Real Liouvillian Extensions of Partial Differential Fields
In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally -adic differential fields with a -adically closed field of constants. For an integrable partial differential system defi...
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| description | In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally -adic differential fields with a -adically closed field of constants. For an integrable partial differential system defined over such a field, we prove that there exists a formally real (resp. formally -adic) Picard-Vessiot extension. Moreover, we obtain a uniqueness result for this Picard-Vessiot extension. We give an adequate definition of the Galois differential group and obtain a Galois fundamental theorem in this setting. We apply the obtained Galois correspondence to characterise formally real Liouvillian extensions of real partial differential fields with a real closed field of constants by means of split solvable linear algebraic groups. We present some examples of real dynamical systems and indicate some possibilities for further development of algebraic methods in real dynamical systems.
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| format | Article |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 095, 16 pages
Real Liouvillian Extensions
of Partial Differential Fields
Teresa CRESPO a, Zbigniew HAJTO b and Rouzbeh MOHSENI b
a) Departament de Matemàtiques i Informàtica, Universitat de Barcelona,
Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
E-mail: teresa.crespo@ub.edu
URL: http://www.ub.edu/tn/personal/crespo.php
b) Faculty of Mathematics and Computer Science, Jagiellonian University,
ul. Lojasiewicza 6, 30-348 Kraków, Poland
E-mail: Zbigniew.Hajto@uj.edu.pl, Rouzbeh.Mohseni@doctoral.uj.edu.pl
Received February 28, 2021, in final form October 25, 2021; Published online October 29, 2021
https://doi.org/10.3842/SIGMA.2021.095
Abstract. In this paper, we establish Galois theory for partial differential systems defined
over formally real differential fields with a real closed field of constants and over formally
p-adic differential fields with a p-adically closed field of constants. For an integrable partial
differential system defined over such a field, we prove that there exists a formally real
(resp. formally p-adic) Picard–Vessiot extension. Moreover, we obtain a uniqueness result
for this Picard–Vessiot extension. We give an adequate definition of the Galois differential
group and obtain a Galois fundamental theorem in this setting. We apply the obtained Galois
correspondence to characterise formally real Liouvillian extensions of real partial differential
fields with a real closed field of constants by means of split solvable linear algebraic groups.
We present some examples of real dynamical systems and indicate some possibilities of
further development of algebraic methods in real dynamical systems.
Key words: real Liouvillan extension; real and p-adic Picard–Vessiot theory; split solvable
algebraic group; gradient dynamical systems; integrability
2020 Mathematics Subject Classification: 12H05; 37J35; 12D15; 14P05
1 Introduction
Galois theory for linear differential equations is the differential counterpart of classical Galois
theory. The idea of Galois of characterising those polynomial equations solvable by radicals
by means of the group of permutations of the roots which preserve the relations between them
was paralleled in the work of Picard and Vessiot who characterised linear differential equa-
tions solvable by quadratures by means of the group of linear automorphisms of the vector
space of solutions that preserve the differential relations between them. In a similar way as
the work of Galois was later formalized by Artin, the one of Picard and Vessiot was formali-
zed by using differential algebra, more precisely introducing the notion of a differential field K
to be a field endowed with a derivation. The constants of K are defined as the elements on
which the derivation vanishes and form a subfield C of K. A satisfactory Galois theory for
linear differential equations defined over a differential field K with algebraically closed field
of constants C was established by Kolchin, under the name of Picard–Vessiot theory (see [17]).
For such a differential equation, Kolchin proved the existence and uniqueness up toK-differential
isomorphism of a Picard–Vessiot field, the analog of the splitting field in classical Galois theo-
ry. The differential Galois group is defined as the group of differential automorphisms of the
mailto:teresa.crespo@ub.edu
http://www.ub.edu/tn/personal/crespo.php
mailto:Zbigniew.Hajto@uj.edu.pl
mailto:Rouzbeh.Mohseni@doctoral.uj.edu.pl
https://doi.org/10.3842/SIGMA.2021.095
2 T. Crespo, Z. Hajto and R. Mohseni
Picard–Vessiot field which fix the base field K and it has the structure of a linear algebraic
group defined over the field of constants C. The fundamental theorem of Picard–Vessiot theory
establishes a bijective correspondence between intermediate differential fields of the Picard–
Vessiot extension and the closed subgroups of the differential Galois group. We note that
the hypothesis that the field of constants C of K is algebraically closed was essential in the
work of Kolchin. His results can be extended to integrable partial differential systems de-
fined over a partial differential field K with algebraically closed field of constants (see [24,
Appendix D]).
In [9], T. Crespo, Z. Hajto and M. van der Put show that the condition that the field of
constants is algebraically closed may be relaxed. They consider formally real and formally p-
adic fields, whose definition is given in Section 2. For a homogeneous linear differential equation
defined over a formally real (resp. formally p-adic) ordinary differential field K with a real closed
(resp. p-adically closed) field of constants CK , they prove the existence of a formally real (resp.
formally p-adic) Picard–Vessiot extension. Moreover, they obtain a result of uniqueness of the
formally real (resp. formally p-adic) Picard–Vessiot extension up to K-differential isomorphism.
In model theoretic language, formally real (resp. formally p-adic) differential fields with a real
closed (resp. p-adically closed) field of constants are instances of differential fields K such that
the field of constants CK is existentially closed in K. Under this hypothesis, the existence of
a Picard–Vessiot extension is proved in [12, Theorem 2.2]. In [15] the results in [9] are generalised
to the case when CK is existentially closed in K, large and bounded.
The standard example of a real closed field is the field R of real numbers whereas the field
of real rational functions in one or several variables is an example of a formally real field.
Picard–Vessiot theory for formally real differential fields with real closed field of constants al-
lows then to characterise some aspects of the behaviour of real functions. In [7] T. Crespo
and Z. Hajto characterised real Liouvillian extensions of ordinary differential fields in terms of
real Picard–Vessiot theory. Their result answers an earlier question of A. Khovanskii, namely
“Is it true that a necessary and sufficient condition for solvability of a real differential equation
by real Liouville functions follows from real Picard–Vessiot theory?”, which originated in the
theory developed in [11]. For a generalisation and further development of this theory one can
consult [16].
