The Ramificant Determinant
We give an introduction to the transalgebraic theory of simply connected log-Riemann surfaces with a finite number of infinite ramification points (transalgebraic curves of genus 0). We define the base vector space of transcendental functions and establish, by elementary methods, some transcendental...
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| Cite this: | The Ramificant Determinant / K. Biswas, R. Pérez-Marco // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 19 назв. — англ. |
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| description | We give an introduction to the transalgebraic theory of simply connected log-Riemann surfaces with a finite number of infinite ramification points (transalgebraic curves of genus 0). We define the base vector space of transcendental functions and establish, by elementary methods, some transcendental properties. We introduce the Ramificant determinant constructed with transcendental periods, and we give a closed-form formula that gives the main applications to transalgebraic curves. We prove an Abel-like theorem and a Torelli-like theorem. Transposing to the transalgebraic curve, the base vector space of transcendental functions, they generate the structural ring from which the points of the transalgebraic curve can be recovered algebraically, including infinite ramification points.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 086, 28 pages
The Ramificant Determinant
Kingshook BISWAS † and Ricardo PÉREZ-MARCO ‡
† Indian Statistical Institute, Kolkata, India
E-mail: kingshook@isical.ac.in
‡ CNRS, IMJ-PRG, University Paris 7, Paris, France
E-mail: ricardo.perez-marco@imj-prg.fr
Received March 13, 2019, in final form October 31, 2019; Published online November 05, 2019
https://doi.org/10.3842/SIGMA.2019.086
Abstract. We give an introduction to the transalgebraic theory of simply connected log-
Riemann surfaces with a finite number of infinite ramification points (transalgebraic curves
of genus 0). We define the base vector space of transcendental functions and establish by
elementary methods some transcendental properties. We introduce the Ramificant deter-
minant constructed with transcendental periods and we give a closed-form formula that
gives the main applications to transalgebraic curves. We prove an Abel-like theorem and
a Torelli-like theorem. Transposing to the transalgebraic curve the base vector space of
transcendental functions, they generate the structural ring from which the points of the
transalgebraic curve can be recovered algebraically, including infinite ramification points.
Key words: transalgebraic theory; Ramificant determinant; log-Riemann surface; Dedekind–
Weber theory; ramified covering; exponential period; Liouville theorem
2010 Mathematics Subject Classification: 30F99; 30D99
1 Introduction
The authors defined the notion of log-Riemann (and tube-log, see [6]) surfaces in the seminal
manuscript [5] (see also [3, 4]) as a proper formalization of classical Riemann surfaces and infinite
ramification points as mathematicians of the XIXth century understood them, in particular
Bernhard Riemann.
These Riemann surfaces are endowed with distinguished charts and provide a direct link
to classical special functions. Log-Riemann surfaces are Riemann domains over C. Lifting
the flat Euclidean metric defines the log-Euclidean metric, and studying the completion of the
associated length space, we can define properly the notion of ramification locus, and in particular
of infinite ramification points. The original approach from [5] is by explicit construction of the
canonical chart by “cut and paste” techniques. Then we obtain a Riemann surface with a local
diffeomorphism π : S → C. Conversely, as presented in [3], we can start with π and define
log-Riemann surfaces. The set of points R added in the completion S∗ = S t R of S for the
log-Euclidean metric on S is the ramification locus R. Points in R are at finite distance and
the completion S∗ is a complete metric space, but is no longer a surface in general, it may not
even be a locally compact space.
Isolated points in R are called ramification points. We only consider in this article the case
where this ramification locus is discrete. Then the local inverse of π composed with a local chart
is fluent in the sense of Ritt (see [16] and the forthcoming Ph.D. thesis by Y. Levagnini). Also in
this case, the mapping π extends continuously to the ramification points p ∈ R, and is a covering
of a punctured neighborhood of p onto a punctured disk in C. The point p is a ramification point
of S and its order is equal to the degree of the covering π near p. The finite order ramification
This paper is a contribution to the Special Issue on Algebraic Methods in Dynamical Systems. The full
collection is available at https://www.emis.de/journals/SIGMA/AMDS2018.html
mailto:kingshook@isical.ac.in
mailto:ricardo.perez-marco@imj-prg.fr
https://doi.org/10.3842/SIGMA.2019.086
https://www.emis.de/journals/SIGMA/AMDS2018.html
2 K. Biswas and R. Pérez-Marco
points may be added to S and give a Riemann surface S×, called the finite completion of S.
When the number of ramification points (finite or infinite order) is finite, and the fundamental
group is finitely generated we talk about transalgebraic curves that is a generalization of classical
algebraic curves allowing infinite ramification points.
Our goal is to develop an algebraic theory of the function spaces on this transalgebraic curves
as is classically done with algebraic curves. Algebraic functions, and the field of meromorphic
functions, form the backbone of the classical theory of R. Dedekind and H. Weber (that will be
referred as Dedekind–Weber theory), originally developed in [7], that represents the historical
precursor of the modern commutative algebra and algebraic geometry approach. For transalge-
braic curves, the base function spaces are formed by transcendental functions as we will see in
this article.
We study this problem in the simplest situation of genus 0, i.e., we assume that S× is simply
connected. Then S× is parabolic and biholomorphic to C (see [4, 5]). Also we proved there (see
also the early work by R. Nevanlinna [13, 14] and M. Taniguchi [18, 19]) that we have an explicit
formula for the uniformization F̃ : C→ S× that is given by an entire function F = π ◦ F̃ of the
form
F (z) =
∫
Q(z)eP (z) dz, (1.1)
where P and Q are polynomials of respective degrees d1 and d2, where d1, resp. d2, is the number
of infinite order, resp. finite order, ramification points. Conversely, given P,Q ∈ C[z] polynomials
of degrees d1, d2 and F an entire function of the form (1.1) there exists a log-Riemann surface S
with d1 infinite order ramification points and d2 finite order ramification points (counted with
multiplicity) such that F lifts to a biholomorphism F̃ : C→ S×. This can be proved by seeing F
appear as a limit of Schwarz–Christoffel uniformizations (see [4] and [5, Section II.5.4]).
We limit our study to the simpler situation with no finite order ramification points, so d2 = 0
and Q = 1, and we denote d = d1 ≥ 1. For k ≥ 0, we consider the functions
Fk(z) =
∫ z
0
tkeP0(t) dt
and in particular F0 whose lift is the uniformization of the log-Riemann surface under consider-
ation. The C-vector space VP0 of transcendental functions
F (z) =
∫ z
z0
Q(t)eP0(t) dt,
where Q ∈ C[z] and z0 ∈ C plays the same role for the associated S = SP0 than the vector
space of polynomials C[z] for the complex plane C. It was proved in [5, Section III.3] that the
functions in this vector space can be characterized by their growth at infinite (“infinite” in S
being understood as its Alexandrov one-point compactification) by a Liouville type theorem
that we recall in Section 5.5. Without any reference to log-Riemann surface theory, we further
study in Section 2 by elementary methods a transcendental base for the ring generated by these
functions. We follow the classical path traced by N.H. Abel and other mathematicians of the
XIXth century to search for a minimal base of transcendentals in order to compute all these
integrals, as was done in Abel’s study of Abelian integrals (for the historical development of
this ideas one can consult Chapter IX of [12]). We show that the d transcendentals F0, . . . , Fd−1
are algebraically independent and are sufficient to compute the remaining integrals. These
functions define Picard–Vessiot extensions of Liouville type of C(z). We also study the Liouville
classification of these transcendentals from the old pre-differential algebra Liouville classification
(see [10, 11]).
The Ramificant Determinant 3
After these preliminaries in Section 2, we turn to study the asymptotic values of F0, . . . , Fd−1,
that are transcendental exponential periods (as defined by D. Zagier and M. Kontsevich in [9])
Ωkl(P0) =
∫ +∞.ωl
0
tk−1eP0(t) dt
for k = 1, . . . , d, where we normalize P0(t) = −1
d t
d+· · · and (ωl)1≤l≤d are the d-th roots of unity.
These periods are in general non-computable integrals. We define the Ramificant determinant by
∆(P0) =
∣∣∣∣∣∣∣∣∣
Ω11 Ω12 . . . Ω1d
Ω21 Ω22 . . . Ω2d
...
...
. . .
...
Ωd1 Ωd2 . . . Ωdd
∣∣∣∣∣∣∣∣∣ .
Even if the Ωkl’s are non-computable, one of the fundamental results established in Section 3
is that the Ramificant determinant is computable, and we give a closed-form formula:
Theorem 1.1. For d ≥ 1, there exists Πd, a universal polynomial with rational coefficients on
the coefficients of P0, such that
∆(P0) =
(2πd)
d
2
√
2π
exp(Πd).
In particular we get the trivial, but fundamental, corollary that the Ramificant determinant
never vanishes, ∆(P0) 6= 0. From this non-vanishing result, we obtain in Section 4 an Abel-like
theorem, that can be seen as a criterion for integrability in finite terms à la Abel and Liouville.
