Darboux-Jouanolou Integrability for Arbitrary Fields

TL;DR

This paper extends the Darboux-Jouanolou integrability theorem to differential 1-forms over any field, broadening the scope of algebraic integrability criteria.

Contribution

It generalizes the Darboux-Jouanolou theorem to arbitrary fields, providing a new theoretical framework for algebraic integrability.

Findings

Proves a Darboux-Jouanolou type theorem over arbitrary fields

Establishes conditions for algebraic integrability of differential 1-forms

Broadens the applicability of integrability criteria to all fields

Abstract

We prove a Darboux-Jouanolou type theorem on the algebraic integrability of differential 1-forms over arbitrary fields.

Equations

$$N_K(n,d-1,2)+2<\left(\genfrac{}{}{0ex}{}{d-1+n}{n}\right)\cdot \left(\genfrac{}{}{0ex}{}{n}{2}\right)+2$$

$$f(v_{\sigma (1)},...,v_{\sigma (r)})=f(v_1,...,v_r)$$

$$f(v_{\sigma (1)},...,v_{\sigma (r)})=(sgn\sigma )f(v_1,...,v_r)$$

$$(Sf)(v_1,...,v_r)=\sum _{\sigma \in S_r}f(v_{\sigma (1)},...,v_{\sigma (r)})$$

$$(Af)(v_1,...,v_r)=\sum _{\sigma \in S_r}(sgn\sigma )f(v_{\sigma (1)},...,v_{\sigma (r)})$$

$$(f\otimes g)(v_1,...,v_{r+s})=f(v_1,...,v_r)g(v_{r+1},...,v_{r+s})$$

$$f\wedge g=A(f\otimes g)$$

$$A_{\ast }(V)=\underset{r=0}{\overset{\infty }{⨁}}{A}_r(V)=\underset{r=0}{\overset{n}{⨁}}{A}_r(V)$$

$$X(fg)=fX(g)+gX(f)$$

$$X=\sum _{i=1}^{n}X(z_i)\frac{\partial }{\partial z_i}$$

$$\frac{d{z}_i}{dt}=X_i(z)$$

$$\omega =\sum _{i=1}^n\omega (\frac{\partial }{\partial z_i})d{z}_i$$

$$df=\sum _{i=1}^n\frac{\partial f}{\partial z_i}d{z}_i$$

$${\Theta }_F=\omega \wedge \frac{dF}{F}$$

$$K(z^p)=\{f:f\in K(z),df=0\}$$

$$\eta ={\lambda }_1\frac{d{F}_1}{F_1}+...+{\lambda }_m\frac{d{F}_m}{F_m}$$

$$Res(\alpha \cdot \frac{dg}{g},0)=0$$

$$\alpha (x)=x^{sp}\frac{A(x)}{B(x)}$$

$$\alpha \cdot \frac{dg}{g}=x^{sp}\frac{A(x)g^{\prime }(x)}{B(x)g(x)}dx$$

$$d(x^{-sp})=(-sp)x^{-sp-1}dx=0$$

$$K(z^p{)}^{(\mathcal{S})}⟶{\Omega }_{K(z)/K}^1$$

$$({\lambda }_j{)}_{f_j\in \mathcal{S}}⟼\sum _{f_j\in \mathcal{S}}{\lambda }_j\frac{d{f}_j}{f_j}$$

$$\sum _{j=1}^r{\lambda }_j\frac{d{f}_j}{f_j}=0$$

$$p_j\in (V_j-{\cup }_{i\ne j}{V}_i)$$

$${\alpha }_1\frac{d{g}_1}{g_1}+...+{\alpha }_j\frac{d{g}_j}{g_j}+...+{\alpha }_r\frac{d{g}_r}{g_r}=0$$

$$\frac{d{g}_j}{g_j}=-\sum _{i\ne j}\frac{{\alpha }_i}{{\alpha }_j}\frac{d{g}_i}{g_i}$$

$$1=-\sum _{i\ne j}Res(\frac{{\alpha }_i}{{\alpha }_j}\frac{d{g}_i}{g_i})=0$$

$${\alpha }_1{\Theta }_{F_1}+{\alpha }_2{\Theta }_{F_2}+...+{\alpha }_m{\Theta }_{F_m}=0$$

$$\eta ={\alpha }_1\frac{d{F}_1}{F_1}+{\alpha }_2\frac{d{F}_2}{F_2}+...+{\alpha }_m\frac{d{F}_m}{F_m}\ne 0$$

$$\omega \wedge \eta =\sum _{i=1}^m{\alpha }_i\omega \wedge \frac{d{F}_i}{F_i}=\sum _{i=1}^m{\alpha }_i{\Theta }_{F_i}=0$$

Full text

1. Introduction

Ordinary differential equations appear in many branches of Mathematics and its applications. For example, occasionally the differential equations are given by a differential $1$-form or a vector field on $\mathbb{R}^{n}$ or $\mathbb{C}^{n}$, and in that case to better understand their solutions (in real or complex time, respectively) we can search for a first integral (a constant function along the solutions).