In this paper, we establish Galois theory for partial differential systems over formally real or
formally p-adic partial differential fields. We prove the existence of a formally real (resp. formally
p-adic) Picard–Vessiot extension for a partial differential system defined over a formally real
(resp. formally p-adic) partial differential field K with a real closed (resp. p-adically closed) field
of constants. We establish as well a uniqueness result of the formally real (resp. formally p-adic)
Picard–Vessiot extension up toK-differential isomorphism. We give an adequate definition of the
differential Galois group in this setting and prove a Galois correspondence theorem. Due to the
fact that real Liouville functions over the field R of real numbers have very interesting topological
properties (see [11, 16]), we restrict to formally real fields in our study of Liouvillianity questions.
We characterise formally real Liouvillian extensions for formally real partial differential fields
by means of the differential Galois group. Previously, we recall in Section 2 the concepts of
formally real field, real closed field, formally p-adic field and p-adically closed field as well as
the main definitions and known results of Picard–Vessiot theory. It is worth noting that the
algebraic characterisation of real Liouville functions allows us to expect an algebraic version of
the theory developed by Gel’fond and Khovanskii in [11, 16]. Similar ideas already appear in
Grothendieck’s “Esquisse d’un programme” (see [13, p. 272]), where he proposed to consider
constant functions with values in the real closure Qr
of the field of rational numbers Q in
the study of integration of Pfaff systems. In the last section of the paper we present some
questions related to the integrability of real dynamical systems showing the interest of this
further development.
Real Liouvillian Extensions of Partial Differential Fields 3
2 Preliminaries
2.1 Ordered fields, formally real fields
In this section we recall the concept of formally real field and real closed field. More details on
these topics may be found in [1, 22].
Definition 2.1. An ordering on a field k is a total order relation ≤ on k satisfying, for any
x, y, z ∈ k,
(i) x ≤ y ⇒ x+ z ≤ y + z,
(ii) 0 ≤ x and 0 ≤ y ⇒ 0 ≤ xy.
An ordered field (k,≤) is a field k equipped with an ordering ≤.
Remark 2.2 ([1, Example 1.1.2]). Given an ordered field (k,≤), there is exactly one ordering
on the field of rational functions k(t), extending the one on k, such that t is positive and smaller
than any positive element in k. If P (t) = ant
n + an−1t
n−1 + · · · + amtm ∈ k[t], with am ̸= 0,
then P (t) > 0 ⇔ am > 0. If P (t)/Q(t) ∈ k(t), then P (t)/Q(t) > 0 ⇔ P (t)Q(t) > 0.
Theorem 2.3 ([1, Theorem 1.1.8]). Let k be a field. The following properties are equivalent:
(i) k can be ordered,
(ii) −1 is not a sum of squares in k,
(iii) for x1, . . . , xn ∈ k,
∑n
i=1 x
2
i = 0 ⇒ x1 = · · · = xn = 0.
Definition 2.4. A field satisfying the properties of the preceding theorem is called a formally
real field.
We note that a formally real field always has characteristic 0.
Definition 2.5. If k is a formally real field, a formally real extension of k is a field extension ℓ/k
such that ℓ is a formally real field. A real closed field is a formally real field that has no nontrivial
formally real algebraic extension.
Theorem 2.6 ([1, Theorem 1.2.2]). Let k be a field. The following properties are equivalent:
(i) k is a real closed field,
(ii) k has a unique ordering,
(iii) the ring k[i] = k[X]/
(
X2 + 1
)
is an algebraically closed field.
Examples 2.7. Q, R are formally real fields, R is a real closed field.
Definition 2.8. A real algebraic closure of an ordered field (k,≤) is an algebraic field exten-
sion kr of k such that kr is a real closed field and the unique ordering of kr restricts to ≤
on k.
Theorem 2.9 ([22, Theorem 3.10]). Any ordered field (k,≤) has a unique (up to k-isomorphism)
real algebraic closure.
4 T. Crespo, Z. Hajto and R. Mohseni
2.2 Valued fields, p-adic fields
In this section we recall the concept of formally p-adic field and p-adically closed field. More
details on these topics may be found in [23].
Definition 2.10. A valuation of a field k is a map
v : k → Γ ∪ {∞},
where Γ is a totally ordered abelian group, such that, for all a, b in k,
(1) v(a) = ∞ ⇔ a = 0,
(2) v(ab) = v(a) + v(b),
(3) v(a+ b) ≥ min{v(a), v(b)}, with equality if v(a) ̸= v(b).
We recall that, for v a valuation of a field k, the valuation ring O is defined as O := {a ∈
k : v(a) ≥ 0} and O has a unique maximal ideal m := {a ∈ k : v(a) > 0}. The residue field is
then defined as the quotient O/m.
Definition 2.11. Let p be a prime number. A p-valuation of a field k of characteristic 0 is
a valuation v of k such that v(p) is minimal positive in the value group and the residue field is
isomorphic to Z/pZ.
A p-valued field (k, v) is a characteristic 0 field k equipped with a p-valuation v. A formally
p-adic field is a characteristic 0 field which can be endowed with a p-valuation.
Remark 2.12. If (k, v) is a p-valued field, the p-valuation v may be extended to a p-valuation
of the field of rational functions k(t) (see [23, Example 2.2]).
Definition 2.13. If k is a formally p-adic field, a formally p-adic extension of k is a field
extension ℓ/k such that ℓ is a formally p-adic field. A p-adically closed field is a formally p-adic
field that has no nontrivial formally p-adic algebraic extension.
Example 2.14. Qp is a p-adically closed field.
Remark 2.15. What we call “p-adic” is called “p-adic of rank one” in [23]. The case of higher
rank p-adic fields can be treated in the same way.
Definition 2.16. A p-adic algebraic closure of a p-valued field (k, v) is an algebraic field exten-
sion kv of k such that kv is a p-adically closed field and the valuation of k extends to kv.
Theorem 2.17 ([23, Corollary 3.11]). Any p-valued field (k, v) has a p-adic algebraic closure.
Let ℓ1 and ℓ2 be two p-adic closures of k. Then ℓ1 and ℓ2 are k-isomorphic if and only if
k ∩ ℓn1 = k ∩ ℓn2 for all n ∈ N (where ℓni := {an|a ∈ ℓi}, i = 1, 2).
2.3 Known results on Picard–Vessiot extensions
LetK be a field endowed with a set ∆ = {∂1, . . . , ∂m} of pairwise commuting derivations. Let CK
denote the field of constants of K. In particular, if m = 1, K is an ordinary differential field.