Also it follows a Torelli-like theorem that proves that the periods determine the polynomial P0.
These results were extended by the first author to finite type log-Riemann surfaces (see [1, 2]).
Another corollary is that the period mapping is étale, and a transalgebraic version of fundamental
symmetric formulas. In Section 5 we develop applications to the transalgebraic theory of log-
Riemann surfaces. To VP0 it corresponds the vector space of functions on S, VS , that generates
the structural ring ÂS . We prove that this ring of functions separates points on the log-Riemann
surface S, including the infinite ramification points in the completion. The transcendental
functions in the structural ring have Stolz limits at the infinite ramification points, thus the
algebraic theory extends to these points also (a Stolz limit corresponds to a limit through an
angular sector and this is an important notion in the theory of conformal representation, see
[15, p. 6]) The points of S∗ are identified with some maximal ideals of the structural ring. We
also explain how to distinguish algebraically the finite ramification points from the infinite ones.
Functions in the vector space VS can be characterized by their growth at infinite, i.e., by an
extension of the classical Liouville theorem to this setting.
Most of the results presented in this article are collected from the algebraic part (Section III)
of the original manuscript [5] that dates back to 2003–2005.
2 A ring of special functions
2.1 Definitions
Let P0(z) ∈ C[z] be a polynomial of degree d ≥ 1
P0(z) = adz
d + ad−1z
d−1 + · · ·+ a1z + a0.
We consider the entire functions
F0(z) =
∫ z
0
eP0(t) dt, F1(z) =
∫ z
0
t eP0(t) dt, . . . , Fd−1(z) =
∫ z
0
td−1eP0(t) dt.
4 K. Biswas and R. Pérez-Marco
and the C-vector space generated by these transcendental functions and constant functions
UP0 = 〈C, F0, . . . , Fd−1〉.
Proposition 2.1. We have
eP0 ∈ UP0 .
Proof. Since
eP0(z) − eP0(0) =
∫ z
0
P ′0(t)eP0(t) dt
we get
eP0 = eP0(0) · 1 + a1F0 + 2a2F1 + · · ·+ (d− 1)ad−1Fd−2 + dadFd−1. �
We prove in the next sections that 1, F0, . . . , Fd−1 are C-linearly independent and also alge-
braically independent.
Definition 2.2. We consider the ring generated by polynomials C[z] adjoining F0, . . . , Fd−1,
AP0 = C[z][F0, . . . , Fd−1].
Let KP0 be the field of fractions of AP0 , thus KP0 is the extension of the field of rational
functions C(z) adjoining F0, . . . , Fd−1,
KP0 = C(z)(F0, . . . , Fd−1) = C(z, F0, . . . , Fd−1).
Our first goal is to prove:
Theorem 2.3. The field KP0 has transcendence degree d+ 1 over C.
2.2 Asymptotics at infinite
The following asymptotic estimate is key in the proofs of the algebraic results.
Proposition 2.4. For j = 0, 1, . . . , d− 1 we have
Fj(z) ∼
zj
P ′0(z)
eP0(z),
when z → +∞.a−1/d
d , that is when z →∞ in a direction given by a d-root of a−1
d .
Proof. In these directions P0 and P ′0 tends to +∞, thus we can assume that P ′0 is non zero at 0
by changing the origin of integration (i.e., by a translation change of variables in the integrals).
Performing two integration by parts we get
Fj(z) =
∫ z
0
tjeP0(t) dt =
∫ z
0
tj
P ′0(t)
P ′0(t)eP0(t) dt
=
[
tj
P ′0(t)
eP0(t)
]z
0
−
∫ z
0
(
jtj−1P ′0(t)− tjP ′′0 (t)
(P ′0(t))2
)
eP0(t) dt
=
zj
P ′0(z)
eP0(z) −
∫ z
0
O
(
1(
a
1/d
d t
)d−j
)
eP0(t) dt
=
zj
P ′0(z)
eP0(z) −
[
O
(
1(
a
1/d
d t
)d−j
)
1
P ′0(t)
eP0(t)
]z
0
+
∫ z
0
O
(
1(
a
1/d
d t
)2d−j−2
)
eP0(t) dt.
Now the two last terms in the last equation are dominated by the first one. �
The Ramificant Determinant 5
2.3 Linear independence
Proposition 2.5. The constant function 1 and the special functions F0, F1, . . . , Fd−1 are linearly
independent over C.
We give different proofs of this Proposition.
1st proof. Consider a non-trivial linear combination
b−1 + b0F0 + b1F1 + · · ·+ bd−1Fd−1 = 0,
and take one derivative. Dividing by eP0 we get
b0 + b1z + · · ·+ bd−1z
d−1 = 0.
Thus we get b0 = b1 = · · · = 0 and then b−1 = 0 also. �
Now we give an analytic proof.
2nd proof. Consider a non-trivial linear combination
b−1 + b0F0 + b1F1 + · · ·+ bd−1Fd−1 = 0
and let 0 ≤ k ≤ d − 1 be the largest index such that bk 6= 0. If k = −1 we are done. If not,
when z → +∞.a−1/d
d we have
b−1 + b0F0 + b1F1 + · · ·+ bd−1Fd−1 ∼ bk
zk
P ′0(z)
eP0(z) →∞.
We have a contradiction. �
Finally we give a more algebraic proof.
3rd proof. First we show that F0, . . . , Fd−1 are C-linearly independent. Choose d distinct
points z0, z1, . . . , zd−1 ∈ C. If a linear combination b0F0 + b1F1 + · · · + bd−1Fd−1 vanishes at
z0, z1, . . . , zd−1 ∈ C then we have
∆(z0, . . . , zd−1) =
∣∣∣∣∣∣∣∣∣
F0(z0) F0(z1) . . . F0(zd−1)
F1(z0) F1(z1) . . . F1(zd−1)
...
...
. . .
...
Fd−1(z0) Fd−1(z1) . . . Fd−1(zd−1)
∣∣∣∣∣∣∣∣∣ = 0.
But we have
∂zd−1
· · · ∂z1∂z0∆ = eP0(z0).eP0(z1) · · · eP0(zd−1).
∣∣∣∣∣∣∣∣∣
1 1 . . . 1
z0 z1 . . . zd−1
...
...
. . .
...
zd−1
0 zd−1
1 . . . zd−1
d−1
∣∣∣∣∣∣∣∣∣
and the Vandermonde determinant is not zero, thus ∂zd−1
· · · ∂z1∂z0∆ 6= 0 and ∆ is not identical-
ly 0. Contradiction.
In order to show that 1, F0, . . . , Fd−1 are C-linearly independent we proceed in a similar way
evaluating the linear combination at d distinct points z0, z1, . . . , zd−1. We consider the same
determinant ∆ adding a first column and a first row of ones. Next we apply the differential
operator ∂z1,z2,...,zd−1
to ∆ and develop the resulting determinant through the first row and we
get a contradiction as before. �
6 K. Biswas and R. Pérez-Marco
We can now prove more.
Proposition 2.6. The special functions F0, F1, . . . , Fd−1 and the constant function 1 are linearly
independent over the ring of polynomials C[z].
Proof. By contradiction consider a non-trivial linear combination with polynomial coefficients
A−1(z) +A0(z)F0(z) + · · ·+Ad−1(z)Fd−1(z) = 0.
Taking one derivative we get
A′−1(z) +A′0(z)F0(z) + · · ·+A′d−1(z)Fd−1(z) = Q1(z)eP0(z),
where Q1(z) = −A0(z) − zA1(z) − · · · − zd−1Ad−1(z). Iterating this procedure and taking k
derivatives, we get
A
(k)
−1(z) +A
(k)
0 (z)F0(z) + · · ·+A
(k)
d−1(z)Fd−1(z) = Qk(z)e
P0(z),
where Qk(z) ∈ C[z]. Choose k ≥ 0 minimal such that all A
(k)
j are constant but not all 0. Let
−1 ≤ l0 ≤ d− 1 be the largest index such that A
(k)
l 6= 0. If l0 = −1, we have
A
(k)
−1 +A
(k)
0 F0(z) + · · ·+A
(k)
d−1Fd−1(z) = A
(k)
−1 = Qk(z)e
P0(z),
so Qk = 0 and A
(k)
−1 = 0.
If l0 ≥ 0, then when z → +∞.a−1/d
d , using Proposition 2.4, we have the asymptotics
A
(k)
−1 +A
(k)
0 F0(z) + · · ·+A
(k)
d−1Fd−1(z) ∼ A(k)
l0
zl0
P ′0(z)
eP0(z).