In that sense, the Darboux theory of integrability is a classical approach. The foundations of this was given by J.-G. Darboux (1878) in his memory [3], showing how can be constructed a first integral for a polynomial vector field, on $\mathbb{R}^{2}$ or $\mathbb{C}^{2}$, from a large enough number of invariant algebraic curves. In fact, he proved that, if $d$ is the degree of the planar vector field, with $\binom{d+1}{2}+1$ invariant algebraic curves we can compute a first integral using that curves. H. Poincaré (1891) noticed the difficulty to obtain algorithmically that invariant curves (see [11]).

Improving the Darboux’s result, J.-P. Jouanolou (1979) showed in [6] that if the number of invariant curves for a planar vector field of degree $d$ on $\mathbb{R}^{2}$ or $\mathbb{C}^{2}$ is at least $\binom{d+1}{2}+2$, then it admits a rational first integral, and it can be computed from the invariant curves. In fact, Jouanolou’s Theorem says that if $K$ is a field of characteristic [math] and $\omega$ is a polynomial $1$-form of degree $d$ on $K^{n}$ with at least $\binom{d-1+n}{n}\cdot\binom{n}{2}+2$ invariant irreducible algebraic hypersurfaces, then $\omega$ has a rational first integral, and that one can also be computed in terms of the invariant hypersurfaces.

The Darboux Integration Method can be used in Physics. For example, C. G. Hewitt (1991) in [5] reveals by this method some new solutions to the Einstein field equations. Many others physical models have been successfully studied in this framework (see [8], [9], [13], [15]).

E. Ghys (2000) in [4] extended Jouanolou’s result to holomorphic foliations on compact complex manifolds. In the local analytic context, under some assumptions, B. Scárdua (2011) shows in [12] a Darboux-Jouanolou type theorem for germs of integrable $1$-forms on $(\mathbb{C}^{n},0)$. Over fields of characteristic zero, a Darbou-Jouanolou type theorem is proved in [2] for polynomial $r$-forms, $r\geq 0$.

M. Brunella and M. Nicollau (1999) proved in [1] a positive characteristic version of Darboux-Jouanolou theorem: if $\omega$ is a rational $1$-form on a smooth projective variety over a field $K$ of positive characteristic $p>0$ with infinitely many invariant hypersurfaces, then $\omega$ admits a rational first integral. The proof uses different tools of algebraic geometry and it is not constructive.

J. V. Pereira (2001) shows in [10] that a generic vector field on an affine space of positive characteristic admits an invariant algebraic hypersurface. This theorem is in sharp contrast with characteristic [math] case where a theorem of Jouanolou says that a generic vector field on the complex plane does not admit any invariant algebraic curve.

Our goal in this paper is to give a general account of Darboux-Jouanolou integrability for polynomial $1$-forms on $K^{n}$, where $K$ is an arbitrary field. In particular, we obtain a general version of Darboux-Jouanolou Criterion ([6], Théorème 3.3, p. 102). For each polynomial $r$-form $\omega$ of degree $d$ on $K^{n}$, with $K$ of characteristic $p\geq 0$, we define a natural number $N_{K}(n,d,r)$ (see Definition 6.3) that depends only on $n$, $r$, $d$ and $p$, and we prove the following

Theorem 1.1**.**

Let $\omega\in\Omega^{1}(K^{n})$ be a polynomial $1$-form of degree $d$ over an arbitrary field $K$. If $\omega$ has $N_{K}(n,d-1,2)+2$ invariant hypersurfaces, then $\omega$ admits a rational first integral.

The number $N_{K}(n,d-1,2)+2$ has the property that in characteristic [math] holds $N_{K}(n,d-1,2)+2=\binom{d-1+n}{n}\cdot\binom{n}{2}+2$ but in characteristic $p>0$ we have

[TABLE]

2. The Grassmann Algebra

For the generalities about Grassmann Algebra see Chapter 2 in [14].

