb N}$. Then the $U_d$ form a countable partition of $\mathbb F$, so at least one of the $U_d$, say $U_{n}$, is uncountable.
We may assume that $n>0$, as the cases $n=-\infty$ and $n=0$ are similar and simpler. Let $y_0,y_1, \ldots y_{n}$ be $n+1$ distinct values in $\mathbb F$, this is possible because $\mathbb F$ is uncountable. (if the characteristic of $\mathbb F$ is zero, we can simply take $y_i=i$). Using Lagrange interpolation, let us put
$$L_k(y)=\frac{\prod_{j \neq k}{(x-x_j)}}{\prod_{j \neq k}{(x_k-x_j)}}$$
for $0 \leq k \leq n$. Then one has, for any polynomial $P$ of degree $\leq n$ and any $y\in{\mathbb F}$,
$$ P(y)=P(y_0)L_0(y)+P(y_1)L_1(y)+ \ldots +P(y_n)L_n(y) $$
In particular, one has for any $(x,y)\in U_n \times {\mathbb F}$,
$$ (1) \ f(x,y)=f(x,y_0)L_0(y)+f(x,y_1)L_1(y)+f(x,y_2)L_2(y)+ \ldots +f(x,y_n)L_n(y) $$ The right-hand side is a fixed bivariate polynomial, let us denote it by $Q(x,y)$. Let $y\in {\mathbb F}$. Then the two univariate polynomials $f(.,y)$ and $Q(.,y)$ coincide on the uncountable set $U_n$, so they must coincide everywhere. Finally $f=Q$ everywhere and we are done.