On the non-existence of rational first integrals for systems of linear differential equations

On the non-existence of rational first integrals for systems of linear differential equations

Andrzej Nowicki

N. Copernicus University, Faculty of Mathematics and Computer Science, ul. Chopina 12–18, 87–100 Toru´ n, Poland

(e-mail: anow@mat.uni.torun.pl) March 11, 1994

Abstract

We present a description of all systems of linear differential equations which do not admit any rational first integral.

1 Introduction.

Let k[X] = k[x₁, . . . , x_n] be the polynomial ring in n variables over a field k of char- acteristic zero, and let k(X) = k(x1, . . . , xn) be the quotient field of k[X]. Assume that f = (f₁, . . . , f_n) ∈ k[X]ⁿ and consider a system of polynomial ordinary differential equa- tions

dx_i(t)

dt = f_i(x₁(t), . . . , x_n(t)), (1.1) where i = 1, . . . , n.

This system has a clear meaning if k is a subfield of the field C of complex numbers.

When k is arbitrary then there also exists a meaning. It is well known and easy to be proved that there exists a solution of (1.1) in k[[t]], the ring of formal power series over k in the variable t.

An element ϕ of k[X] rk (resp. of k(X)rk) is said to be a polynomial (resp. rational) first integral of the system (1.1) if the following identity holds

n

X

i=1

f_i∂ϕ

∂x_i = 0. (1.2)

We would like to describe all the sequences f ∈ k[X]ⁿ such that the system (1.1) has no any polynomial (or rational) first integral. The problem is known to be difficult even for n = 2.

In this paper we study the above problem in the case when f1, . . . , fnare homogeneous linear polynomials.

Let us assume that

f_i =

n

X

j=1

a_ijx_j, i = 1, . . . , n, (1.3) where each a_ij belong to k. Denote by λ₁, . . . , λ_n the n eigenvalues of the matrix [a_ij] in an algebraic closure of k. Moreover, let Z and Z⁺ denote respectively the ring of integers and the set of non-negative integers.

The following two theorems are the main results of the paper.

Theorem 1.1. Let f = (f₁, . . . , f_n) ∈ k[X]ⁿ where f₁, . . . , f_n are linear polynomials of the form (1.3). The following conditions are equivalent:

(1) The system (1.1) does not admit any polynomial first integral.

(2) The eigenvalues λ₁, . . . , λ_n of the matrix [a_ij] are Z⁺-independent.

Theorem 1.2. Let f = (f₁, . . . , f_n) ∈ k[X]ⁿ where f₁, . . . , f_n are linear polynomials of the form (1.3). The following conditions are equivalent:

(1) The system (1.1) does not admit any rational first integral.

(2) The Jordan matrix of the matrix [a_ij] has one of the following two forms:

(a)







λ₁ 0

. ..

0 λ_n





,

where the eigenvalues λ₁, . . . , λ_n are Z-independent; or

(b)



























λ₁ 0

. ..

λ_i−1

 λ_i 1 0 λ_i+1

 λ_i+2

. ..

0 λ_n



























for some i ∈ {1, . . . , n − 1} where λ_i = λ_i+1 and the eigenvalues λ₁, . . . , λ_i, λ_i+2, . . . , λ_n are Z-independent.

2 Derivations and rings of constants.

Throughout the paper we use the vocabulary of differential algebra (see for example [3] or [4]).

Let us assume that R is a commutative ring containing the field k and d is a k- derivation of R, that is, d : R −→ R is a k-linear mapping such that d(ab) = ad(b) + d(a)b for all a, b ∈ R. We denote by R^d the ring of constants of d, i. e., R^d= {a ∈ R; d(a) = 0}.

Let k ⊂ k⁰ be a field extension and let R ⊗_k k⁰ be the tensor product of R and k⁰ over k. Let us consider the k⁰-linear mapping d ⊗ 1 : R ⊗_k k⁰ −→ R ⊗_k k⁰ such that (d ⊗ 1)(a ⊗ α) = d(a) ⊗ α, for all a ∈ R and α ∈ k⁰. It is obvious that R ⊗k k⁰ is a commutative ring containing k⁰ and d ⊗ 1 is a k⁰-derivation of R ⊗_kk⁰.