But since l0 ≤ d− 1,
Qk(z) ∼ A
(k)
l0
zl0
P ′0(z)
is only possible when l0 = d − 1. Thus l0 = d − 1, and the degree of Aj is at most the degree
of Ad−1. When z → +∞.a−1/d
d we have that Ad−1Fd−1 dominates AjFj for j < d− 1. Thus if c
is the leading coefficient of Ad−1(z) and m is its degree then, when z → +∞.a−1/d
d , we have
A−1(z) +A0(z)F0(z) + · · ·+Ad−1(z)Fd−1(z) ∼ cz
m+d−1
P ′0(z)
eP0(z).
On the other hand A−1 + A0F0 + · · · + Ad−1Fd−1 = 0, so c must be 0, Ad−1 is zero, as well as
all the other Aj . We have a contradiction. �
2.4 Algebraic independence
We prove now Theorem 2.3, i.e., that the field KP0 has transcendence degree d over C(z). Clearly
the transcendence degree is at most d. That it is exactly d follows from the next result:
Lemma 2.7. For k = 1, . . . , d − 1, Fk is transcendental over C(z, F0, . . . , Fk−1), and F0 is
transcendental over C(z).
Before proving the lemma, we give a definition.
The Ramificant Determinant 7
Definition 2.8. The exponential degree, resp. the polynomial degree, of a monomial expression
zmFn0
0 Fn1
1 · · ·F
nd−1
d−1
are |n| = n0 + n1 + · · ·+ nd−1, resp. m+ n1 + 2n2 + · · ·+ (d− 1)nd−1 = m+ (d− 1).n, where
(d− 1) denotes the vector (0, 1, 2, . . . , d− 1), and n the vector (n0, n1, . . . , nd−1).
Lemma 2.9. In a vanishing C-linear combination of monomials in z, F0, . . . , Fd−1 each sub-
linear combination of monomials with the same exponential and polynomial degree must vanish.
Proof. We have the asymptotics when z → +∞.a−1/d
d ,
zmFn0
0 Fn1
1 · · ·F
nd−1
d−1 ∼
zm+n1+2n2+···+(d−1)nd−1
(P ′0(z))n0+n1+···+nd−1
e(n0+n1+···+nd−1)P0(z)
∼ zm+(d−1).n−|n|(d−1)e|n|.P0(z).
Now consider a vanishing C-linear combination of monomials
0 =
∑
m,n
am,nz
mFn0
0 Fn1
1 · · ·F
nd−1
d−1 =
∑
N≥0
∑
m,n
|n|=N
am,nz
mFn0
0 Fn1
1 · · ·F
nd−1
d−1 .
The different exponential asymptotics show that for each N ≥ 0,
0 =
∑
m,n
|n|=N
am,nz
mFn0
0 Fn1
1 · · ·F
nd−1
d−1 =
∑
m≥0
∑
n
|n|=N
am,nz
mFn0
0 Fn1
1 · · ·F
nd−1
d−1 .
Again the same argument using the different asymptotics for monomials with the same expo-
nential degree but different polynomial degree gives the result, that is, for each N ≥ 0 and
m ≥ 0,∑
n
|n|=N
am,nz
mFn0
0 Fn1
1 · · ·F
nd−1
d−1 = 0. �
Lemma 2.10. Let N ≥ 1. The monomials Fn0
0 Fn1
1 · · ·F
nk
k of exponential degree N are C[z]-
linearly independent.
Proof. We prove the result by induction on N ≥ 1. For N = 1 we have the result by Proposi-
tion 2.6. Assume the result for N − 1 and consider, by contradiction, a non-trivial C[z] linear
dependence relation∑
n
An(z)Fn0
0 Fn1
1 · · ·F
nk
k = 0.
We can assume using the previous lemma that each term in this sum has the same polynomial
degree (we could also assume for the same reasons that each polynomial An(z) is a monomial,
but we don’t need that). This means that there exists a constant K such that for each n
degAn + k.n = K,
where k = (0, 1, 2, . . . , k). Taking one more derivative to the precedent relation we get∑
n
A′n(z) Fn0
0 Fn1
1 · · ·F
nk
k = −
∑
n
j=0,1,...,k
zjAn(z)Fn0
0 · · ·F
nj−1
j · · ·Fnkk eP0 .
8 K. Biswas and R. Pérez-Marco
Note that the exponential degree of the terms on the right hand side is the same as the one on
the left side, but the polynomial degrees are greater by 1, therefore∑
n
A′n(z)Fn0
0 Fn1
1 · · ·F
nk
k = 0.
We continue taking derivatives and stop one step before all A
(l+1)
n vanish, that is when∑
n
A
(l)
n F
n0
0 Fn1
1 · · ·F
nk
k = 0,
is a non-trivial C-linear combination of homogeneous monomials on the Fj ’s. Observe now that
taking one more derivative in this last relation and dividing by eP0 gives∑
n
j=0,1,...,k
A
(l)
n z
jFn0
0 · · ·F
nj−1
j−1 F
nj−1
j F
nj+1
j+1 · · ·F
nk
k = 0.
Observe that each monomial in z, F0, . . . , Fk in this sum comes from exactly one monomial in
F0, . . . , Fk of the relation we have differentiated. And this last relation is a non-trivial C[z]-linear
combination between monomials of exponential degree N − 1. By induction assumption this is
impossible. �
Proof of Theorem 2.3. It is enough to prove Lemma 2.7. If Fk is not transcendental over
C(z, F0, . . . , Fk−1), then we have a non-trivial polynomial relation between z, F0, . . . , Fk. Iso-
lating parts of the same exponential degree we are lead to a non-trivial C[z]-linear relation
between homogeneous monomials in F0, . . . , Fk which contradicts the previous Lemma 2.10. �
2.5 Computation of integrals
We adopt here a similar point of view to Abel and his contemporaries on elliptic functions and,
in general, Abelian integrals. The special functions F0, F1, . . . , Fd−1 are all we need in order to
compute a large class of integrals, or “transcendentals” as Abel would put it. As for Abelian
integrals, next theorem shows that computable integrals have finite codimension in the family
of integrals considered.
Theorem 2.11. We consider the C-vector space
VP0 = VP0(C) = C[z].eP0(z) ⊕ C.F0 ⊕ · · · ⊕ C.Fd−2
= zC[z].eP0(z) ⊕ C.1⊕ C.F0 ⊕ · · · ⊕ C.Fd−1.
For Q(z) ∈ C[z], any primitive∫ z
0
Q(t)eP0(t) dt
is in the vector space VP0. Conversely, any point of the hyperplane of VP0 of functions vanishing
at 0 is such a primitive{∫ z
0
Q(t)eP0(t) dt; Q(z) ∈ C[z]
}
= {F ∈ VP0 ; F (0) = 0}.
We have
VP0 =
{∫ z
z0
Q(t)eP0(t) dt; z0 ∈ C, Q(z) ∈ C[z]
}
.
The Ramificant Determinant 9
Proof. First note that the equality of the two sums results from the fact that eP0 is a C-linear
combination of F0, . . . , Fd−1, and the direct sums result from the algebraic independence proved
in the previous section. We prove the result by induction on the degree of Q. The result is
clear for degQ ≤ d − 2 because then
∫
QeP0 dt is a linear combination of 1, F0, . . . , Fd−2. For
degQ ≥ d− 1, we take the Euclidean division of Q by P ′0,
Q = AP ′0 +B,
where A,B ∈ C[z] and degB < d− 1. Then, by integration by parts it follows∫ z
0
Q(t)eP0(t) dt =
∫ z
0
(A(t)P ′0(t) +B(t))eP0(t) dt
=
[
A(t)eP0(t)
]z
0
−
∫ z
0
A′(t)eP0(t) dt+
∫ z
0
B(t)eP0(t) dt
= A(z)eP0(z) −A(0)eP0(0) −
∫ z
0
A′(t)eP0(t) dt+
∫ z
0
B(t)eP0(t) dt.
Now we have A(z)eP0(z) ∈ C[z]eP0(z), −A(0)eP0(0) ∈ C, and the primitive
∫ z
0 B(t)eP0(t) dt is
a linear combination of F0, . . . , Fd−2. Moreover, we have degA′ < degQ so the result follows by
induction.
For the converse, let F ∈ VP0 vanishing at 0 and write
F (z) = zP (z)eP0(z) + c0 + c1F0 + · · ·+ cdFd−1,
where P (z) ∈ C[z] and c0, c1, . . . , cd ∈ C. Since F (0) = 0 we have c0 = 0. Also
c1F0 + . . . cdFd−1 =
∫ z
0
(
c1 + c2t+ · · ·+ cdt
d−1
)
eP0(t) dt,
and
zP (z)eP0(z) =
∫ z
0
(
P (t) + tP ′(t) + tP (t)P ′0(t)
)
eP0(t) dt. �
Remark 2.12. 1. Let K ⊂ C be a subfield of the complex numbers. If P0(z) ∈ K[z] and P0 is
normalized such that P0(0) = 0, then any primitive∫ z
0
Q(t)eP0(t) dt,
where Q(z) ∈ K[z] belongs to the K-vector space
VP0(K) = zK[z]eP0(z) ⊕K⊕KF0 ⊕ · · · ⊕KFd−1.