We consider a differential system
∂jY = AjY, 1 ≤ j ≤ m, (2.1)
where Aj is an (r × r)-matrix with entries in K, 1 ≤ j ≤ m.
A solution for the system (2.1) is a column vector v in the vector space Lr, for L some
differential field extension of K, such that ∂jv = Ajv, 1 ≤ j ≤ m. A fundamental matrix for the
Real Liouvillian Extensions of Partial Differential Fields 5
system (2.1) is an r × r invertible matrix M with entries in some differential field extension L
of K such that ∂jM = AjM , 1 ≤ j ≤ m. If M ∈ GLr(L) is a fundamental matrix for (2.1),
then the set of fundamental matrices for (2.1) with entries in L consists in the matrices of the
form MC for C ∈ GLr(CL), for CL the constant field of L. We note that, if a fundamental matrix
exists for (2.1), then the commutation of the derivatives implies that the matrices Aj satisfy
∂jAi +AiAj = ∂iAj +AjAi, 1 ≤ i, j ≤ m. (2.2)
The relations (2.2) are therefore a necessary condition for the existence of a common solution to
the equations in the system (2.1). We say that the differential system (2.1) is integrable if the
matrices Aj satisfy (2.2).
Definition 2.18. A Picard–Vessiot extension for the differential system (2.1) is a field exten-
sion L of K such that:
1. L is equipped with a set of pairwise commuting derivations extending the ones in ∆.
2. There exists a fundamental matrix Z for (2.1) with entries in L.
3. L is (as a field) generated over K by the entries of Z.
4. The field of constants of L is the same as the field of constants of K.
When CK is algebraically closed, a classical result in differential Galois theory states that
a Picard–Vessiot field exists for the system (2.1) and it is unique up to a K-differential isomor-
phism (see, e.g., [5, Theorems 5.6.5 and 5.6.9] for the ordinary case and [24, Appendix D] for
the partial case).
In [9], Crespo, Hajto and van der Put obtained the following theorem on existence and
uniqueness of Picard–Vessiot fields for formally real ordinary differential fields with real closed
field of constants and for formally p-adic ordinary differential fields with p-adically closed field
of constants.
Theorem 2.19 ([9, Theorem 2]). Let K be a formally real (resp. formally p-adic) ordinary
differential field with real closed (resp. p-adically closed) field of constants. We consider a dif-
ferential system
Y ′ = AY, (2.3)
where A is an (r × r)-matrix with entries in K.
1. Existence. There exists a formally real (resp. formally p-adic) Picard–Vessiot extension
of K for (2.3), i.e., a Picard–Vessiot extension of K for (2.3) which is also a formally
real (resp. formally p-adic) field extension of K.
2. Unicity for the real case. Let L1, L2 denote two formally real Picard–Vessiot extensions
of K for (2.3). Suppose that L1 and L2 have orderings which induce the same ordering
on K. Then L1 and L2 are K-differentially isomorphic.
3. Unicity for the p-adic case. Let L1, L2 denote two formally p-adic Picard–Vessiot
extensions of K for (2.3). Suppose that L1 and L2 have p-adic closures L+
1 and L+
2 such
that the p-valuations of L+
1 and L+
2 induce the same p-valuation on K and such that
K ∩
(
L+
1
)n
= K ∩
(
L+
2
)n
for every integer n ≥ 2. Then L1 and L2 are K-differentially
isomorphic.
6 T. Crespo, Z. Hajto and R. Mohseni
3 Galois theory for partial differential systems
over formally real or formally p-adic differential fields
In this section we consider an integrable partial differential system (2.1) defined over a formally
real (resp. formally p-adic) differential field K with real closed (resp. p-adically closed) field
of constants CK . Let CK denote an algebraic closure of CK . The differential Galois group G
for (2.1) is a CK/CK-form of the differential Galois group G for (2.1) considered as defined over
K := K⊗CK CK , which is a partial differential field with algebraically closed field of constants CK .
However the differential Galois group G for (2.1) over K gives more information than G on the
behaviour of the solutions to (2.1). In the case when K is a formally real ordinary differential
field and CK is real closed, we obtained in [7] that the property for a system Y ′ = AY defined
over K to have solutions which are real Liouville functions is characterised by the differential
Galois group G. Therefore it is interesting to extend this result to partial differential fields.
3.1 Picard–Vessiot extensions
In this section, we will use the approach of Kolchin in [18] to show how the Picard–Vessiot theory
of formally real and formally p-adic partial differential fields can be deduced from the ordinary
case. Let us note that Kolchin’s definition of Picard–Vessiot extension for partial differential
fields is different from the one given in Definition 2.18. In [6, Theorem 1] the equivalence of
both definitions is proved.
Remark 3.1. Following Kolchin [18], to a partial differential field k with pairwise commu-
ting derivations ∂1, . . . , ∂m we associate an ordinary differential field kD := k⟨u1, . . . , um⟩ with
u1, . . . , um independent differential indeterminates, endowed with the derivation D := u1∂1 +
· · ·+um∂m. As remarked by Kolchin, kD and k have the same field of constants. Let us observe
that, as a field, kD is a purely transcendental extension of k. Therefore, if k is a formally real
field (resp. a formally p-adic field), then using induction and Remark 2.2 (resp. Remark 2.12),
the ordering (resp. the p-valuation) of k may be extended to kD, i.e., kD is a formally real (resp.
formally p-adic) field.
We consider now a system of the form (2.1) defined over a formally real (resp. formally p-adic)
partial differential field. We shall prove the following theorems on the existence and uniqueness
of Picard–Vessiot extensions in this setting.
Theorem 3.2.
1. Let us suppose that K is a formally real partial differential field with real closed field of
constants CK . Then for an integrable differential system (2.1) defined over K, there exists
a formally real differential field L with field of constants CK such that L = K({yij}1≤i,j≤r),
for (yij)1≤i,j≤r ∈ GLr(L) a fundamental matrix for (2.1), i.e., L/K is a Picard–Vessiot
extension for (2.1).
2. Let us suppose that K is a formally p-adic partial differential field with p-adically closed
field of constants CK . Then for an integrable differential system (2.1) defined over K,
there exists a formally p-adic differential field L with field of constants CK such that L =
K({yij}1≤i,j≤r), for (yij)1≤i,j≤r ∈ GLr(L) a fundamental matrix for (2.1), i.e., L/K is
a Picard–Vessiot extension for (2.1).