Proposition 2.1. The k⁰-vector spaces R^d⊗_kk⁰ and (R ⊗_kk⁰)^d⊗1 are isomorphic.

Proof. The exact sequence 0 −→ R^d −→ Rⁱ −→ R of vector spaces over k (where^d i(x) = x, for x ∈ R^d) induces the exact sequence

0 −→ R^d⊗_kk^{0 i⊗1}−→ R ⊗_kk^{0 d⊗1}−→ R ⊗_kk⁰

of vector spaces over k⁰. Thus, (R ⊗_kk⁰)^d⊗1= Ker(d ⊗ 1) = Im(i ⊗ 1) ≈ R^d⊗_kk⁰.  If σ : R −→ R is a k-automorphism of R and d is a k-derivation of R then the mapping σdσ⁻¹is also a k-derivation of R. Two k-derivations d and δ of R are said to be equivalent if there exists a k-automorphism σ of R such that d = σδσ⁻¹.

Recall that when R is without zero divisors, the derivation d can be extended in a unique way to its quotient field by setting: d(a/b) = b⁻²(d(a)b − ad(b)).

It is easy to check the following

Proposition 2.2. Let d and δ be equivalent k-derivations of R. Then R^d = k if and only if R^δ = k. Moreover, if R is without zero divisors and R₀ is the quotient field of R, then R^d₀ = k if and only if R^δ₀ = k. 

We shall use the above notations for the ring k[X] and its quotient field k(X). Let us note that a k-derivation of k[X] is completely defined by its values on the variables x1, . . . , xn. If f = (f1, . . . , fn) ∈ k[X]ⁿ then there exists a unique k-derivation d of k[X]

such that d(x_i) = f_i for all i = 1, . . . , n. This derivation is defined by d(ϕ) =

n

X

i=1

f_i∂ϕ

∂x_i, (2.1)

for any ϕ ∈ k[X].

Thus, the set of all polynomial first integrals of (1.1) coincides with the set k[X]^dr k where

k[X]^d = {ϕ ∈ k[X]; d(ϕ) = 0}

and d is the k-derivation defined by (2.1). Moreover, the set of all rational first integrals of (1.1) coincides with the set k(X)^dr k, where

k(X)^d = {ϕ ∈ k(X); d(ϕ) = 0}

and d is the unique extension of the k-derivation (2.1) to k(X).

The rings of constants k[X]^d and k(X)^d are intensively studied from a long time; see for example [7], [6] and [1], where many references on this subject can be found.

Every k-derivation d of k[X] determines two subfields of k(X); namely the subfield k(X)^dand the subfield (k[X]^d)₀, the quotient field of k[X]^d. The following example shows that these subfields could be different.

Example 2.3. Assume that n = 2 and k[X] = k[x, y]. Let d = x_∂x^∂ + y_∂y^∂ . Then (k[X]^d)₀ 6= k(X)^d.

Proof. It is clear that k[X]^d = k. Thus (k[X]^d)₀ = k. However k(X)^d 6= k, since x/y ∈ k(X)^d . Indeed:

d(x

y) = d(x)y − xd(y)

y² = xy − xy y² = 0

y² = 0.  Let us note the following proposition which is easy to be proved.

Proposition 2.4. Let d be a k-derivation of k[X]. If f and g are nonzero relatively prime polynomials from k[X], then d(f /g) = 0 if and only if d(f ) = pf and d(g) = pg for some p ∈ k[X]. 

Note also an immediate consequence of Proposition 2.1.

Proposition 2.5. Let d be a k-derivation of k[X], and let k⁰ be an overfield of k. Denote by d⁰ the k⁰-derivation of k⁰[X] such that d⁰(xi) = d(xi) for i = 1, . . . , n. Then k[X]^d= k if and only if k⁰[X]^d0 = k⁰. 