This results from the previous proof since the Euclidean division of polynomials is well defined
in the ring K[z], and eP0(0) = 1. The proof of the converse statement is analogous.
2. In general, let K be a field and consider the differential ring K[z]. For P0 ∈ K[z], degP0 = d,
we define eP0 as generating the Liouville extension defined by the differential equation
y′ − P0y = 0.
We consider the extension K0 generated by
y′ = eP0 , y′ = zeP0 , . . . , y′ = zd−1eP0 ,
and denote by F0, F1, . . . , Fd−1 these primitives. Then the K-vector space
MP0 = zK[z]eP0 ⊕K.1⊕K.F0 ⊕ · · · ⊕K.Fd−1
coincides with the set of all primitives
∫
QeP0 dt modulo constants.
10 K. Biswas and R. Pérez-Marco
2.6 Differential ring structure
We denote by D = d
dz the differentiation operator in the ring AP0 . Let AN,nP0
be the C-module
generated by those monomials of exponential degree N and polynomial degree n. We have the
graduation
AP0 =
⊕
N,n≥0
AN,nP0
.
The following proposition is immediate.
Proposition 2.13. We have
DAN,nP0
⊂ AN,n−1
P0
⊕
(
AN−1,n
P0
⊕ AN−1,n+1
P0
⊕ · · · ⊕ AN−1,n+d−1
P0
)
eP0 .
In particular, the principal ideal
(
eP0
)
generated by eP0 is absorbing for the derivation, i.e., any
element of AP0 ends up into
(
eP0
)
after a finite number of derivatives.
Next we determine the elements of AP0 without zeros.
Proposition 2.14. The only elements in AP0 without zeros are
C∗ ∪
{
enP0 ; n ≥ 1
}
,
that is, the non-zero constant functions and eP0 , e2P0 , . . . .
The group of units in AP0 is composed by the non-vanishing constant functions
A×P0
= C∗.
Proof. Let F ∈ AP0 without zeros. Since AP0 is a ring of entire functions of order at most d,
and F is zero free, we can find a polynomial of degree ≤ d such that
F = eQ.
Now, when z → +∞.a−1/d
d , using Proposition 2.4, the asymptotics of each F ∈ AP0 is of the
form
F (z) ∼ czaebP0(z),
where c ∈ C, and a, b ∈ N, b ≥ 0. Therefore we must have Q = nP0 for some n ≥ 1 or Q is
a constant polynomial (case b = 0). This proves the first statement.
For the second statement, let F ∈ A×P0
be invertible. Then 1/F belongs to the ring, so it is
holomorphic. Thus F has no zeros. Moreover F cannot be of the form enP0 for n ≥ 0 since
e−nP0(z) → 0,
when z → +∞.a−1/d
d and we know that for any non-constant element G in the ring AP0
G(z)→ +∞,
when z → +∞.a−1/d
d . �
The Ramificant Determinant 11
2.7 Picard–Vessiot extensions
We recall that a Picard–Vessiot extension of a differential ring A is a differential ring extension
A[u1, . . . , un] generated by u1, . . . , un fundamental solutions of an homogeneous linear differential
equation of order n
y(n) + bn−1y
(n−1) + · · ·+ b1y
′ + b0y = 0,
where bj ∈ A and the ring of constants of the extension coincides with the ring of constants
of A.
We recall also that a Liouville extension is a Picard–Vessiot extension generated by successive
adjunctions of integrals or exponentials of integrals (see [8, Chapter III.12, p. 23] and [17]). These
have a solvable differential Galois group [8, Chapter III.13, p. 24].
Theorem 2.15. The field KP0 = C(z, F0, . . . , Fd−1) and the ring AP0 = C[z, F0, . . . , Fd−1] are
Picard–Vessiot extensions of C(z) and C[z] respectively, i.e., they are generated by the fun-
damental solutions of a linear homogeneous differential equation with polynomial coefficients.
Moreover these extensions are Liouville extensions.
The ring of constants are the constant functions. We only need to find the homogeneous linear
differential equation satisfied by F0, . . . , Fd−1. We construct a homogeneous linear differential
equation satisfied by F ′0, . . . , F
′
d−1.
We define a double sequence of functions (yn,m) n∈Z
m≥0
by
• y0,0 = eP0 ,
• for n > m, yn,m = 0,
• for n < 0, yn,m = 0,
• for n ∈ N, m ≥ 0,
yn,m+1 = yn−1,m + y′n,m
(Pascal’s triangle rule with one derivative).
The first lemma is straightforward.
Lemma 2.16. We have
• for n ≥ 0, yn,n = eP0,
• for m ≥ 0, y0,m =
(
eP0
)(m)
,
• for all n ∈ N, m ≥ 0, yn,m = Qn,meP0, where Qn,m is a universal polynomial with positive
integer coefficients on P ′0, P
′′
0 , P
(3)
0 , . . . .
And we need a second lemma:
Lemma 2.17. We define for k ≥ 0, yk(z) = zkeP0(z) = zkyk,k. Then we have
• for 0 ≤ l ≤ k,
y
(l)
k = zky0,l + kzk−1y1,l + k(k − 1)zk−2y2,l + · · ·+ k!
(k − l)!
zk−lyl,l,
• for k ≤ l,
y
(l)
k = zky0,l + kzk−1y1,l + k(k − 1)zk−2y2,l + · · ·+ k!
1
zyk−1,l + k!yk,l.
12 K. Biswas and R. Pérez-Marco
Proof. It results from a direct induction on l observing that y′0,l = y0,l+1 and y0,l+y′1,l = y1,l+1,
and so on. �
Proof of the theorem. We look for polynomials b0, b1, . . . , bd−1 such that y0 = F ′0, y1 = F ′1,
. . . , yd−1 = F ′d−1 are solutions of
y(d) + bd−1y
(d−1) + · · ·+ b1y
′ + b0y = 0.
They will form a fundamental set of solutions since these functions are C-linearly independent.
Once we find these polynomial coefficients, the special functions 1, F0, F1, . . . , Fd−1 will form
a fundamental set of solutions of
y(d+1) + bd−1y
(d) + · · ·+ b1y
′′ + b0y
′ = 0.
We can plug yk into the differential equation and compute y
(l)
k using Lemma 2.17. Then grouping
together the factors of zj , j = 0, . . . , d− 1, we get a triangular system
bjyj,j + bj+1yj,j+1 + · · ·+ bd−1yj,d−1 + yj,d = 0.
Then, since yj,j = eP0 , we get
bj = −bj+1yj,j+1e−P0 − · · · − bd−1yj,d−1e−P0 − yj,de−P0 ,
and the result follows using Lemma 2.16. Note that the extension is a Liouville extension as
claimed since each F0 is the exponential of an integral followed by an integral, and for j ≥ 1 the
special function Fj is an integral over the field generated by eP0 . �
Remark 2.18. The Wronskian of F0, F1, . . . , Fd−1 satisfies the differential equation
W ′ − dP ′0W = 0,
and is equal to W (z) = edP0(z).
Example 2.19. 1. For d = 1, the equation is
y′ − P ′0y = 0.
2. For d = 2, the equation is
y′′ − 2P ′0 y
′ +
[
(P ′0)2 − P ′′0
]
y = 0.
In particular, for P0(z) = z2,
y′′ − 4zy′ +
(
4z2 − 2
)
y = 0.
2.8 Liouville classification
Between 1830 and 1840 J. Liouville developed a classification of transcendental functions ge-
nerated by algebraic expressions, logarithms and exponentials, and proved the non-elementary
character of some natural integrals and solutions of some differential equations. Later he noticed
that his classification can be extended by allowing integrations instead of using the logarithm
function, which constitutes a particular case since any expression log f is the primitive of f ′/f .
We recall Liouville’s classification. Functions of order 0 are algebraic functions of the variab-
le z, that is those functions satisfying a polynomial equation with polynomial coefficients on z.
Assume by induction that order n functions have been defined. Functions of order n + 1 are
The Ramificant Determinant 13
those functions that are not of order n and that can be obtained by taking an exponential or
a primitive of order n functions or that satisfy an algebraic equation with such coefficients.
We refer to J.F. Ritt’s book on elementary integration [16] for more information on this
subject, the precursor of modern differential algebra.
Note that Liouville classification only concerns functions that are multivalued in the complex
plane, i.e., except for isolated singularities and ramifications they can be continued holomorphi-
cally through all the complex plane when avoiding these isolated singularities (these are called
“fluent” functions in Ritt’s terminology [16]).
From this classification we have:
Proposition 2.20. Entire functions in the ring AP0 are functions of order at most 2. Moreover,
if d ≥ 2, we have that F0 is of order 2.
For the proof of the non-elementarity of the integral giving F0 see [16, p. 48].