Let $V$ be a vector space over an arbitrary field $K$.

Definition 2.1**.**

A $r$-linear function ($r$-covector) $f:V^{r}\rightarrow K$ is said to be symmetric if

[TABLE]

and it is said to be alternanting if

[TABLE]

for all permutations $\sigma\in S_{r}$.

The $K$-vector space $A_{r}(V)$ is the space of all alternating $r$-linear functions on $V$.

Definition 2.2**.**

Given a $r$-linear function $f:V^{r}\rightarrow K$, we define

[TABLE]

as the symmetrizing operator; and

[TABLE]

as the alternating operator.

Definition 2.3**.**

Let $f$ be a $r$-linear function and $g$ a $s$-linear function on the $K$-vector space $V$. Their tensor product is the ($r+s$)-linear function $f\otimes g$ defined by

[TABLE]

Definition 2.4**.**

Let $f\in A_{r}(V)$ and $g\in A_{s}(V)$. We define their wedge product, also called exterior product, by

[TABLE]

Remark 2.5**.**

This wedge product just defined can be different by a constant factor from the usual one in differential geometry, and we use that in order to make sense in a field of positive characteristic.

For a finite-dimensional vector space $V$, say of dimension $n$, define

[TABLE]

With the wedge product of multicovectors as multiplication, $A_{*}(V)$ becomes an anticommutative graded algebra, called the exterior algebra or the Grassmann algebra of multicovectors on the vector space $V$.

3. Rational and Polynomial Vector Fields

Let $K$ be an arbitrary field. Denote by $K^{n}$ the $n$-dimensional $K$-vector space (with coordinates $z_{1}$, $z_{2}$,…, $z_{n}$). Also denote by $K[z]$ the polynomial ring $K[z_{1},...,z_{n}]$ and by $K(z)$ the field of fractions $K(z_{1},...,z_{n})$.

Definition 3.1**.**

A rational vector field $X$ on $K^{n}$ is a $K$-derivation of $K(z)$, that is, a $K$-linear operator on $K(z)$ that satisfies Leibniz’s rule, i. e.,

[TABLE]

A vector field $X$ can be written as

[TABLE]

where $X(z_{i})=X_{i}\in K(z)$ and $\frac{\partial}{\partial z_{i}}$ is de natural derivation such that $\frac{\partial}{\partial z_{i}}(z_{j})=\delta_{ij}$ (the Kronecker’s delta is $\delta_{ij}=1$ if $i=j$ and $\delta_{ij}=0$ if $i\neq j$).

If $X(z_{i})=X_{i}\in K[z]$ is polynomial, $i=1$,…, $n$, then we say that $X$ is a polynomial vector field on $K^{n}$, and in this case it is a $K$-derivation of $K[z]$. The $K$-vector space of polynomial vector fields will be denoted by $\mathfrak{X}(K^{n})$.

In case of $K=\mathbb{R}$ or $\mathbb{C}$, the system

[TABLE]

$i=1,...,n$, defines a system of differential equations on $K^{n}$.

Definition 3.2**.**

Like in theory of differential equations if $f\in K(z)$ is a rational function such that $df\neq 0$ and $X(f)=0$ we say that $f$ is a first integral or a non-trivial constant of derivation of $X$. In case of $K=\mathbb{R}$ or $\mathbb{C}$ the leaves $\{f(z)=constant\}$ are invariant sets for the system $\frac{dz_{i}}{dt}=X_{i}(z)$, $i=1,...,n$. We say that an irreducible polynomial $F\in K[z]$ is invariant by a polynomial vector field $X$ if $F$ divides $X(F)$ (we also say that $\{F=0\}\subset(K^{a})^{n}$ is an invariant hypersurface for $X$ over the algebraic closure $K^{a}$ of $K$). In case of $K=\mathbb{R}$ or $\mathbb{C}$, $\{F=0\}$ is an invariant set for the system $\frac{dz_{i}}{dt}=X_{i}(z)$, $i=1,...,n$.

4. Rational and Polynomial Differential Forms

A rational differential $1$-form $\omega$ on $K^{n}$ is a $K(z)$-linear map from the set of rational vector fields to $K(z)$:

[TABLE]

where $\omega(\frac{\partial}{\partial z_{i}})\in K(z)$ and $dz_{i}$ is de natural $1$-form dual to $\frac{\partial}{\partial z_{i}}$, that is, $dz_{i}(\frac{\partial}{\partial z_{j}})=\delta_{ij}$.