The next proposition shows that the field k(X)^d has a similar property.

Proposition 2.6. Let d be a k-derivation of k[X], and let k⁰ be an overfield of k. Denote by d⁰ the k⁰-derivation of k⁰[X] such that d⁰(xi) = d(xi) for i = 1, . . . , n. Then k(X)^d= k if and only if k⁰(X)^d0 = k⁰.

For the proof of this proposition we need three lemmas.

Lemma 2.7. Assume that L(t) is the field of rational functions in one variable t over a field L containing k. Let d be a k-derivation of L such that L^d = k, and let δ be the k-derivation of L(t) such that δ | L = d and δ(t) = 0. Then L(t)^δ = k(t).

Proof. It is obvious that k(t) ⊆ L(t)^δ. Let w = f /g ∈ L(t)^δ with f, g ∈ L[t], and put p = deg(g). If p = 0 then it is clear that w ∈ k[t] ⊂ k(t). Let p > 1. We may assume that g is monic and that p is minimal. Set g = t^p+ b₁t^p−1+ · · · + b_p, where b₁, . . . , b_p ∈ L.

Then we have: δ(g) = d(b1)t^p−1+ · · · + d(bp).

Observe that δ(g) = 0. Indeed, if δ(g) 6= 0 then w = δ(f )/δ(g) and we have a contradiction with the minimality of p.

Thus, d(b1) = · · · = d(bp) = 0 and hence, g ∈ k[X] (because L^d = k). Consequently, δ(f ) = δ(g)w = 0 · w = 0, that is, f ∈ k[X]. Therefore, w = f /g ∈ k(t). 

Lemma 2.8. Let L be a field containing k and let S be a set of algebraically independent elements over L. Let d be a k-derivation of L such that L^d = k. If δ is the k-derivation of L(S) such that δ | L = d and δ(s) = 0, for every s ∈ S, then L(S)^δ = k(S).

Proof. It is obvious that k(S) ⊆ L(S)^δ. Let w ∈ L(S)^δ. Since w ∈ L(S), there exists a finite subset S⁰ of S such that w ∈ L(S⁰). Denote by δ⁰ the restriction of the derivation δ to L(S⁰). Then w ∈ L(S⁰)^δ0 and hence, by Lemma 2.7 and by induction, w ∈ k(S⁰) ⊆ k(S). 

Lemma 2.9. Let F be a field containing k and let F ⊆ k⁰ be an algebraic field extension.

Assume that δ is an F -derivation of F [X] and let d⁰ be the k⁰-derivation of k⁰[X] such that d⁰(xi) = δ(xi) for i = 1, . . . , n. If F (X)^δ= F then k⁰(X)^d0 = k⁰.

Proof. Suppose that k⁰(X)^d0 6= k⁰ and let w ∈ k⁰(X)^d0 r k⁰. Since k⁰(X) is algebraic over F (X), there exist a₁, . . . , a_p ∈ F (X) such that w^p + a₁w^p−1+ · · · + a_p = 0. Let us assume that p is minimal. Then δ(a₁)w^p−1+ · · · + δ(a_p) = d⁰(w^p + a₁w^p−1+ · · · + ap) = d⁰(0) = 0 and hence, by the minimality of p, δ(a1) = · · · = δ(ap) = 0, that is, a₁, . . . , a_p ∈ F (because F (X)^δ = F ). Thus, w is algebraic over F . This implies that w is algebraic over k⁰. But k⁰ is algebraically closed in k⁰(X). Hence, w ∈ k⁰ and we have a contradiction. 

Proof of Proposition 2.6. If k⁰(X)^d0 = k⁰ then it is clear that k(X)^d= k. Assume now that k(X)^d= k. Let S be a subset of k⁰ such that the extension k ⊂ k(S) is purely transcendental and the extension k(S) ⊂ k⁰ is algebraic. Put L = k(X), F = k(S). Then, by Lemma 2.8, F (X)^δ = L(S)^δ = k(S) = F , where δ is the F -derivation of F [X] such that δ(x_i) = d(x_i) for all i = 1, . . . , n. Now Lemma 2.9 implies that k⁰(X)^d0 = k⁰. 