3 The Ramificant determinant
3.1 Definition of the Ramificant determinant
From now on we normalize P0 to have leading coefficient −1/d. We denote ω1, . . . , ωd the d
roots of 1, for k = 1, . . . , d,
ωk = e
2π
d
i(k−1).
From the normalization of P0, the functions Fk have d asymptotic values in the directions given
by the (ωl). We denote these values by
Ωkl = Ωkl(P0) = Fk(+∞.ωl) =
∫ +∞.ωl
0
tk−1eP0(t) dt.
These asymptotic values are transcendental periods (see [9] for the terminology), and also loca-
tions of infinite ramification points in the associated log-Riemann surfaces. They have a deep
transalgebraic meaning.
Definition 3.1. The Ramificant determinant associated to P0 is
∆(P0) =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
∫ +∞.ω1
0
eP0(z) dz
∫ +∞.ω1
0
zeP0(z) dz . . .
∫ +∞.ω1
0
zd−1eP0(z) dz∫ +∞.ω2
0
eP0(z) dz
∫ +∞.ω2
0 zeP0(z) dz . . .
∫ +∞.ω2
0
zd−1eP0(z) dz
...
...
. . .
...∫ +∞.ωd
0
eP0(z) dz
∫ +∞.ωd
0
zeP0(z) dz . . .
∫ +∞.ωd
0
zd−1eP0(z) dz
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
.
If we write
P0(t) = −1
d
td + ad−1t
d−1 + · · ·+ a1t+ a0
with (a0, a1, . . . , ad−1) ∈ Cd then the Ramificant determinant is an entire function of d complex
variables and we write
∆(P0) = ∆(a0, a1, . . . , ad−1)
and
Ωkl(a0, a1, . . . , ad−1) = Ωkl(P0).
14 K. Biswas and R. Pérez-Marco
3.2 Formula for the Ramificant determinant
Even if we cannot compute in general the asymptotic values, it turns out that we can compute
the Ramificant determinant. We have the following important result:
Theorem 3.2. For each d ≥ 0, there exists a universal polynomial of d variables with rational
coefficients
Πd(X0, X1, . . . , Xd−1) ∈ Q[X0, . . . , Xd−1]
with Πd(0, . . . , 0) = 0 and such that the Ramificant determinant is given by
∆(a0, a1, . . . , ad−1) =
(2πd)
d
2
√
2π
exp (Πd(a0, a1, . . . , ad−1)) .
A fundamental corollary of this theorem is that the Ramificant determinant is never 0.
Corollary 3.3. The Ramificant determinant does not vanish
∆(a0, a1, . . . , ad−1) 6= 0.
The miracle of the theorem is that among the parameter space Cd there is exactly one point,
namely (0, . . . , 0), where we can explicitly, compute the Ramificant determinant. Then from
∆(0, . . . , 0) we derive the general formula for ∆(a0, a1, . . . , ad). We first compute the period for
P0(t) = −1
d t
d.
Lemma 3.4. Let ω be a d-root of 1, ωd = 1. We have∫ +∞.ω
0
tke−t
d/d dt = ωk+1d
k+1
d
−1Γ
(
k + 1
d
)
,
i.e.,
Ωkl(0, . . . , 0) = ωk+1
l d
k+1
d
−1Γ
(
k + 1
d
)
.
Proof. By a linear change of variables we have∫ +∞.ω
0
tke−t
d/d dt = ωk+1
∫ +∞
0
tke−t
d/d dt.
Now, the change of variables u = sd/d gives
ωk+1
∫ +∞
0
tke−t
d/d dt = ωk+1d
k+1
d
−1
∫ +∞
0
u
k+1
d
−1e−u du = ωk+1d
k+1
d
−1Γ
(
k + 1
d
)
. �
Now we compute ∆(0, . . . , 0).
Lemma 3.5. We have
∆(0, . . . , 0) =
(2πd)
d
2
√
2π
.
The Ramificant Determinant 15
Proof. Using the previous lemma we have
∆(0, . . . , 0) =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
d
1
d
−1Γ
(
1
d
)
ω1 d
2
d
−1Γ
(
2
d
)
ω2
1 . . . d
d
d
−1Γ
(
d
d
)
ωd1
d
1
d
−1Γ
(
1
d
)
ω2 d
2
d
−1Γ
(
2
d
)
ω2
2 . . . d
d
d
−1Γ
(
d
d
)
ωd2
...
...
. . .
...
d
1
d
−1Γ
(
1
d
)
ωd d
2
d
−1Γ
(
2
d
)
ω2
d . . . d
d
d
−1Γ
(
d
d
)
ωdd
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
= d
1
d
(1+2+···+d)−dΓ
(
1
d
)
Γ
(
2
d
)
. . .Γ
(
d
d
) ∣∣∣∣∣∣∣∣∣
ω1 ω2
1 . . . ωd1
ω2 ω2
2 . . . ωd2
...
...
. . .
...
ωd ω2
d . . . ωdd
∣∣∣∣∣∣∣∣∣
= d
1−d
2 (2π)
d−1
2 d
1
2
−d 1
dΓ(1)
∣∣∣∣∣∣∣∣∣
ω1 ω2
1 . . . ωd1
ω2 ω2
2 . . . ωd2
...
...
. . .
...
ωd ω2
d . . . ωdd
∣∣∣∣∣∣∣∣∣
=
1√
2π
(
2π
d
) d
2
∣∣∣∣∣∣∣∣∣
ω1 ω2
1 . . . ωd1
ω2 ω2
2 . . . ωd2
...
...
. . .
...
ωd ω2
d . . . ωdd
∣∣∣∣∣∣∣∣∣ ,
where we have used Gauss multiplication formula
Γ(z).Γ
(
z +
1
d
)
. . .Γ
(
z +
d− 1
d
)
= (2π)
d−1
2 d
1
2
−dzΓ(dz).
We have that ωdj = 1 and the last determinant is equal to (−1)d−1Vd where Vd is the Vander-
monde determinant
Vd =
∣∣∣∣∣∣∣∣∣
1 ω1 ω2
1 . . . ωd−1
1
1 ω2 ω2
2 . . . ωd−1
2
...
...
...
. . .
...
1 ωd ω2
d . . . ωd−1
d
∣∣∣∣∣∣∣∣∣ =
∏
i 6=j
(ωi − ωj).
Finally, the next lemma applied to the polynomial Q(X) = Xd − 1, shows that
Vd =
∏
i
(
dωd−1
i
)
= dd
(∏
i
ωi
)d−1
= (−1)d−1dd. �
Lemma 3.6. If ξ1, . . . , ξd are the d roots of a monic polynomial Q(X), then we can compute
the Vandermonde determinant V (ξ1, . . . , ξd) of the (ξ1, . . . , ξd) as
V (ξ1, . . . , ξd) =
∣∣∣∣∣∣∣∣∣
1 ξ1 ξ2
1 . . . ξd−1
1
1 ξ2 ξ2
2 . . . ξd−1
2
...
...
...
. . .
...
1 ξd ξ2
d . . . ξd−1
d
∣∣∣∣∣∣∣∣∣ =
∏
i 6=j
(ξi − ξj) =
d∏
i=1
Q′(ξi).
Proof. We have Q′(ξi) =
∏
j 6=i(ξi − ξj) and the result follows. �
16 K. Biswas and R. Pérez-Marco
Now we can prove Theorem 3.2.
Proof of Theorem 3.2. Consider the entire function of several complex variables ∆(a0, a1, . . . ,
ad−1). Observe that by Theorem 2.11 we have that each integral∫ +∞.ωi
0
zneP0(z) dz
is a linear combination with coefficients polynomial integer coefficients on the (aj) of the integrals
for j = 0, 1, . . . , d− 1,∫ +∞.ωi
0
zjeP0(z) dz.
Therefore, differentiating column by column, we observe that for each j = 0, 1, . . . , d − 1, we
have
∂aj∆ = cj∆,
where cj is a polynomial on the (aj) with integer coefficients. We conclude that the logarithmic
derivative of ∆ with respect to each variable is a universal polynomial with integer coefficients
on the variables (aj). This gives the existence of the universal polynomial Πd such that
∆(a0, a1, . . . , ad−1) = c.eΠd(a0,a1,...,ad−1),
with Πd(0, . . . , 0) = 0 and c = ∆(0, . . . , 0) ∈ C. The result follows from Lemma 3.5. �
3.3 The universal polynomials Πd
It is interesting to compute and study the combinatorial properties of the family of universal
polynomials (Πd). We can compute a few first polynomials.
Theorem 3.7. We have
Π1(X0) = X0, Π2(X0, X1) = 2X0 +
1
2
X2
1 ,
Π3(X0, X1, X2) = 3X0 + 2X1X2 +
4
3
X3
2 ,
and for d = 4
Π4(X0, X1, X2, X3) = 4X0 + 3X3X1 + 2X2
2 + 9X2
3X2 + · · · ,
where the remaining term is a polynomial in X3, and, in general, for d ≥ 5,
Πd(X0, X1, . . . , Xd−1) = dX0 + (d− 1)Xd−1X1 +
(
2(d− 2)Xd−2 + (d− 1)2X2
d−1
)
X2 + · · · ,
where the remaining terms are independent of X0, X1 and X2.
