We are grateful to Peter Thompson for pointing out an error in [1, Lemma 3.5, p. 1848]. The original proof worked only under the assumption that is a vector of constants. However, some of the components of could be the states of the dynamic under consideration, and the lemma was used in such a setup (i.e., with involving states) later in [1, Proposition 3.4].
We give a more explicit version of the statement and provide a correct proof. The desired statement will be deduced from the following:
Lemma 1.Consider a system of differential equations
(1)
where
and
are tuples of differential indeterminates,
are scalar parameters, and
. Let
be the LCM of the denominators of
. Let
be a nonzero differential polynomial. Then there exist nonzero
and
such that, for every tuple
and every power series solution
of (1) with parameters
in
such that
we have
Proof.Consider the following differential ideal
We claim that
I contains a nonzero polynomial of the form
such that
and
. First we will show that, if the claim is true, then
P1 and
P2 satisfy the condition of the lemma. Assume the contrary, that there is a power series solution
of (1) with parameters
such that the constant term of
is nonzero but
. Since
is a zero of differential polynomials
P and
for every
, it is a zero of the ideal
Since
, every element in
I, which is the saturation of the above ideal at
Q, also vanishes at
. In particular,
vanishes at
, so we arrive at the contradiction with
.
Now we will prove the claim. Consider the ring . Let J be the ideal generated by in R. The definition of I via the saturation at Q implies that
Thus, it is sufficient to prove that there is an element of the form
with
and
in
J. We define a derivation
on
R (which is basically the Lie derivative) by
Since
and
I is a differential ideal,
J is invariant under
.
Let be the localization of R with respect to and be the ideal generated by J in this localization. The derivation can be naturally extended to , and is also -invariant. It is sufficient to prove that . Consider a nonzero element of with the smallest number of monomials and, among such elements, an element of the smallest total degree. We will call it S. If , we are done. Otherwise, one of u appears in S, say u1. Let .
Since is a Noetherian ring, there exists such that
We have
for
and
Therefore, we have
If
S were divisible by
, then
would have the same number of monomials but smaller degree, this contradicts to the choice of
S. Therefore,
has fewer monomials than
S thus contradicting the choice of
S.
The following corollary is equivalent to [1, Lemma 3.5, p. 1848] but explicitly highlights that some of the entries of may be initial conditions, not only system parameters.
Corollary 1. (Clarified version of [[1], Lemma 3.5, p. 1848])In the notation of [1, Section 2.2], let be nonzero. Then there exist nonempty Zariski open subsets and such that, for every , , and the corresponding , the function is a nonzero element of .
Proof.We apply Lemma 1 to the model Σ and the polynomial P as in the statement, and obtain polynomials and P2(u). We define Zariski open sets Θ and U by and , respectively. Then the lemma implies that, for and , will be a nonzero function.