Hence $dz_{1}$,…, $dz_{n}$ is the canonical dual basis to the basis of rational vector fields $\frac{\partial}{\partial z_{1}}$,…, $\frac{\partial}{\partial z_{n}}$.

If $\omega(\frac{\partial}{\partial z_{i}})$ is polynomial, $i=1$,…, $n$, we say that $\omega$ is a polynomial differential $1$-form on $K^{n}$, and in this case it is a $K[z]$-linear map $\mathfrak{X}(K^{n})\rightarrow K[z]$, $X\mapsto\omega(X)$. The $K$-vector space of polynomial differential $1$-forms will be denoted by $\Omega^{1}(K^{n})$

If $f$ is a rational function in $K(z)$, we write the exact $1$-form

[TABLE]

Definition 4.1**.**

If $\omega$ is a rational $1$-form and $f$ is a rational function such that $df\neq 0$ and $\omega\wedge df=0$ we say that $f$ is a first integral or a non-trivial constant of derivation of $\omega$. In case of $K=\mathbb{R}$ or $\mathbb{C}$ then $\{f(z)=constant\}$ defines a foliation of $K^{n}$. If $\omega$ is a polynomial $1$-form we say that an irreducible polynomial $F$ is invariant by $\omega$ if $F$ divides $\omega\wedge dF$ (we also say that $\{F=0\}\subset(K^{a})^{n}$ is an invariant hypersurface for $\omega$ over the algebraic closure $K^{a}$ of $K$). In case of $K=\mathbb{R}$ or $\mathbb{C}$ then $\{F=0\}$ is a leaf of $\omega$.

Analogously we can define differential $r$-forms for $r\geq 1$ and the $K$-vector space of polynomial differential $r$-forms will be denoted by $\Omega^{r}(K^{n})$.

Definition 4.2**.**

Let $F$ be an invariant polynomial for the $1$-form $\omega$. We say that the polynomial $2$-form

[TABLE]

is a cofactor of $F$.

Definition 4.3**.**

Let $K$ be a field of arbitrary characteristic $p$ ($p=0$ or $p$ is a prime integer). The field of differential constants is the sub-field of $K(z)$ given by

[TABLE]

Note that in characteristic [math] we have $K(z^{p})=K(z^{0})=K$. Otherwise, in prime characteristic $p>0$ we obtain $K(z^{p})$ as the $K$-vector subspace of $K(z)$ generated by $\{g^{p}:g\in K(z)\}$.

We call the elements of $K(z^{p})$ of $\partial$-constants.

Remark 4.4**.**

Note that $K(z^{p})$ is the kernel of the $d$ operator, and $K(z)$ is finite dimensional as a $K(z^{p})$-vector space.

Definition 4.5**.**

Let $F_{1}$,…, $F_{m}$ be a collection of irreducible polynomials and $\lambda_{1}$,…, $\lambda_{m}\in K(z^{p})$ be $\partial$-constants. The linear combination

[TABLE]

is called a logarithmic $1$-form.

Note that every logarithmic $1$-form $\eta$ is closed, that is, $d\eta=0$.

5. Logarithmic 1-forms and Residues

Remember the Hilbert’s Nullstellensatz:

Theorem 5.1** ([7], Theorem 1.5, p. 380).**

Let $I$ be an ideal of $K[z]$ and let $V(I)=\{z\in(K^{a})^{n}:f(z)=0,\forall f\in I\}$ be the algebraic variety associated to $I$, where $K^{a}$ is the algebraic closure of $K$. Let $P$ be a polynomial in $K[z]$ such that $P(c)=0$ for every zero $(c)=(c_{1},...,c_{n})\in V(I)$. Then there is an integer $m>0$ such that $P^{m}\in I$.

We will need something about residues.

Lemma 5.2**.**

Let $g\in K[x]$, $g(0)\neq 0$, be a polynomial function in one variable, where $K$ is a field of positive characteristic $p>0$. If $\alpha\in K(x^{p})$, then

[TABLE]

Proof.

We can write $g(x)=a_{0}+a_{1}x+...+a_{r}x^{r}$, where $a_{0}\neq 0$, and

[TABLE]

where $s$ is an integer and $A(0)\neq 0$, $B(0)\neq 0$.