More generally, Πd is of degree 1 in Xk for k < d/2.
Proof. For d ≥ 1 the dependence of the Ramificant determinant ∆ on a0 is straightforward by
direct factorization of ea0 in the integrals, which gives
Πd(X0, . . . , Xd−1) = dX0 + · · ·
The Ramificant Determinant 17
with remaining terms are independent of X0. Also this can be seen by differentiation column
by column of ∆,
∂a0∆ = d∆,
which also gives the result. For the dependence on a1 we use this last approach. For d ≥ 2, we
have
∂a1∆ = (d− 1)ad−1∆.
This is because the differentiation of the first d − 1 columns yields 0. Also for the last column
we have
zd = −zP ′0(z) + (d− 1)ad−1z
d−1 + (d− 2)ad−2z
d−2 + · · ·+ a1z.
And the integrals corresponding to the term −zP ′0(z) contribute 0 because∫
−zP ′0(z)eP0(z) dz =
[
−zeP0
]
+
∫
eP0(z) dz.
And by linearity of the integrals in the last column the lower order terms (d−2)ad−2z
d−2 + · · ·+
a1z contribute 0. Thus the only contribution comes from the term (d− 1)ad−1z
d−1 which gives
(d− 1)ad−1∆. Now this last equation gives for d = 2,
∂a1∆ = a1∆,
and we have Π2(X0, X1) = 2X0 + 1
2X
2
1 .
For d ≥ 3 we get
Πd(X0, X1, . . . , Xd−1) = dX0 + (d− 1)Xd−1X1 + · · · ,
where the remaining terms are independent of X0 and X1. Now we assume d ≥ 3 and
we determine the dependence on a2. We proceed as before and differentiate column by co-
lumn ∂a2∆. Only the last two columns give a contribution. The last but one column contributes
by (d− 2)ad−2∆ because
zd = −zP ′0(z) + (d− 1)ad−1z
d−1 + (d− 2)ad−2z
d−2 + · · ·+ a1z,
and the last column contributes by
[
(d− 2)ad−2∆ + (d− 1)2a2
d−1
]
∆ because
zd+1 = −z2P ′0(z) + (d− 1)ad−1z
d + (d− 2)ad−2z
d−1 + · · ·+ a1z
2,
and modulo P ′0 we have
zd+1 =
[
(d− 2)ad−2∆ + (d− 1)2a2
d−1
]
zd−1 + · · ·+ [P ′0],
where the dots denote lower order terms. Thus we have
∂a2∆ =
(
2(d− 2)ad−2 + (d− 1)2a2
d−1
)
∆.
When d = 3 this gives
∂a2∆ =
(
2a1 + 4a2
2
)
∆,
therefore
Π3(X0, X1, X2) = 3X0 + 2X2X1 +
4
3
X3
2 .
18 K. Biswas and R. Pérez-Marco
When d = 4 we get
∂a2∆ =
(
4a2 + 9a2
3
)
∆.
So
Π4(X0, X1, X2, X3) = 4X0 + 3X3X1 + 2X2
2 + 9X2
3X2 + · · · ,
where the remaining term is a polynomial in X3. When d ≥ 5 we get
Πd(X0, X1, . . . , Xd−1) = dX0+ (d− 1)Xd−1X1 +
(
2(d− 2)Xd−2+ (d− 1)2X2
d−1
)
X2+ · · · ,
where the remaining terms are independent of X0, X1 and X2.
A close inspection of the procedure (for a complete analysis see what follows next) shows
that if k < d/2 then
∂ak∆ = c∆,
where c is a polynomial on ad−1, ad−2, . . . , ad−k thus the last result follows. �
The next results provide an algorithm to compute the universal polynomial Πd.
Theorem 3.8. Let d ≥ 2. For n ≥ 0 we define (An,k)0≤k≤d−1 to be the coefficients of the
remainder when dividing zn by zP ′0:
zn = An,d−1z
d−1 +An,d−2z
d−2 + · · ·+An,1z +An,0 [zP ′0].
For n ≤ d− 1 and k 6= n, we have An,k = 0, and An,n = 1.
For n = d,
Ad,k = kak.
And for n ≥ d+ 1, we can compute the sequence (An,k) by induction using
An+1,k = (d− 1)ad−1An,k + (d− 2)ad−2An−1,k + · · ·+ a1An−d+2,k.
Proof. For the induction relation, we use
zn+1 = −zn−d+2P ′0 + (d− 1)ad−1z
n + (d− 2)ad−2z
n−1 + · · ·+ a1z
n−d+2.
The rest is clear. �
Corollary 3.9. For d ≥ 2, 0 ≤ k ≤ d − 1, and n ≥ d, An,k is a polynomial with integer
coefficients on a0, a1, . . . , ad−1 of total degree n− d+ 1.
Proof. This is straightforward from the induction relations. �
Now we can compute the polynomial Πd using the polynomials (An,k).
Corollary 3.10. For d ≥ 2, the polynomial Πd is uniquely determined by the equations, for
0 ≤ k ≤ d− 1,
∂akΠd(a0, . . . , ad−1) = Ad−1+k,d−1 +Ad−2+k,d−2 + · · ·+Ad,d−k.
Proof. Differentiating, column by column, we get (this is clear from the above computations)
∂ak∆ =
(
Ad−1+k,d−1 +Ad−2+k,d−2 + · · ·+Ad,d−k
)
∆,
and the result follows. �
The Ramificant Determinant 19
4 Applications of the Ramificant determinant
4.1 Integrability and Abel-like theorem
The non-vanishing of the Ramificant determinant immediately gives the following result:
Theorem 4.1. In the C-vector space 〈F0, . . . , Fd−1〉C the only function with all asymptotic
values vanishing is the 0 function. In the C-vector space UP0 = 〈1, F0, . . . , Fd−1〉C the subspace
of functions with vanishing asymptotic values is the complex line generated by eP0.
A primitive
∫
QeP0 dt is integrable in finite terms in the sense of Abel and Liouville if we
can compute this primitive and it is an element of the ring C
[
z, eP0
]
. Therefore, in this context
of elementary integration, we say that an holomorphic 1-form ω is exact if there is a function
f ∈ C
[
z, eP0
]
such that df = ω.1 We give a simple criterion for integrability in finite terms.
Theorem 4.2 (integrability criterion). A necessary and sufficient condition for a primitive
F (z) =
∫ z
0
Q(t)eP0(t) dt
to be computable in finite terms is that the d asymptotic values for l = 1, . . . , d,
F (+∞.ωl) = Ωl(F ) =
∫ +∞.ωl
0
Q(t)eP0(t) dt
are all the same constant Ω(F ).
In that case, the differential Q(t)eP0(t) dt is exact,
Q(t)eP0(t) dt = d
(
A(t)eP0(t)
)
for some A ∈ C[t] such that AP ′0 +A′ = Q.
Proof. Note that we know from Theorem 2.11 that such a function F is of the form
F (z) = A0(z)eP0(z) + b−1 + b0F0(z) + · · ·+ bd−1Fd−1(z),
where A0 ∈ C[z], A0(0) = 0, and b−1, b0, . . . , bd−1 ∈ C. Making z = 0, we have b−1 = 0 and
F (z) = A0(z)eP0(z) + b0F0(z) + · · ·+ bd−1Fd−1(z). (4.1)
So we have for l = 1, . . . , d,
d−1∑
k=0
bkΩkl = Ω(F ). (4.2)
We can look at these equations as a linear system on (b0, . . . , bd−1). The non-vanishing of the
Ramificant determinant shows that there is exactly one solution. But if we choose (b0, b1, . . . ,
bd−2, bd−1) = (a1, 2a2, . . . , (d− 1)ad−1,−1), then we have for l = 1, . . . , d,
d−1∑
k=0
bkΩkl =
[
eP0(t)
]+∞.ωl
0
= −eP0(0).
1We thank the referee for pointing out this precision to avoid confusion with the usual notion of exact form in
differential geometry.
20 K. Biswas and R. Pérez-Marco
Therefore, the only solution to the system (4.2) is
(b0, b1, . . . , bd−2, bd−1) = (a1, 2a2, . . . , (d− 1)ad−1,−1).