3 Linear derivations.

In the present section we recall from [5] some useful facts on linear derivations.

Let us make precise some notions. Assume that d is a k-derivation of k[X] such that d(xi) =

n

X

j=1

aijxj for i = 1, . . . , n, (3.1) where each a_ij belongs to k. Let λ₁, . . . , λ_nbe the n eigenvalues (belonging to an algebraic closure k of k) of the matrix [aij]. Moreover, let the Jordan matrix (in k) of the matrix [a_ij] be of the form:







J_m1(ρ₁) 0 . ..

0 J_ms(ρ_s)





 (3.2)

where s > 1, m¹ + · · · + ms = n, {ρ1, . . . , ρs} = {λ1, . . . , λn} and each Jmi(ρi) is the m_i× m_i block:

Jmi =







ρ_i 1 . .. 1

ρ_i





. (3.3)

The homogeneity of polynomials d(x1), . . . , d(xn) together with the fact that they are of the same degree 1 has the following consequences.

Proposition 3.1 ([5]). Let d be the k-derivation defined by (3.1).

(1) If f ∈ k[X]^d then all homogeneous components of f belong also to k[X]^d.

(2) Let f, p ∈ k[X] and assume that f 6= 0 and d(f ) = pf . Then p ∈ k and d(g) = pg for all homogeneous components g of f . 

The following proposition will be played an important role in our considerations.

Proposition 3.2 ([5] Lemma 2.3). Let d be the k-derivation of k[X] defined by (3.1).

Assume that f is a nonzero homogeneous polynomial from k[X] satisfying the equality d(f ) = pf for some p ∈ k. Then, there exist non-negative integers i1, . . . , in such that

i₁λ₁ + · · · + i_nλ_n = p i₁ + · · · + i_n = deg f



.  (3.4)

Using the above propositions one can deduce the following

Corollary 3.3. Let d be a k-derivation of k[X] defined by (3.1) with the Jordan matrix (3.2). Assume that the elements ρ₁, . . . , ρ_s are Z-independent. Then the following three properties hold:

(1) If f and g are nonzero homogeneous polynomials from k[X] such that d(f ) = pf and d(g) = pg for some p ∈ k, then deg f = deg g.

(2) If f is a polynomial from k[X] such that d(f ) = pf for some p ∈ k, then f is homogeneous.

(3) If d(f /g) = 0, where f and g are nonzero relatively prime polynomials from k[X], then f and g are homogeneous of the same degree.

Proof. (1). We know that {ρ₁, . . . , ρ_s} = {λ₁, . . . , λ_n}. Hence, by Proposition 3.2, there exist non-negative integers i₁, . . . , i_s and j₁, . . . , j_s such that

i₁ρ₁ + · · · + i_sρ_s = p, i₁ + · · · + i_s = deg f, j₁ρ₁ + · · · + j_sρ_s = p, j₁ + · · · + j_s = deg g.

Then (i1− j1)ρ1+ · · · + (is− js)ρs = 0 and hence, i1 = j1, . . . , is = jsbecause the elements ρ₁, . . . , ρ_s are Z-independent. Therefore, deg f = i1+ · · · + i_s = j₁+ · · · + j_s = deg g.

(2). Suppose that f contains two nonzero homogeneous components f₁ and f₂ with deg f1 6= deg f2. Then d(f1) = pf1, d(f2) = pf2 (by Proposition 3.1) and so, using (1), we obtain a contradiction: degf₁ = deg f₂.

(3). It is a consequence of (1), (2) and Proposition 2.4. 

4 Proof of Theorem 1.1.

In view of Section 2 we must prove that if k is an arbitrary field of characteristic zero and d is the k-derivation of k[X] defined by (3.1), then the following two conditions are equivalent:

(1) k[X]^d = k;

(2) The n eigenvalues λ₁, . . . , λ_n of the matrix [a_ij] are Z⁺-independent.