Proof. First we consider an auxiliary differential system
DY = ADY, (3.1)
Real Liouvillian Extensions of Partial Differential Fields 7
where D = u1∂1 + · · · + um∂m and AD = u1A1 + · · · + umAm ∈ Mr×r(KD), with u1, . . . , um
independent differential indeterminates. Since (KD, D) is a formally real (resp. formally p-adic)
ordinary differential field, with real closed (resp. p-adically closed) field of constants CK , by Theo-
rem 2.19, there exists a formally real (resp. formally p-adic) Picard–Vessiot extension LD/KD
for the system (3.1). Let Z denote a fundamental matrix for (3.1) such that LD is generated
over KD by the entries of Z. Let CK denote an algebraic closure of CK . The derivation D
extends uniquely to KD := KD ⊗CK CK and LD := LD ⊗CK CK , with field of constants CK .
Since the extensions CK/CK and KD/CK are linearly disjoint, KD and LD are fields and LD is
generated over KD by the entries of Z. Hence LD/KD is a Picard–Vessiot extension for (3.1).
On the other hand, since K := K⊗CK CK is a partial differential field with algebraically closed
field of constants CK , we have a Picard–Vessiot extension L/K for the system (2.1) (see [24,
Appendix D]).
The field extensions we have considered up to now are shown in the following diagram:
L LD
K KD
LD
K KD
We define LD := L⟨u1, . . . , um⟩ and endow it with the derivation D = u1∂1 + · · · + um∂m.
By [18, Theorem 1], LD is a Picard–Vessiot extension of KD for (3.1). By the uniqueness of
the Picard–Vessiot extension in the ordinary case, LD is isomorphic to LD and we can assume
L ⊂ LD.
The Galois group G := Gal(CK/CK) acts on the field LD = LD ⊗CK CK by acting on the
second factor. We note that this action commutes with the derivation D and is continuous.
Let V be the CK-vector space of solutions to (2.1) contained in L. By restricting the action of G
on LD to V we obtain a continuous semi-linear action. Let V G := {v ∈ V | σ(v) = v, ∀σ ∈ G}.
Clearly V G is a CK-subspace of the CK-vector space V . We want to show that the CK-vector
space V G has dimension equal to the dimension of V over CK .
Let v1, . . . , vd be a CK-basis of V G. We shall prove first that v1, . . . , vd are linearly independent
over CK . Assume otherwise and let a1v1+· · ·+advd = 0, with ai ∈ CK , 1 ≤ i ≤ d be a dependence
relation with a minimal number of non-zero coefficients. We may assume a1 ̸= 0 and, by scaling,
a1 = 1. By applying σ ∈ G to the dependence relation and subtracting the obtained relation
from the first one, we obtain from the minimality condition σ(ai)−ai = 0 for 2 ≤ i ≤ d. Since σ
is any element in G, we obtain that the coefficients ai belong to CK . This gives a contradiction,
hence v1, . . . , vd are CK-linearly independent.
In order to prove that v1, . . . , vd generate V over CK , we shall see that a vector v ∈ V can be
written as a linear combination of vectors in V G, with coefficients in CK . In the real case, by
Theorem 2.6, we have CK = CK(i), where i =
√
−1, and G = {Id, c} with c defined by c(i) = −i.
We may then write v = (1/2)(v + c(v)) + i (1/2i)(v − c(v)) and (1/2)(v + c(v)), (1/2i)(v − c(v))
belong to V G. In the p-adic case, G is a pro-finite group. For v ∈ V ⊂ LD = LD ⊗CK CK , there
exists a finite extension C̃/C such that v ∈ LD ⊗CK C̃ and we may assume C̃/CK Galois. Let
n :=
[
C̃ : CK
]
and G̃ := Gal
(
C̃/CK
)
= {σ1, . . . , σn}. By the linear independence of characters
8 T. Crespo, Z. Hajto and R. Mohseni
and the equality of dimensions, the map
C̃ ⊗CK C̃ → C̃n, x⊗ y 7→ (xσ1(y), . . . , xσn(y))
is an isomorphism. We may then find elements x1, . . . , xn, y1, . . . , yn in C̃ such that
n∑
i=1
xiyi = 1,
n∑
i=1
xiσ(yi) = 0, for σ ∈ G̃, σ ̸= Id .
We have then v =
∑
σ∈G̃
∑n
i=1 xiσ(yi)σ(v) =
∑n
i=1 xi
(∑
σ∈G̃ σ(yiv)
)
and
∑
σ∈G̃ σ(yiv) ∈ V G,
for i = 1, . . . , n.
We have then obtained that the CK-vector space V G has dimension r = dimCK V . Now V G is
clearly a vector space of solutions to (2.1). Let us define L := K
〈
V G
〉
the subfield of L generated
by the elements in V G. By construction, L is a Picard–Vessiot extension of K for (2.1). Since
L ⊂ LD, the field L is a formally real (resp. formally p-adic) field.
The complete diagram of field extensions is as follows:
L LD ≃ LD
K KD
L LD
K KD
■
Theorem 3.3. Let L1, L2 denote two formally real Picard–Vessiot extensions of K for (2.1).
Suppose that L1 and L2 have orderings which induce the same ordering on K. Then L1 and L2
are K-differentially isomorphic.
Proof. We have that (L1)D and (L2)D are two formally real Picard–Vessiot extensions of KD
for the system (3.1). We extend the ordering from K to KD and from Li to (Li)D, i = 1, 2, in
such a way that the ordering of (Li)D restricts to the ordering in KD, i = 1, 2 (see Remark 3.1).
By Theorem 2.19, we have a differential KD-isomorphism
φ : L1⟨u1, . . . , um⟩ → L2⟨u1, . . . , um⟩.
If L1 = K(yij) and L2 = K
(
ỹij
)
for fundamental matrices (yij) and
(
ỹij
)
of the system (2.1),
then (φ(yij))1≤i,j≤r is a fundamental matrix of (2.1) which gives (φ(yij)) =
(
ỹij
)
C, with C ∈
Mr×r(CK). Therefore φ restricts to a differential K-isomorphism
φ|L1
: L1 → L2. ■
Theorem 3.4. Let L1, L2 denote two formally p-adic Picard–Vessiot extensions of K for (2.1).
