First of all, a happy new year. Be it better than 2015,
healthy, wealthy, fruitful and cross-fertilizing
for you, familly and friends.
In order to cope with families of solutions of evolution equations, I had to prove the following lemma
Lemma: Let $Z=\{z_n\}_{n\in \mathbb{N}}$ be a set of indeterminates, then
$[e^{z_0},e^{z_1}]$ is algebraically independent on $\mathbb{C}[Z]$
within $\mathbb{C}[[Z]]$. In other words, if a finitely supported sum
$$
\sum_{n,m}P_{n,m}[(z_i)_{i\geq 0}]e^{nz_0}e^{mz_1}
$$
is zero, then every polynomial $P_{n,m}\in \mathbb{C}[Z]$ is zero.
I cannot imagine it is unknown among specialists. My question is :
Does someone have a reference for this property ?
Thanks in advance.
On request, a proof below (I have withdrawn the – too long – previous one, using "orders of infinity".)
All relies on the following proposition which is characteristic free.
Proposition Let $(\mathcal{A},d)$ be a commutative differential ring without zero divisor, and $R=ker(d)$ be its subring of constants. Let $z\in \mathcal{A}$ such that $d(z)=1$ and $S=\{e_\alpha\}_{\alpha\in I}$ be a set of eigenfunctions of $d$ all different ($I\subset R$) i.e.
- $e_\alpha\not=0$
- $d(e_\alpha)=\alpha e_\alpha\ ;\ \alpha\in I$
Then the family $(e_\alpha)_{\alpha\in I}$ is linearly free over $R[z]$ (the subring generated by $R\cup \{z\}$, see remark).
From this, one can show the
Corollary Let $\mathcal{A}$ be a $\mathbb{Q}$-algebra (associative, commutative and unital) and $z$ an indeterminate, then $\{z,e^z\}\subset \mathcal{A}[[z]]$ are algebraically independent over $\mathcal{A}$.
Remark
If $\mathcal{A}$ is a $\mathbb{Q}$-algebra or only of characteristic zero, i.e.
$$
n1_\mathcal{A}=0\Rightarrow n=0
$$
then $d(z)=1$ implies that $z$ is transcendent over $R$. This is never the case in characteristic $p$ where $z^p$ is a constant.
End of remark
One finishes the job proving, by successive extensions, that the sets
$$
\{e^{z_0},e^{z_1},\cdots e^{z_n},z_0,z_1,\cdots ,z_n\}
$$
are algebraically independent over $\mathcal{A}$. Which is stronger than the desired result.
Best Answer
First of all, I am not technically answering your question because you ask specifically for a reference. However, here is a nice algebraic proof.
The result should follow immediately from the following lemma, assumed known for $k = \mathbb C$.
Lemma 1. Let $k$ be a field of characteristic $0$. Then the elements $z, e^z \in k[[z]]$ are algebraically independent over $k$.
Proof. If they are algebraically dependent, there exists a finitely generated (over $\mathbb Q$) subfield $k'\subseteq k$ over which they are algebraically dependent. But any finitely generated field of characteristic $0$ embeds into $\mathbb C$, contradicting the fact that the lemma is true over $\mathbb C$. $\square$
(This type of argument is known in algebraic geometry as the Lefschetz principle. I believe the first place where it appeared was Lefschetz's Algebraic Geometry textbook from the fifties, but I have not been able to verify that it didn't appear somewhere before. See also this answer on MO for a survey.)
Recall the following lemmata from transcendence theory.
Lemma 2. Let $\Omega$ be a big overfield, and let $k \subseteq \Omega$ be some small base field. Let $\alpha_1, \ldots, \alpha_n \in \Omega$. Then $\alpha_1, \ldots, \alpha_n$ are algebraically independent over $k$ if and only if $\operatorname{tr.deg} (k(\alpha_1,\ldots,\alpha_n)/k) = n$.
(Reference: this follows immediately from Theorem VIII.1.1 of Lang's Algebra.)
Lemma 3. Let $k \subseteq l \subseteq m$ be a tower. Then $\operatorname{tr.deg} (m/k) = \operatorname{tr.deg} (m/l) + \operatorname{tr.deg} (l/k)$.
(Reference: exercise in [loc. cit.].)
Proposition. The elements $z_0, \ldots, z_n, e^{z_0}, e^{z_1}$ are algebraically independent.
Proof. By Lemma 2, we have to prove that $\operatorname{tr.deg} (\mathbb C(z_0, \ldots, z_n, e^{z_0}, e^{z_1})/\mathbb C) = n+3$. We clearly have $$\operatorname{tr.deg}(\mathbb C(z_2,\ldots,z_n)/\mathbb C) = n-1,$$ since the $z_i$ are assumed to be algebraically independent. Letting $k = \mathbb C(z_2,\ldots,z_n)$, Lemma 1 tells us that $$\operatorname{tr.deg}(k(z_1, e^{z_1})/k) = 2.$$ Hence, by Lemma 3, we get $$\operatorname{tr.deg}(\mathbb C(z_1,\ldots,z_n,e^{z_1})/\mathbb C) = n+1.$$ Another application of Lemma 1 and Lemma 3 gives the result. $\square$
The result now follows, as you noted, because an algebraic dependence is defined over a finitely generated subextension.