(2) ⇒ (1). Suppose that there exists f ∈ k[X] such that d(f ) = 0 and f 6∈ k. We may assume (by Proposition 3.1) that f is homogeneous. Then d(f ) = 0f and hence, by Proposition 3.2, there exist i₁, . . . , i_n ∈ Z⁺satisfying (3.4). Since i₁λ₁+ · · · + i_nλ_n= 0 and λ1, . . . , λn are Z⁺-independent, i1 = · · · = in = 0. Therefore, deg f = i1 + · · · + in = 0, which contradicts with the fact that f 6∈ k.

(1) ⇒ (2). Assume now that k[X]^d = k. Let k⁰ be an algebraically closed field containing k and let d⁰ be the k⁰-derivation of k⁰[X] such that d⁰(x_i) = d(x_i) for all i = 1, . . . , n. Then (by Proposition 2.5) k⁰[X]^d0 = k⁰ so, we may assume that the field k is algebraically closed. Moreover, using Proposition 2.2, we may assume that [a_ij] is the Jordan matrix (3.2).

Therefore, there exist a subset {y1, . . . , ys} contained in {x1, . . . , xn} such that d(yi) = ρ_iy_i for all i = 1, . . . , s.

Suppose now that the eigenvalues λ₁, . . . , λ_n are not Z⁺-independent. Then there exists a nonzero sequence (j1, . . . , js) of non-negative integers such that j1ρ1+ · · · + jsρs = 0. Put f = y^j₁¹· · · y_s^js. Then f ∈ k[X] r k and

d(f ) = (j₁ρ₁+ · · · j_sρ_s)f = 0f = 0.

But it is a contradiction. This completes the proof ot Theorem 1.1. 

5 Proof of Theorem 1.2.

In view of Proposition 2.6 we may assume that k is an algebraically closed field of characteristic zero. Let d be a k-derivation of k[X] defined by (3.1) with the Jordan matrix (3.2). We must show that the following two conditions are equivalent:

(1) k(X)^d= k;

(2) The elements ρ₁, . . . , ρ_s are Z-independent and s > n − 1.

In view of Proposition 2.2 without loss of generality we may assume that the matrix [a_ij] of the derivation d coincides with the Jordan matrix (3.2). Therefore, there exist a subset {y₁, . . . , y_s} contained in {x₁, . . . , x_n} such that d(y_i) = ρ_iy_i for all i = 1, . . . , s.

(1) ⇒ (2). Let us assume that k(X)^d= k.

First we shall show that ρ₁, . . . , ρ_s are Z-independent. Suppose that there exists a nonzero sequence (j₁, . . . , j_s) of integers such that j₁ρ₁+· · ·+j_sρ_s= 0. Put ϕ = y₁^j1· · · y_s^js. Then we have a contradiction: ϕ ∈ k(X) r k and d(ϕ) = (j1ρ₁+ · · · j_sρ_s)ϕ = 0ϕ = 0.

Now we shall prove that s > n − 1. Let us suppose that s < n − 1 and consider the numbers m₁, . . . , m_s defined in (3.2). We know that m₁+ · · · + m_s = n. Therefore, we must consider only two the following cases:

Case I: There exists j ∈ {1, . . . , s} such that m_j > 3.

In this case there exist three variables x, y, z ∈ {x₁, . . . , x_n} such that d(x) = ax, d(y) = ay + x and d(z) = az + y, where a = ρ_j. Put ϕ = f /g, where f = x² and g = y² − 2xz. Then d(f ) = 2af , d(g) = 2ag and hence, d(ϕ) = 0 and ϕ 6∈ k. It is a contradiction with the fact that k(X)^d = k.

Case II: There exist i, j ∈ {1, . . . , s} such that i 6= j, m_i > and mj > 2.

In this case there exist four variables x, y, z, t ∈ {x1, . . . , xn} such that d(x) = ax, d(y) = ay + x, d(z) = bz and d(t) = bt + z, where a = ρ_i and b = ρ_j. Then (xt − yz)/xz ∈ k(X)^dr k.