Suppose that L1 and L2 have K-isomorphic p-adic closures L+
1 and L+
2 . Then L1 and L2 are
K-differentially isomorphic.
Proof. As u1, . . . , um are independent differential indeterminates, the K-isomorphism from L+
1
to L+
2 may be extended to a field isomorphism from
(
L+
1
)
D
to
(
L+
2
)
D
over KD. Now we may
extend the p-adic valuation to
(
L+
1
)
D
and define a p-valuation on
(
L+
2
)
D
by transferring to
Real Liouvillian Extensions of Partial Differential Fields 9
it the valuation of
(
L+
1
)
D
by means of the isomorphism. Next we choose a p-adic closure M1
of
(
L+
1
)
D
. We can extend the isomorphism from
(
L+
1
)
D
to
(
L+
2
)
D
to an embedding of M1 into
an algebraic closure of
(
L+
2
)
D
. Its image, equipped with the valuation induced by the valuation
of
(
L+
1
)
D
via the embedding is a p-adic closure M2 of
(
L+
2
)
D
. Now M1 and M2 are isomorphic
as valued fields over KD and are p-adic closures of (L1)D and (L2)D. By Theorem 2.19, we have
a differential KD-isomorphism
φ : L1⟨u1, . . . , um⟩ → L2⟨u1, . . . , um⟩.
The rest of the proof is like for Theorem 3.3. ■
3.2 Galois correspondence
In the case when the field of constants of the differential field K is algebraically closed, the diffe-
rential Galois group of a Picard–Vessiot extension L/K is defined as the group of K-differential
automorphisms of L. The next example illustrates that, when the field of constants is not
algebraically closed, the group of K-differential automorphisms of L may be “too small” to
obtain a satisfactory differential Galois theory.
Example 3.5. Let K := R(t1, t2) be the field of rational functions in the variables t1, t2 over
the field R of real numbers endowed with the usual derivations ∂1 := ∂/∂t1 and ∂2 := ∂/∂t2.
Clearly the field of constants of K is R. We consider the differential system
∂1Y = t2 Y, ∂2Y = t1 Y, (3.2)
defined over K. A solution to the system (3.2) is et1t2 , hence L := K
(
et1t2
)
is a formally real
Picard–Vessiot extension of K for (3.2). If σ is a K-differential automorphism of L, we have
σ
(
et1t2
)
= λet1t2 , with λ ∈ R \ {0} and composing two such automorphisms amounts to multi-
plying the corresponding factors λ. Hence the group DAutK L of K-differential automorphisms
of L is isomorphic to the multiplicative group of R. We consider now the intermediate field
F := K
(
e3t1t2
)
. An F -differential automorphism τ of L is given by τ
(
et1t2
)
= λet1t2 , with
λ3 = 1, since τ must fix e3t1t2 . Hence the group of F -differential automorphisms of L is trivial.
The subgroups of DAutK L corresponding to F and L by the Galois correspondence will then
be equal whereas F and L are not equal.
Let K be a formally real (resp. formally p-adic) partial differential field with real closed (resp.
p-adically closed) field of constants CK and let L/K be a Picard–Vessiot extension. Let CK denote
an algebraic closure of CK and consider the fields K := K ⊗CK CK and L := L ⊗CK CK . As in
the ordinary case (see [8]), we shall consider the set DHomK
(
L,L
)
of K-differential morphisms
from L into L and transfer the group structure from DAutK L to DHomK
(
L,L
)
by means of
the bijection
DAutK L → DHomK
(
L,L
)
,
τ 7→ τ|L.
Definition 3.6. For the Picard–Vessiot extension L/K we define the differential Galois group
as the set DHomK
(
L,L
)
endowed with the group structure given above and denote it by
DGal(L|K).
Proposition 3.7. The map DGal(LD|KD) → DGal(L|K) which to each KD-differential mor-
phism from LD into LD = LD ⊗CK CK assigns its restriction to L is an isomorphism of groups.
10 T. Crespo, Z. Hajto and R. Mohseni
Proof. Since L/K is a Picard–Vessiot extension, we have that L = K
(
{yij}1≤i,j≤r
)
, for
(yij)1≤i,j≤r a fundamental matrix for some differential system S defined over K. If σ is a KD-
differential morphism from LD into LD, then σ((yij)) is a fundamental matrix for S. Therefore
there exists an invertible matrix C with entries in the field of constants CK of LD such that
σ((yij)) = (yij)C, hence σ((yij)) has entries in L. By restricting σ to L, we obtain then
a K-morphism from L to L. For an element x in L, we have Dx =
∑m
i=1 ui∂ix and σ(Dx) =∑m
i=1 uiσ(∂ix), since ui ∈ KD, 1 ≤ i ≤ m. On the other hand D(σ(x)) =
∑m
i=1 ui∂i(σ(x)). Since
the elements ui are algebraically independent, we obtain that σ|L commutes with ∂i, 1 ≤ i ≤ m.
Hence σ|L is a K-differential morphism from L to L. Reciprocally, a K-differential morphism
from L to L may be extended to a KD-differential morphism from LD to LD. ■
Theorem 3.8. The differential Galois group DGal(L|K) is a CK-defined closed subgroup of
some general linear group defined over CK , i.e., a linear algebraic group defined over CK . The
CK-valued points of DGal(L|K) correspond to the K-differential automorphisms of L.
Proof. The first assertion follows from Proposition 3.7 and [8, Proposition 1]. The second one
is clear from the definition of DGal(L|K). ■
For a closed subgroup H of DGal(L|K), LH is a partial differential subfield of L containingK.
If E is an intermediate partial differential field, i.e., K ⊂ E ⊂ L, then L/E is a Picard–Vessiot
extension and DGal(L|E) is a CK-defined closed subgroup of DGal(L|K). We obtain a Galois
correspondence theorem.
Theorem 3.9. Let K be a formally real (resp. formally p-adic) partial differential field with real
closed (resp. p-adically closed) field of constants CK , let L/K be a Picard–Vessiot extension and
DGal(L|K) be its differential Galois group.
1. The correspondences
H 7→ LH , E 7→ DGal(L|E)
define inclusion inverting mutually inverse bijective maps between the set of CK-defined
closed subgroups H of DGal(L|K) and the set of partial differential fields E with K⊂E⊂L.