Hence

[TABLE]

Since $B(0)g(0)\neq 0$ note that $\alpha\cdot\frac{dg}{g}$ has a pole at [math] if and only if $A(x)g^{\prime}(x)$ has a term $cx^{-sp-1}$ with $c\neq 0$. Also note that

[TABLE]

Hence $A(x)g^{\prime}(x)$ has no term $cx^{-sp-1}$ with $c\neq 0$. We conclude that the residue is $Res(\alpha\cdot\frac{dg}{g},0)=0$.

∎

The following lemma is the positive characteristic version of Jouanolou’s Lemma ([6], Lemme 3.3.1, p. 102).

Lemma 5.3**.**

If $\mathcal{S}$ is a finite representative system of primes in $K[z]=K[z_{1},...,z_{n}]$ (that is, $\mathcal{S}$ is a finite collection of distinct irreducible polynomials), then the $K(z^{p})$-linear map

[TABLE]

[TABLE]

is injective.

Proof.

Let $\lambda_{1}$,…, $\lambda_{r}\in K(z^{p})$ and $f_{1}$,…, $f_{r}\in\Omega^{1}_{K(z)/K}$ be such that

[TABLE]

For each $i\in\{1,2,...,r\}$, define $V_{i}=\{f_{i}=0\}\subset(K^{a})^{n}$. By Hilbert’s Nullstellensatz, there is a point

[TABLE]

(Suppose that for every $p\in V_{j}$ there is a $V_{i}$, $i\neq j$, such that $p\in V_{i}$; that is, $V_{j}\subset\cup_{i\neq j}V_{i}$. Consider $I$ the ideal generated by $\Pi_{i\neq j}f_{i}$. We have $V(I)=\cup_{i\neq j}V_{i}$. Then, by the Hilbert theorem, there exists an integer $n$ such that $f_{j}^{n}$ is in the ideal $I$, and then it $f_{j}$ has a common factor with $\Pi_{i\neq j}f_{i}$, and it is a contradiction since every $f_{i}$ is irreducible.)

We can suppose that $p_{j}$ is a smooth point of $V_{j}$. Let $L_{j}$ be an affine line through $p_{j}$ transversal to $V_{j}$. Consider $u:K^{a}\rightarrow L_{j}$ be a parametrization of $L_{j}$ with $u(0)=p_{j}$ and $g_{i}=f_{i}\circ u$. For all $i\neq j$, the function $\frac{dg_{i}}{g_{i}}$ is regular at [math], but $\frac{dg_{j}}{g_{j}}$ has a pole at [math] with $Res(\frac{dg_{j}}{g_{j}},0)=1$. Then, from the above relation, we have

[TABLE]

where $\alpha_{i}=\lambda_{i}\circ u$ for $i=1$,…, $r$.

If $\alpha_{j}\neq 0$, we can put

[TABLE]

and taking residues at [math], using Lemma 5.2 we obtain

[TABLE]

Hence $\alpha_{j}=0$, that is, $\lambda_{j}\mid_{L_{j}}=0$. Since $p_{j}$ and $L_{j}$ are generic, we conclude that $\lambda_{j}=0$.

∎

6. Darboux Integration Method

We explore the method of integration developed by J.-G. Darboux in [3]. The original context was differential equation on $\mathbb{C}^{2}$. J.-P. Jouanolou in [6] extends the argument for $1$-forms on $K^{n}$, where $K$ is an algebraically closed field of characteristic [math]. Here we extend this method for an arbitrary field $K$.

Proposition 6.1**.**

Let $\omega\in\Omega^{1}(K^{n})$ be a $1$-form, where $K$ is an arbitrary field. If there are invariant irreducible polynomials $F_{1}$,…, $F_{m}$ in $R$ and constants $\alpha_{1}$,…, $\alpha_{m}$ in $K(z^{p})$ such that

[TABLE]

then there is a logarithmic $1$-form $\eta\neq 0$ such that $\omega\wedge\eta=0$. (In that case we say that $\eta$ is tangent to $\omega$.)

Proof.

Consider the rational $1$-form

[TABLE]

(see Lemma 5.3).

Obviously $\eta$ is closed ($d\eta=0$). Also $\eta$ satisfies

[TABLE]

∎

Remark 6.2**.**

In case of $K=\mathbb{R}$ or $\mathbb{C}$, from the logarithmic $1$-form $\eta$ above, by integration we obtain a (multivaluated, analytic or holomorphic) first integral for $\omega$.

