Assuming in Lemma 1
and changing, if necessary, the numbering of equations, equalities (
4) can be written in the form
where
A is a matrix whose elements are analytic functions, generally speaking, are ambiguous.
Proof of Lemma 3. Let
where
is the smallest possible. Differentiating equality (
6), we obtain
Since the number
n is minimal, the linear combinations on the left-hand sides equalities (
6) and (
7) must be proportional. In this way,
Hence, and, since , then , . □
Proof of Lemma 4. If functions (
9) are algebraically independent over
, then it is convenient to carry out all operations with them formally, as with their corresponding variables
The fundamental matrix
takes the form
where
is a rational function of variables (
10), defined by the equation
, equivalent to
from where
where
are algebraic complements the corresponding elements of the matrix
, which are polynomials in variables (
10). Note that for a different choice of functions (
9) included in
, the function
, generally speaking, is multiplied by some factor from
.
The derivatives with respect to
z of variables (
10) can be calculated formally, proceeding from the systems of Equation (
8) and equalities (
11).
Let the function
v satisfy the equation
and belongs to the field
, that is, it can be represented in the form
where
T is a rational function over
of functions (
9) and
z,
P and
Q are polynomials in the same functions,
.
Replacing functions (
9) in equality (
13) by variables (
10) and differentiating it with respect to
z, we obtain
where
,
are polynomials in variables (
10) and
z,
. In view of equality (
12)
identically by (
10) and
z. This implies, that if in equality (
13) instead of
,
we substitute any other linearly independent solutions of the corresponding systems
, such that
, then the function
will be a solution of Equation (
12) and, therefore,
,
. Let
T really depend on the variables included in the matrix
, and
. Substitute in
T instead of variables
functions
,
, where
is a new variable, and instead of the remaining variables (
10), the corresponding functions (
9). Obviously, the Jacobian
will not change in this case. Then
In view of the algebraic independence of functions (
9), this equality is preserved when replacing functions (
9) with the corresponding variables (
10). Differentiating after such a change equality (
14) with respect to
and then setting
, we get
We define the degree of a rational function with respect to any set of variables as the difference between the degrees of the numerator and denominator for this population. It is easy check that, with such a definition, the degree of the product of rational functions is equal to the sum of the powers of the factors, the degree of the sum does not exceed the maximum degrees of terms, and when taking a partial derivative with respect to some variable from the selected population, the degree decreases. Hence, the degree of the right-hand side of equality (
15) with respect to the set of variables
is strictly less than the degree of the left side, except for the case when
T does not depend on these variables, and
. Exactly the same reasoning shows that in the case of
T does not depend on
,
, and in case
T does not depend on
. It remains to prove that for
T does not depend on
,
. Repeating the previous reasoning taking into account that according to Liouville’s formula
, instead of (
15) we get
Comparing the degrees in the set of variables
,
left and right sides of the resulting equality, we conclude that
T does not depend on
and
. Thus,
T is independent on variables (
10) included in
, and therefore, on all variables (
10). □
The Corollary 1 of Lemma 4 is obtained using Lemma 3 and the fact that any product of powers of functions whose logarithmic derivatives belong to , is a function with the same property.
Proof of Lemma 5. From Lemma 7 of Chapter 3 in [
3] it follows that if
and the rank of the set of linear forms
is less than
q, then there is at least one nontrivial solution of system (
16), under whose substitution all linear forms
,
vanish. If
, where
, this means that
. Then, in view of the conditions of the Lemma, we obtain that for any fixed
s and
t the functions
are linearly independent linear forms of
. The coefficients of these linear forms are independent of
s and therefore constitute the matrix
, and
. □
Proof of Theorem 1. It is enough to show what if equalities (
1) are not satisfied with conditions (
3), then they are not satisfied with the condition
.
It is easy to check that the matrix
will be the fundamental matrix of the system
, where
The matrix satisfies the conditions of Lemma 5, therefore the matrix
is the fundamental matrix of the differential equation
moreover,
,
,
. If equalities (
1) with conditions (
3) are not satisfied for the matrices
for
, then, according to Lemma 2, they also fail for
for
. Then, conditions (
4) of Lemma 1, in which we put
, cannot hold for matrices
. Therefore, from Lemma 1 we obtain that
of functions
are algebraically independent over
. Then, in view of Corollary 2, are algebraically independent over
functions
Since
, hence, taking into account Corollary 1, we conclude that equalities (
1) are impossible. □