2. The intermediate partial differential field E is a Picard–Vessiot extension of K if and only
if the subgroup DGal(L|E) is normal in DGal(L|K). In this case, the restriction morphism
DGal(L|K) → DGal(E|K),
σ 7→ σ|E
induces an isomorphism
DGal(L|K)/DGal(L|E) ≃ DGal(E|K).
Proof. The proof follows the same steps as the one of [8, Theorem 1]. In the present case, we
use the Galois correspondence theorem for partial differential fields with an algebraically closed
field of constants (see [24, Appendix D]). ■
3.3 Liouvillian extensions
In this section we restrict to formally real partial differential fields. We note that real Liouville
functions may be characterised by topological properties (see [11]) and it is therefore interesting
to characterise real Liouville solutions to differential systems defined over formally real fields by
means of the differential Galois group.
Real Liouvillian Extensions of Partial Differential Fields 11
Definition 3.10. Let K be a field endowed with pairwise commuting derivations ∂1, . . . , ∂m.
Let L/K be a partial differential field extension, α an element in L. We say that α is
− an integral over K if ∂kα = ak ∈ K, 1 ≤ k ≤ m, and ak is not a derivative in K for all k;
− the exponential of an integral over K if ∂kα/α ∈ K \ {0}, 1 ≤ k ≤ m.
Remark 3.11. If ∂kα = ak ∈ K, for 1 ≤ k ≤ m, then Dα =
∑m
k=1 ukak ∈ KD and if ak is not
a derivative in K for all k, then
∑m
k=1 ukak is not a derivative in KD. So, if α is an integral
over K, it is also an integral over KD. Analogously, if ∂kα/α ∈ K \ {0}, for 1 ≤ k ≤ m, then
Dα/α ∈ KD \ {0}. So, if α is the exponential of an integral over K, it is also the exponential of
an integral over KD.
Let now K be a formally real field with real closed field of constants CK . From Proposition 3.7
and the corresponding results in the ordinary case [7, Examples 7 and 8], we obtain that, if α is
an integral over K, then K⟨α⟩/K is a real Picard–Vessiot extension and its differential Galois
group DGal(K⟨α⟩|K) is isomorphic to the additive group Ga; and, if α is the exponential of
an integral and the field K⟨α⟩ is real and with field of constants equal to CK , then K⟨α⟩/K is
a Picard–Vessiot extension and DGal(K⟨α⟩|K) is isomorphic to the multiplicative group Gm,
or a finite subgroup of it.
Definition 3.12. A partial differential field extension L/K is called a Liouvillian extension
(resp. a generalised Liouvillian extension) if there exists a chain of intermediate partial differen-
tial fields K = F1 ⊂ F2 ⊂ · · · ⊂ Fn = L such that Fi+1 = Fi(αi), where αi is either an integral
or the exponential of an integral over Fi (resp. or αi is algebraic over Fi), 1 ≤ i ≤ n− 1.
Lemma 3.13. If L/K is a (generalised) Liouvillian extension of partial differential fields then
the ordinary differential field extension LD/KD is a (generalised) Liouvillian extension.
Proof. Let K = F1 ⊂ F2 ⊂ · · · ⊂ Fn = L be a chain of intermediate partial differential fields
such that Fi+1 = Fi(αi). We consider the chain of ordinary differential fields KD ⊂ (F2)D ⊂
· · · ⊂ LD. We have (Fi+1)D = Fi+1⟨u1, . . . , um⟩ = Fi(αi)⟨u1, . . . , um⟩ = Fi⟨u1, . . . , um⟩(αi) =
(Fi)D(αi). Now, by Remark 3.11, if αi is an integral or the exponential of an integral or
algebraic over Fi, then αi is an integral or the exponential of an integral or algebraic over (Fi)D,
respectively. ■
Definition 3.14. Let G be a connected solvable linear algebraic group defined over a field C.
We say that G is C-split if it has a composition series G = G1 ⊃ G2 ⊃ · · · ⊃ Gs = 1 consisting of
connected C-defined closed subgroups such that Gi/Gi+1 is C-isomorphic to Ga or Gm, 1 ≤ i < s.
From the results obtained in the ordinary case [7, Section 3, Theorems 17 and 18], we obtain
the characterisation of real Liouvillian extensions of real partial differential fields.
Theorem 3.15. Let K be a real partial differential field with real closed field of constants CK ,
L/K be a formally real Picard–Vessiot extension, DGal(L|K) be its differential Galois group.
The following conditions are equivalent:
1. L/K is a generalised Liouvillian extension.
2. L is contained in a generalised Liouvillian extension M of K, such that the field of con-
stants of M is CK and M is a formally real field.
3. The identity component of DGal(L|K) is solvable and CK-split.
12 T. Crespo, Z. Hajto and R. Mohseni
Proof. Clearly (1) implies (2). By Lemma 3.13, (2) implies that LD is contained in a formally
real generalised Liouvillian extension of KD, not adding constants to KD. Then by Proposi-
tion 3.7 and [7, Theorem 18], we obtain (3).
We assume now that the identity component G0 of G := DGal(L|K) is solvable and CK-split
and let L0 = LG0
. Since
[
G : G0
]
is finite, L0/K is a finite extension. By [2, Chapter V,
Theorem 15.4], in G0 is triangularizable by a matrix with entries in CK . We may then assume
that there exist elements v1, . . . , vr ∈ L such that L = L0⟨v1, . . . , vr⟩ and for every σ ∈ G0
we have
σvj = a1jv1 + · · ·+ aj−1,jvj−1 + ajjvj , j = 1, . . . , r, (3.3)
with aij constants in L(i) (depending on σ). The first equality is σv1 = a11v1 which implies
σ(∂kv1/v1) = ∂kv1/v1, for all σ ∈ G0 and all k = 1, . . . ,m, hence v1 is the exponential of an
integral over L0. Now dividing the equations in (3.3) for j = 2 to r by the first equation and
applying ∂k, we obtain
σ∂k
(
vj
v1
)
=
a2j
a11
∂k
(
v2
v1
)
+ · · ·+ aj−1,j
a11
∂k
(
vj−1
v1
)
+
ajj
a11
∂k
(
vj
v1
)
, j = 2, . . . , r,
for k = 1, . . . ,m. By induction hypothesis on r, we obtain that the extension
L1 := L0⟨v1⟩
〈{
∂k
(
vj
v1
)}
k=1,...,m;j=2,...,r
〉/
L0
is a Liouvillian extension. Now the elements vj/v1 are integral over L1 and we obtain that L/L0
is a Liouvillian extension and L/K is a generalised Liouvillian extension. ■
Remark 3.16. From Lemma 3.13, Proposition 3.7 and Theorem 3.15, we obtain that L/K is
a formally real (generalised) Liouvillian extension of partial differential fields if and only if the
ordinary differential field extension LD/KD is a formally real (generalised) Liouvillian extension.