This completes the proof of implication (1) ⇒ (2).

For the proof of implication (2) ⇒ (1) we need two lemmas. Let us recall that if δ is a k-derivation of a ring R then δ is called locally nilpotent if for ach r ∈ R there exists a natural number m such that δ^m(r) = 0.

Lemma 5.1 ([8]). Let R be a commutative ring without zero divisors containing k and let δ be a locally nilpotent k-derivation of R. If δ(a) = ua for some u, a ∈ R then δ(a) = 0.

Proof. If r ∈ R then we define deg_δ(r) as follows:

deg_δ(r) = l if r 6= 0, δ^l(r) 6= 0 and δ^l+1 = 0;

deg_δ(r) = −∞ if r = 0.

The function deg_δ behaves like the usual function degree over a polynomial ring (see [2]).

In particular, if a, b ∈ R then deg_δ(ab) = deg_δ(a) + deg_δ(b).

Assume now that δ(a) = ua and suppose that δ(a) 6= 0. Then a 6= 0, u 6= 0, deg_δδ(a) = deg_δa − 1 and we have a contradiction:

deg_δa − 1 = deg_δδ(a) = deg_δ(ua) = deg_δa + deg_δu > degδa.  Lemma 5.2. Let δ be a k-derivation of k[x₁, x₂] such that

 δ(x₁) = ax1

δ(x₂) = ax₂+ x₁ where a ∈ k. If a 6= 0 then k(x₁, x₂)^δ = k.

Proof. Let δ_aand δ₀ be the k-derivations of k[x₁, x₂] such that δ_a(x₁) = ax₁, δ_a(x₂) = ax₂, δ₀(x₁) = 0 and δ₀(x₂) = x₁. Then δ = δ_a+ δ₀, δ₀ is locally nilpotent, and k[x₁, x₂]^δ0 = k[x₁].

Suppose that f and g are nonzero relatively prime polynomials from k[x₁, x₂] such that δ(f /g) = 0. It follows from Corollary 3.3 that f and g are homogeneous of the same degree; set m = deg f = deg g. Moreover (by Proposition 2.4), δ(f ) = pf and δ(g) = pg for some p ∈ k. Observe that δ_a(f ) = maf . Indeed, if f =P

i+j=mu_ijxⁱ₁x^j₂ with u_ij ∈ k, then

δ_a(f ) = X

i+j=m

u_ijδ_a(xⁱ₁x^j₂) = X

i+j=r

u_ijmaxⁱ₁x^j₂ = maf.

Hence,

δ₀(f ) = δ(f ) − δ_a(f ) = pf − maf = (p − ma)f

and hence, Lemma 5.1 implies that δ₀(f ) = 0. This means that f ∈ k[x₁], that is, f = ux^m₁ for some u ∈ k r {0}. Using the same argument we see that g = vx^m1 with v ∈ k r {0}.

Therefore, f /g = u/v ∈ k. 

Now we may continue our proof of Theorem 1.2.

(2) ⇒ (1). Assume that ρ₁, . . . , ρ_s are Z-independent and s > n − 1. We must consider two cases: ”s = n” and ”s = n − 1”.

Case (a): s = n. In this case the derivation d is diagonal; d(x_i) = ρ_ix_i for all i = 1, . . . , n.

Let f and g be nonzero relatively prime polynomials from k[X] such that d(f /g) = 0.

We know from Corollary 3.3 that f and g are homogeneous of the same degree. Moreover, d(f ) = pf and d(g) = pg for some p ∈ k (Proposition 2.4). Since ρ1, . . . , ρn are Z- independent, it is easy to see that f and g are monomials. Put f = axⁱ₁¹· · · xⁱ_nⁿ and g = bx^j₁¹· · · x^j_nⁿ, with a, b ∈ k r {0}. Then we have:

i₁ρ₁+ · · · + i_nρ_n= p = j₁ρ₁+ · · · + j_nρ_n

and hence, (i₁− j₁)ρ₁+ · · · + (i_n− j_n)ρ_n= 0. Thus, i₁ = j₁, . . . , i_n= j_n, that is, f /g ∈ k.