(
−Ω(F )e−P0(0)
)
and plugging this value in equation (4.1), we get
F (z) = A0(z)eP0(z) +
[
eP0(t)
]z
0
(
−Ω(F )e−P0(0)
)
=
(
A0(z)− Ω(F )e−P0(0)
)
eP0(z) + Ω(F ),
thus F is computable in finite terms. The exactness of the differential follows by differentiation
of this equation with A(t) = A0(t)− Ω(F )e−P0(0). �
This result can be reformulated as an Abel’s theorem in this setting. We consider paths
(γl)1≤l≤d going to ∞ in C starting in the direction given by ωl and ending in the direction given
by ωl+1 (the index l is taken modulo d). Then we consider the transcendental periods∫
γl
Q(t)eP0(t) dt = F (+∞.ωl+1)− F (+∞.ωl).
The condition of Theorem 4.2 that all asymptotic values are equal is equivalent to have all
periods vanishing∫
γl
Q(t)eP0(t) dt = 0
and then the conclusion of Theorem 4.2 is that the differential form ω = Q(t)eP0(t) dt is exact.
Note that the integral over the path γl only depends on the homotopy class of γl relative to
the asymptotic directions. The converse is clear: If the holomorphic differential form ω is exact
then all periods are zero, for 1 ≤ l ≤ d we have∫
γl
ω dt = 0.
The C-vector space H1 of holomorphic differential forms of the type ω = Q(t)eP0(t) dt modulo
exact differentials (de Rham cohomology space-type) has a base (Abelian differentials)
ω0 = eP0(t) dt, ω1 = teP0(t) dt, . . . , ωd−1 = td−1eP0(t) dt.
We consider the C-vector space H1 of formal C-linear combination of paths (γl)1≤l≤d (C-
homology space). What we proved is the following Abel-like theorem:
Theorem 4.3 (Abel-like theorem). The pairing H1 ×H1 → C given by
(ω, γ) 7→
∫
γ
ω dt
is non-degenerate.
A generalization of this result to non-simply connected finite type log-Riemann surfaces is
proved in [2].
The Ramificant Determinant 21
4.2 The period mapping is étale
Definition 4.4. The period mapping Υ: Cd → Cd is
Υ(a0, a1, . . . , ad−1) = (F0(+∞.ω1), F0(+∞.ω2), . . . , F0(+∞.ωd)).
Theorem 4.5. The period mapping Υ is a local diffeomorphism everywhere.
Remark 4.6. The period mapping is not a global diffeomorphism as is easily seen construc-
ting two distinct log-Riemann surfaces with d ramification points with the same images by the
projection mapping π.
Proof. The computation of the determinant of the differential of the period mapping at a point
gives the value of the Ramificant determinant at this point,
detDa0,...,ad−1
Υ = ∆(a0, . . . , ad−1).
Then we use the local inversion theorem using the non-vanishing of the determinant. �
4.3 Separation of asymptotic directions
Using the functions F0, . . . , Fd−1 we can distinguish the different asymptotic directions.
Theorem 4.7. Let ωk and ωl be roots of 1 such that for all j = 0, 1, . . . , d− 1, we have
Fj(+∞.ωk) = Fj(+∞.ωl)
then
ωk = ωl.
Proof. Otherwise the Ramificant determinant will have two identical rows and will vanish. �
4.4 Transalgebraic symmetric formulas
The natural transalgebraic philosophy is to think of the transcendental periods (Fk(+∞.ωl))
as transalgebraic numbers when P0(z) ∈ Q[z]. Then it is natural to ask what is the relation
between these periods and the coefficients of P0 that define them, similar to the fundamental
symmetric formulas for the roots of an algebraic equation. We have the following:
Theorem 4.8. For j = 1, . . . , d − 1 (note that j = 0 is excluded), we have that e−a0aj is
a universal rational function on
(Fk(+∞.ωl)) k=0,...,d
l=1,...,d
.
More precisely, e−a0aj∆ (where ∆ is the Ramificant determinant) is a universal polynomial
function of degree d− 1 on (Fk(+∞.ωl)) k=0,...,d
l=1,...,d
.
Proof. Observe that for l = 1, . . . , d, we have
−Fd−1(+∞.ωl) + (d− 1)ad−1Fd−2(+∞.ωl) + · · ·+ a1F0(+∞.ωl)
=
∫ +∞.ωl
0
P ′0(z)eP0(z) dz =
[
eP0(z)
]+∞.ωl
0
= −ea0 .
22 K. Biswas and R. Pérez-Marco
Therefore if we consider the matrix
M =
F0(+∞.ω1) F1(+∞.ω1) . . . Fd−1(+∞.ω1)
F0(+∞.ω2) F1(+∞.ω2) . . . Fd−1(+∞.ω2)
...
...
. . .
...
F0(+∞.ωd) F1(+∞.ωd) . . . Fd−1(+∞.ωd)
we have
M.
a1
2a2
...
(d− 1)ad−1
−1
= −ea0
1
1
...
1
1
.
Thus
a1
2a2
...
(d− 1)ad−1
−1
= −ea0M−1
1
1
...
1
1
and by Cramer’s formulas the coefficients of M−1 are polynomials on the entries of M divided
by the Ramificant ∆ = detM . �
4.5 Torelli-like theorem
As we have observed, only the location of the ramification points, i.e., the values (F0(+∞.ωl))
are not enough to characterize the polynomial P0 (or the associated log-Riemann surface). This
changes if we consider all values (Fk(+∞.ωl)) as the next corollary shows. Thus we obtain that
the periods determine the log-Riemann surface, which is a Torelli-like theorem.
Corollary 4.9 (Torelli-like theorem). Let P0 and Q0 be two normalized polynomials,
P0(z) = −1
d
zd + ad−1z
d−1 + · · ·+ a1z + a0,
Q0(z) = −1
d
zd + bd−1z
d−1 + · · ·+ b1z + b0.
Consider the associated functions
Fk(z) =
∫ z
0
tkeP0(t) dt, Gk(z) =
∫ z
0
tkeQ0(t) dt.
If for k = 0, . . . , d− 1 and l = 1, . . . , d we have
Fk(+∞.ωl) = Gk(+∞.ωl),
then for k = 1, . . . , d− 1, we have ea0ak = eb0bk, i.e.,
eP0(0)(P0(z)− P0(0)) = eQ0(0)(Q0(z)−Q0(0)).
So the polynomials are determined up to their constant term. In particular, if the polynomials
have the same constant term, then
P0 = Q0.
The Ramificant Determinant 23
5 Introduction to transalgebraic Dedekind–Weber theory
5.1 Transalgebraic curves of genus 0
We refer to [3, 4, 5] for background on log-Riemann surfaces.
Definition 5.1. A transalgebraic curve S of genus 0 is a simply connected log-Riemann surface
with a finite set of ramification points.
Then the underlying Riemann surface is parabolic and biholomorphic to C (see [4] or [5] for
a proof). We prove in [4] the following basic uniformization theorem
Theorem 5.2. Let S be a transalgebraic curve of genus 0, and z0 ∈ S a base point with
π(z0) = 0. Let F̃ : C → S be the unique uniformization such that F̃ (0) = z0 and F ′(0) = 1.
Then we have that
F (z) = π ◦ F̃ (z) =
∫ z
0
Q(t)eP0(t) dt
for some polynomials Q,P0 ∈ C[t]. The number of finite (resp. infinite) ramification points is
degQ (resp. degP ).
From now on we consider a transalgebraic curve S of genus 0 without finite ramification
points, corresponding to polynomials Q = 1 and P = P0 so that its uniformization is the lift
of F0. The degree of P0 is d and S has exactly d distinct infinite ramification points that project
by π to finite values on C that are equal to the asymptotic values of F0.
5.2 The structural ring
We define a ring of functions that play the same role for S than polynomials for the complex
plane C.
Let P0 ∈ C[z] be the polynomial such that d = degP0 and the uniformization of S is the lift of
F0(z) =
∫ z
0
eP0(t) dt,
i.e., the uniformization F̃0 : (C, 0)→ (S, z0) is such that F0 = π ◦ F̃0. We define as in Section 2
the transcendental functions F1, . . . , Fd−1, and the ring AP0 and its field of fractions KP0 . We
consider the natural sub-ring of AP0 of holomorphic functions in S and having finite asymptotic
values, i.e., finite functions in S∗.
Definition 5.3. We consider the sub-ring ÂP0 ⊂ AP0
ÂP0 = zC[z, F0, . . . , Fd−1]eP0(z) ⊕ C[F0, . . . , Fd−1],
and its associated field of fractions K̂P0 ⊂ KP0 .
The sub-ring ÂP0 ⊂ AP0 is the subspace of AP0 of holomorphic functions with finite asymp-
totic values.
To justify this definition, observe that if
G1 = A1eP0 +B1, G2 = A2eP0 +B2
with A1, A2 ∈ zC[z, F0, . . . , Fd−1] and B1, B2 ∈ C[F0, . . . , Fd−1], then we have
F1.F2 = AeP0 +B
24 K. Biswas and R. Pérez-Marco
with A = A1A2eP0 + A1B2 + A2B1 ∈ zC[z, F0, . . . , Fd−1] (using Proposition 2.1), and B =
B1.B2 ∈ C[F0, . . . , Fd−1], so we have a well defined sub-ring. We are discarding from AP0 the
non-constant polynomials that have infinite asymptotic values.