Example 3.17. Let K := R(t1, t2) be the field of rational functions in the variables t1, t2 over
the field R of real numbers endowed with the usual derivations ∂1 := ∂/∂t1 and ∂2 := ∂/∂t2.
The field of constants of K is R. We consider the differential system
∂1Y =
(
1/t1 0
0 1/t1
)
Y, ∂2Y =
(
0 1
−1 0
)
Y, (3.4)
defined over K. A fundamental matrix for the system (3.4) is(
t1 sin t2 −t1 cos t2
t1 cos t2 t1 sin t2
)
,
hence L := K⟨t1 sin t2, t1 cos t2⟩ = K⟨sin t2, cos t2⟩ is a Picard–Vessiot extension of K for (3.4).
An element σ in the Galois group DGal(L/K) is determined by σ(sin t2) and σ(cos t2). If σ(sin t2)
= a sin t2 + b cos t2, then σ(cos t2) = σ(∂2(sin t2)) = ∂2(σ(sin t2)) = a cos t2 − b sin t2. Moreover
(sin t2)
2+(cos t2)
2 = 1 implies (σ(sin t2))
2+(σ(cos t2))
2 = 1, which gives a2+b2 = 1. We obtain
then DGal(L/K) ≃ SO2.
We consider now the differential system
∂1Y =
(
1/t1 0
0 1/t1
)
Y, ∂2Y =
(
0 1
1 0
)
Y, (3.5)
Real Liouvillian Extensions of Partial Differential Fields 13
defined over K. A fundamental matrix for the system (3.5) is(
t1 sinh t2 t1 cosh t2
t1 cosh t2 t1 sinh t2
)
,
hence L := K⟨t1 sinh t2, t1 cosh t2⟩ = K⟨sinh t2, cosh t2⟩ = K
〈
et2
〉
is a Picard–Vessiot extension
of K for (3.5). An element σ in the Galois group DGal(L/K) is determined by σ
(
et2
)
and
σ
(
et2
)
= cet2 , with c ∈ R \ {0}. We obtain then DGal(L/K) ≃ Gm.
We note that SO2 is isomorphic to Gm over C but not over R. Considering the systems (3.4)
and (3.5) as defined over the field R(t1, t2) we obtain that the Picard–Vessiot extension for (3.5)
is a real Liouvillian extension whereas the Picard–Vessiot extension for (3.4) is not a real gene-
ralised Liouvillian extension, since SO2 is not R-split.1
4 Comments on the integrability of real dynamical systems
In this section we present the relationship between the concept of real Liouville function and
the results on Liouvillian extensions of formally real differential fields treated in [7] and Sec-
tion 3.3 of this paper. The solutions of differential equations which until now were abstract
elements are considered in this section as real functions in real variables and we study their
properties as such functions. Considering them as elements of a field of real functions, which
is a formally real differential field, the results of the preceding sections apply. We comment as
well on some questions related with the integrability of real dynamical systems going beyond
the Liouviallinity questions. It is worth noting that real Liouville functions are a particular case
of Pfaff functions which A. Khovanskii studies in his monograph [16]. However Pfaff functions
may appear as solutions to non-linear differential equations and at present a Galois theory for
non-linear differential equations defined over a formally real differential field has not been es-
tablished. Khovanskii’s theory uses essentially the completeness of the field R of real numbers.
It would be interesting to give an algebraic approach to this theory allowing to consider also
real closed fields different from the field of real numbers.
On the other hand, Theorem 3.15 shows that real Liouvillianity of an extension L/K of
formally real differential fields is codified in the structure of the differential Galois group
DGal(L/K). A condition of this type is suggested by A. Grothendieck in his proposal of an
axiomatic definition of tame topology (cf. [13, p. 272]). In our approach we use differential
Galois groups defined over the real closed field of constants.
4.1 Real Liouvillianity of solutions
To any open and connected subset U ⊂ Rn we assign the integral domain of real analytic func-
tions O(U) equipped with standard partial derivations ∂1 = ∂
∂x1
, . . . , ∂n = ∂
∂xn
. Let M(U)
denote the fraction field Fr(O(U)). We identify the elements of M(U) with the complete mero-
morphic fractions, i.e., the functions
ϕ : U \ {g = 0} ∋ x 7→ f(x)
g(x)
∈ R,
which are maximal elements with respect to inclusion. From now on, we consider only differential
fields which are subfields of differential fields of the form (M(U), ∂1, . . . , ∂n). Besides, we identify
the field R with the trivial differential structure with the subfield of constant functions in M(U).
1We note that Remark 3 in [11] is incorrectly translated from the Russian original. The third sentence must
read: “This circle does not have a normal tower of subgroups with quotient groups isomorphic either to the
additive or the multiplicative group of the field R”.
14 T. Crespo, Z. Hajto and R. Mohseni
Definition 4.1. Let f : U → R be a real analytic function defined in an open and connected
subset U ⊂ Rn. f is called a real Liouville function (resp. generalised real Liouville function)
if it lies in some formally real Liouvillian extension (resp. generalised formally real Liouvillian
extension) K ⊂ M(U) of the field R ⊂ M(U).
Example 4.2. Let us consider the following dynamical system in R2.
Y ′ =
(
0 1
1/x2 0
)
Y,
which corresponds to the second order linear differential equation y′′−
(
1/x2
)
y = 0, defined over
the differential field K := R(x), with the usual derivation d/dx. Applying Kovacic’s algorithm
(see [5, Section 7.3]), we obtain that the two following real functions form a basis of the vector
space of solutions
y1 = x(1+
√
5)/2, y2 = x(1−
√
5)/2.
A Picard–Vessiot extension for this system overK is thenK⟨y1, y2⟩ = K⟨y1⟩, since y1y2 = x∈K.