Definition 6.3**.**

Let $K$ be a field of positive characteristic $p>0$, and define $S_{K}(d)$ to be the $K$-vector space of polynomials of degree less than or equal to $min\{p-1,d\}$. In case that $K$ is a field of characteristic [math], define $S_{K}(d)=S_{d}$ as the $K$-vector space of polynomials of degree less than or equal to $d$. Define

[TABLE]

and

[TABLE]

Corollary 6.4**.**

Let $\omega\in\Omega^{1}(K^{n})$ be a $1$-form of degree $d$. If $\omega$ has $N_{K}(n,d-1,2)+1$ invariant irreducible polynomials, then there is a logarithmic $1$-form $\eta\neq 0$ such that $\omega\wedge\eta=0$.

Proof.

Suppose that there are $m=N_{K}(n,d-1,2)+1$ invariant irreducible polynomials $F_{1}$, $F_{2}$,…, $F_{m}$.

The cofactors associated to the invariant polynomials are differential $2$-forms of degree less than or equal to $d-1$. Since the $K(z^{p})$-vector space of differential $2$-forms of degree less than or equal to $d-1$ has $K(z^{p})$-dimension equal to $N_{K}(n,d-1,2)$, we have that the cofactors associated to the polynomials $F_{i}$, $i=1$,…, $m$, are $K(z^{p})$-linearly dependent. Therefore, there are $\partial$-constants $\alpha_{1}$,…, $\alpha_{m}$ in $K(z^{p})$ such that

[TABLE]

By the above proposition, the corollary follows.

∎

Example 6.5**.**

Suppose $\omega=\alpha ydx+\beta xdy$, where $\alpha$ and $\beta$ are constants in $K^{*}$. We can easily see that $F=x$ and $G=y$ are invariant. The cofactor associated to $x$ is given by

[TABLE]

while the cofactor associated to $y$ is

[TABLE]

Hence

[TABLE]

Consequently $\eta=\alpha\frac{dx}{x}+\beta\frac{dy}{y}$ is a logarithmic $1$-form tangent to $\omega$ (in characteristic $p=2$ the $1$-form $\eta$ is linear).

Proposition 6.6**.**

Let $\omega\in\Omega^{1}(K^{n})$ be a $1$-form. If there are invariant irreducible polynomials $F_{1}$,…, $F_{m}$ in $K[z]$ and constants $\delta_{1}$,…, $\delta_{m}$ in $\mathbb{Z}$ (or in the prime subfield $\mathbb{Z}_{p}$ of $K$ if $char(K)=p>0$) such that

[TABLE]

then $\omega$ has a rational (polynomial) first integral.

Proof.

Consider the rational $1$-form

[TABLE]

Obviously $\eta$ is closed ($d\eta=0$) and also $\eta$ satisfies

[TABLE]

Define the rational function (or polynomial)

[TABLE]

Note that $dF_{i}^{\delta_{i}}=\delta_{i}F_{i}^{\delta_{i}}\frac{dF_{i}}{F_{i}}$ and then $dG=G\eta$, hence

[TABLE]

∎

7. Darboux-Jouanolou Criterion: Proof of Theorem 1.1

Let $F_{1}$,…, $F_{m}$ be $N_{K}(n,d-1,2)+2$ invariant irreducible polynomials (for the $1$-form $\omega$). By the using of the cofactors associated to the invariant polynomials $F_{1}$,…, $F_{m-1}$, we can construct, by Corollary 6.4, the logarithmic $1$-form

[TABLE]

such that $\omega\wedge\eta_{1}=0$. Analogously, we also can obtain

[TABLE]

such that $\omega\wedge\eta_{2}=0$ with $\beta_{m}\neq 0$.

As we suppose $\beta_{m}\neq 0$, the "polar divisors" of $\eta_{1}$ and $\eta_{2}$ are distinct. Hence, by Lemma 5.3, there is a rational function $f\in K(z)-K(z^{p})$ such that $\eta_{1}=f\eta_{2}$. Taking the exterior derivative, we obtain

[TABLE]

Since $\eta_{1}$ and $\eta_{2}$ are closed, we conclude that $df\wedge\eta_{2}=0$. In this way, since $\omega\wedge\eta_{2}=0$ we have that $\omega\wedge df=0$. Therefore $f$ is a rational first integral for $\omega$.

∎