Case (b): s = n − 1. Using an induction on n we shall show that k(X)^d= k. It is obvious that n > 2. If n = 2 then our assertion follows from Lemma 5.2.

Let n > 2. Without loss of generality we may assume that the matrix [a_ij] of d is of the form















 a 1 0 a



0 b₁

. ..

0 br















(5.1)

where a, b₁, . . . , b_r are Z-independent elements of k and where r = n − 2. Thus, the derivation d is of the form:



















d(x₁) = ax₁ d(x₂) = ax₂+ x₁ d(x₃) = b₁x₃ d(x₄) = b₂x₄

...

d(x_n) = b_rx_n. Observe that d(k[x1, . . . , xn−1]) ⊆ k[x1, . . . , xn−1].

Put R = k[x₁, . . . , x_n−1] and let δ be the restriction of d to R. Then δ is a linear k-derivation of R. If n = 3 then δ is the derivation as in Lemma 5.2. If n > 3 then the matrix of δ is of the form (5.1) with r = n − 1. Thus, by an induction, R^δ = k.

Assume now that f and g are nonzero relatively prime polynomials from k[X] such that d(f /g) = 0. Then (by Proposition 2.4) d(f ) = pf , d(g) = pg for some p ∈ k, and (by Corollary 3.3) f , g are homogeneous of the same degree; set m = deg f = deg g. Denote by y the variable x_n and put

f = F y^u+ F₁y^u−1+ · · · g = Gy^v+ G₁y^v−1+ · · · ,

where u > 0, v > 0, and where F, G are nonzero homogeneous polynomials from R with deg F = m − u and deg G = m − v.

Comparing the coefficients of y^u in the equality d(f ) = pf we see that d(F ) = (p − ub_r)F , that is, δ(F ) = (p − ub_r)F . Comparing the coefficients of y^v in the equality d(g) = pg we have: δ(G) = (p − vb_r)G.

Now, let us use Proposition 3.2 (for the derivation δ and the polynomials F and G). By this proposition there exist non-negative integers j₁, j₂, i₁, . . . , i_r−1 and j₁⁰, j₂⁰, i⁰₁, . . . , i⁰_r−1 such that

j₁a + j₂a + i₁b₁ + · · · + i_r−1b_r−1 = p − ub_r, j₁⁰a + j₂⁰a + i⁰₁b₁ + · · · + i⁰_r−1b_r−1 = p − vb_r. Then

((j1+ j2) − (j₁⁰ + j₂⁰))a + (i1− i⁰₁)b1+ · · · + (ir−1− i⁰_r−1)br−1+ (u − v)br = 0 and hence, u = v (because the elements a, b₁, . . . , b_r are Z-independent).

Thus, δ(F ) = qF and δ(G) = qG where q = p − ub_r = p − vb_r. This implies that δ(F/G) = 0. Since R^δ = k, F = cG for some nonzero c ∈ k.

Let us now consider the homogeneous polynomial h ∈ k[X] equal to f − cg. It is obvious that d(h) = ph. We shall show that h = 0.

Suppose that h 6= 0. Since f and g are nonzero relatively prime, the polynomials f and h are also nonzero and relatively prime. Repeating the above procedure for the polynomials f and h we see that they have the same degree with respect to y. But deg_yh < deg_yf = u because the coefficient of y^u in h is equal to F − cG = 0. It is a contradiction. Therefore h = 0, that is, f /g = c ∈ k. This completes the proof of Theorem 1.2. 

References

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[2] M. Ferrero, Y. Lequain, A. Nowicki, A note on locally nilpotent derivations, J.Pure Appl.Algebra 79(1992), 45 – 50.

[3] I. Kaplansky, An Introduction to Differential Algebra, Hermann, Paris, 1976.

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