A K-differential isomorphism from K⟨y1⟩ to C(x)⟨y1⟩ sends y1 to λy1 with λ ∈ C∗. Hence the
differential Galois group is the multiplicative group Gm, defined over R.
Let us observe that the real Liouville functions y1 and y2 fulfill Pfaff equations of the form
Y ′ = αY
x , where α = 1
2
(
1 ±
√
5
)
and therefore they can be studied by the methods of Pfaff
geometry. More precisely the graphs of y1 and y2 are leaves of the foliations in R2 defined by
the 1-forms ωα = αydx − xdy. Considering them in the positive half plane {x > 0} we obtain
separating solutions to which Khovanskii’s theory is aplicable (cf. [16, pp. 4 and 5]). In this
paper we do not enter in details of this theory but an interested reader can consult [16].
Our next example is related with real Liouville first integrals of simple gradient systems.
Recently gradient systems are intensively studied in relation with the gradient conjecture and
its generalisations (see [3, 4]).
Example 4.3. T.H. Colding and W.P. Minicozzi II in their recent work [3] have formulated
a generalised version of the famous René Thom conjecture. More precisely Conjecture 1.1
from [3] has been proved in [19], while the following conjecture [3, Conjecture 1.2], called by the
authors Arnold–Thom conjecture, remains open.
Let f be a real analytic function in an open set U ⊂ Rn and grad f be its gradient in the
Euclidean metric.
Arnold–Thom conjecture. If a gradient flow line x(t) has a limit point x0 ∈ U , then the
limit of the unit tangents x′(t)
|x(t)| at x0 exists.
In our second example, we present a polynomial gradient system that admits Liouville first
integral. Its trajectories in the neighborhood of the origin are C1 manifolds with boundary, but
do not admit a C2 extension through their limit point. Thus this example shows that the above
statement of Arnold–Thom conjecture is optimal.
Let us consider a simple quadratic polynomial potential on R2
f(x, y) = λx2 + µy2, λ, µ ∈ R \ {0}
and its gradient dynamical system
dx
dt
= 2λx,
dy
dt
= 2µy. (4.1)
Real Liouvillian Extensions of Partial Differential Fields 15
Solving (4.1), one obtains the following real Liouvillian first integral
I(x, y) =
xµ
yλ
.
It is easy to see that for λ = 2, µ = 3 and I(x, y) = 1, the level curve is an ordinary cusp.
Moreover as the curvature at the origin of
{
(x, y) ∈ R2 : y2 = x3
}
tends to infinity, a C2
prolongation of the branch curve y = x3/2 is impossible (see, e.g., [10, Theorem 5.1.6]).
4.2 Final remarks
The Arnold–Thom conjecture was known at the beginning of the 90’s of last century under the
name “Tangent Problem” (see [20, Section 9]). One more interesting conclusion suggested by the
Example 4.3 is that rational or, more generally, meromorphic integrability is quite exceptional
even among systems that admit Liouville integrals. An interested reader can consult the excellent
survey on the classical integrability problems [21] and some more computational examples of
non-integrable real dynamical systems in [14].
Acknowledgments
We are very thankful to the anonymous referees for their valuable comments which helped us to
improve significantly the presentation of our results. R. Mohseni acknowledges support of the
Polish Ministry of Science and Higher Education. T. Crespo and Z. Hajto acknowledge support
of grant PID2019-107297GB-I00 (MICINN).
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1 Introduction
2 Preliminaries
2.1 Ordered fields, formally real fields
2.2 Valued fields, p-adic fields
2.3 Known results on Picard–Vessiot extensions
3 Galois theory for partial differential systems over formally real or formally p-adic differential fields
3.1 Picard–Vessiot extensions
3.2 Galois correspondence
3.3 Liouvillian extensions
4 Comments on the integrability of real dynamical systems
4.1 Real Liouvillianity of solutions
4.2 Final remarks
References
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| id | nasplib_isofts_kiev_ua-123456789-211432 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T10:41:54Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Crespo, Teresa Hajto, Zbigniew Mohseni, Rouzbeh 2026-01-02T08:31:49Z 2021 Real Liouvillian Extensions of Partial Differential Fields. Teresa Crespo, Zbigniew Hajto and Rouzbeh Mohseni. SIGMA 17 (2021), 095, 16 pages 1815-0659 2020 Mathematics Subject Classification: 12H05; 37J35; 12D15; 14P05 arXiv:2104.09548 https://nasplib.isofts.kiev.ua/handle/123456789/211432 https://doi.org/10.3842/SIGMA.2021.095 In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally -adic differential fields with a -adically closed field of constants. For an integrable partial differential system defined over such a field, we prove that there exists a formally real (resp. formally -adic) Picard-Vessiot extension. Moreover, we obtain a uniqueness result for this Picard-Vessiot extension. We give an adequate definition of the Galois differential group and obtain a Galois fundamental theorem in this setting. We apply the obtained Galois correspondence to characterise formally real Liouvillian extensions of real partial differential fields with a real closed field of constants by means of split solvable linear algebraic groups. We present some examples of real dynamical systems and indicate some possibilities for further development of algebraic methods in real dynamical systems. We are very thankful to the anonymous referees for their valuable comments, which helped us to improve significantly the presentation of our results. R. Mohseni acknowledges the support of the Polish Ministry of Science and Higher Education. T. Crespo and Z. Hajto acknowledge support of grant PID2019-107297GB-I00 (MICINN). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Real Liouvillian Extensions of Partial Differential Fields Article published earlier |
| spellingShingle | Real Liouvillian Extensions of Partial Differential Fields Crespo, Teresa Hajto, Zbigniew Mohseni, Rouzbeh |
| title | Real Liouvillian Extensions of Partial Differential Fields |
| title_full | Real Liouvillian Extensions of Partial Differential Fields |
| title_fullStr | Real Liouvillian Extensions of Partial Differential Fields |
| title_full_unstemmed | Real Liouvillian Extensions of Partial Differential Fields |
| title_short | Real Liouvillian Extensions of Partial Differential Fields |
| title_sort | real liouvillian extensions of partial differential fields |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211432 |
| work_keys_str_mv | AT crespoteresa realliouvillianextensionsofpartialdifferentialfields AT hajtozbigniew realliouvillianextensionsofpartialdifferentialfields AT mohsenirouzbeh realliouvillianextensionsofpartialdifferentialfields |