All functions in ÂP0 have finite asymptotic values since all Fk do have finite asymptotic
values, and any polynomial in C[z] appears multiplied by eP0 . We consider now k0 : S → C, the
inverse of the uniformization F̃0, k0 = F̃−1
0 .
Definition 5.4 (structural ring). The structural ring ÂS of the log-Riemann surface S is the
ring of holomorphic functions f on S of the form
f = F ◦ k0,
where F ∈ ÂP0 . In particular, for k = 0, . . . , d − 1, we define the holomorphic functions
fk : S → C by
fk = Fk ◦ k0.
Observe that f0 = π is the projection mapping of S. The structural ring ÂS is an integral
domain.
We define the structural field K̂S to be the field of fractions of ÂS . Therefore we have
ÂS ≈ ÂP0 , K̂S ≈ K̂P0 .
We define in the same way AS and its field of fractions KS .
Definition 5.5. The coordinate ring C[π], resp. field C(π), is the sub-ring of the structural
ring ÂS , resp. subfield of the structural field K̂S , generated by the coordinate function π.
Observe that we have
C[π] ≈ C[F0] ⊂ ÂP0 , C(π) ≈ C(F0) ⊂ K̂P0 ,
because elements f of the coordinate ring are of the form
f = F ◦ k0,
with F ∈ C[F0].
5.3 Transcendence degree and number of infinite ramification points
The number of infinite ramification points in the log-Riemann surface S can be read algebraically
as the transcendence degree of KS or K̂S over C(π).
Theorem 5.6. The transcendence degree of K̂S over C(π) is[
K̂S : C(π)
]
tr
= d.
Proof. We have that [KP0 : C[F0]]tr = d because 1, z, F0, . . . , Fd−1 are algebraically indepen-
dent. �
The Ramificant Determinant 25
5.4 Stolz limits and refined analytic estimates
By Stolz limit at an infinite ramification point w∗ of S∗ we understand a limit when we converge
to w∗ remaining in a sector with vertex at w∗.
Proposition 5.7. Any function f ∈ ÂS is Stolz continuous in S∗, i.e., it has Stolz limits at the
infinite ramification points.
It is enough to prove this result for f in the vector space VS ⊂ ÂS
VS = k0C[k0]
(
eP0 ◦ k0
)
⊕ C.1⊕ C.f0 ⊕ · · · ⊕ C.fd−1,
i.e., f = F ◦ k0 with F ∈ VP0 .
This Stolz continuity is weaker than continuity for the topology defined by the natural flat
metric on S that gives the completion S∗. We can show that the only continuous functions in VS
for the completion topology are the ones in the coordinate sub-ring C[π]. We have:
Proposition 5.8. Any function f ∈ VS not belonging to the subspace C.1⊕C.f0 has a Stolz con-
tinuous extension to S∗ but not a continuous extension. In particular, the functions f1, . . . , fd−1
do extend Stolz continuously to S∗ but not continuously. The function f0 also extends continu-
ously to S∗ for the metric topology.
This result and a stronger version of Proposition 5.7 is proved in [5, Section III.2] and results
from refined analytic estimates for the functions f ∈ ÂS , but we can also prove it directly by the
same argument used in conformal representation theory to prove that the existence of a radial
limit implies Stolz convergence.
5.5 Liouville theorem
We have growth conditions that characterize the functions in the vector space VS . For the
precise statement and the proof of the following theorem (that we will not use in this article)
we refer to [5, Section III.3].
Theorem 5.9 (general Liouville theorem). Let f : S → C be a holomorphic function which
has a finite Stolz continuous extension to S∗. Let ∞ the end at infinite of the Alexandrov
compactification of S. If f satisfies a precise set of growth conditions on f(w) when w → ∞
(see [5, Section III.3]) we have that f ∈ VS , that is there exists F ∈ VP0 such that f = F ◦ k0.
5.6 Separation of points
The guiding principle of Dedekind–Weber theory is to reconstruct algebraically the Riemann
surface from its function field, that in the case of a compact Riemann surface is the field of
meromorphic functions. A first fact to check is that we can separate points with functions.
In the case of a compact Riemann surface the space of holomorphic functions is reduced to
constants, and it is useless. In our situation we can separate points using holomorphic functions
in our structural ring.
Theorem 5.10. The ring ÂS separates the points of S∗.
Proof. Let w1, w2 ∈ S∗ with w1 6= w2. If both points are regular points (non-ramification
points), w1, w2 ∈ S, take z1, z2 ∈ C such that zi = k0(wi). Then the function f ∈ ÂS ,
f = F ◦ k0, with
F (z) = (z − z1)eP0(z)
vanishes at w1 but not at w2.
26 K. Biswas and R. Pérez-Marco
When one of the points, say w1, is a ramification point, then we can take f = F ◦ k0 with
F (z) = eP0(z)
the function f will vanish at w1 but not at w2. The function corresponding to eP0 separates
infinite ramification points from regular points.
The remaining case is when both points are ramification points w1, w2 ∈ S∗−S. Then, using
Theorem 4.7 we have that there is a function fk that does not vanish simultaneously at both
points, hence it separates w1 and w2. �
Dedekind–Weber theory in the case of the complex plane is elementary. Recall that to each
point on z0 ∈ C we can associate a maximal ideal mz0 of C[z], namely the ideal of functions
vanishing at z0. Conversely, any maximal ideal m of C[z] is of this form since the residual field is C
C[z]/m ≈ C
and z is mapped by this quotient into some z0 ∈ C, thus m = mz0 . In that way the points
of the complex plane C can be reconstructed algebraically from the ring of polynomials C[z],
each point corresponding to a maximal ideal. The ring is of dimension 1 and any prime ideal
is maximal. In the same way we can reconstruct the Riemann sphere identifying points with
discrete valuation rings in the field of fractions C(z).
In our situation, to each point of S∗, including the infinite ramification points, we can asso-
ciate a maximal ideal of ÂS .
Corollary 5.11. There is an embedding S∗ ↪→ Max ÂS , the space of maximal ideals of ÂS , by
w0 7→ mw0 where mw0 =
{
f ∈ ÂS ; f(w0) = 0
}
.
Proof. Observe that any ideal mw0 is maximal because it is the kernel of the ring morphism
ÂS → C,
f 7→ f(w0)
and
ÂS/mw0 ≈ C
is a field, so mw0 is maximal. �
Proposition 5.12. The maximal ideal mw∗ associated to an infinite ramification point is not
principal.
Proof. Observe that eP0 ◦k0 ∈ mw∗ and eP0 ◦k0 has no non-trivial divisors by Proposition 2.14,
hence mw∗ is not principal. �
5.7 Regular vs. infinite ramification points
We define on ÂP0 the differential operator D = d
dz . The following lemma is clear.
Lemma 5.13. The ring ÂP0 endowed with D is a differentiable ring. The ring of constants
are the constant functions. Moreover, the principal ideal generated by eP0 is absorbent for the
derivation:
D(ÂP0) ⊂
(
eP0
)
.
The Ramificant Determinant 27
The differential operator D defines a derivation D̂ on the structural ring ÂS which can be
expressed on the variable w = F0(z) as
D̂ =
(
eP0 ◦ k0
) d
dw
.
Definition 5.14. The infinite ramification divisor is the principal ideal ℵ∞ generated by eP0 ◦k0
ℵ∞ =
(
eP0 ◦ k0
)
.
Next proposition is also clear.
Proposition 5.15. We have that
D̂(ÂS) ⊂ ℵ∞
and ℵ∞ is the intersection of maximal ideals associated to infinite ramification points
ℵ∞ =
⋂
w∗
mw∗ .
Next theorem allows to distinguish regular and infinite ramification points from the position
of their maximal ideal mw0 with respect to the ramification divisor ℵ∞.
Theorem 5.16. Let mw0 be the maximal ideal associated to a point w0 ∈ S∗. We have that
mw0 ∩ D̂−1(mw0) is a sub-ideal of mw0, and
• If w0 ∈ S is a regular point, we have that mw0 ∩ℵ∞ 6= ℵ∞ and also, mw0 ∩ℵ∞ 6= mw0 and
mw0 ∩ D̂−1(mw0) is a strict sub-ideal of mw0, mw0 ∩ D̂−1(mw0) ( mw0.
• If w0 ∈ S∗−S is an infinite ramification point, then ℵ∞ ⊂ mw0, ℵ∞ 6= mw0, and mw0∩ℵ∞ =
ℵ∞ also D̂−1(mw0) = ÂS so mw0 ∩ D̂−1(mw0) = mw0.
Proof. We prove that mw0 ∩ D̂−1(mw0) is an ideal. Let f ∈ mw0 ∩ D̂−1(mw0). We check that
f.h ∈ mw0 ∩ D̂−1(mw0) for any h ