Is induction starting at $n=0$ on $F[x_1,\dots,x_n]$ correct? Suppose you have some polynomial ring $F[x_1,\dots,x_n]$ and you want to prove some fact about it by induction on the number of formal variables $x_i$.
Is it ever wrong to start with $n=0$? Since when $n=0$, we are left with just $F$, which sometimes is not a polynomial ring anymore.
Note: assuming that the fact we are proving make senses for $F$ too.
 A: It may depend on the target statement, but it is often worth a try to start induction one step earlier than the target statement suggests. E.g. to show that the number of subsets of a nonempty finite set is $2^n$, why not start with the empty set instead of with a singleton set? 
Of course this will fail if you want to show tha tthe number of these subsets is even, simply because that is wrong for th eempty set.
And in your concrete setting: Formally, $F$ is a polynomial ring. A polynomial ring over $F$ in a set $S$ of indeterminates, where $F\cap S=\emptyset$, is nothing but a ring $A$ together with a fixed ring hoimomorphism $\iota\colon F\to A$ and a fixed map $i\colon S\to A$ such that for every ring $B$, ring homomorpphism $\phi\colon F\to B$ and map $f\colon S\to B$ there exits a unique ring homomorphism $\eta\colon A\to B$ satisfying $\eta\circ \iota=\phi$ and $\eta\circ i=f$. 
There's nothing wrong with allowing $S=\emptyset$.
Except of course your target statement somehow requires at least one indeterminate (e.g. "$F$ is a proper subring of ...").
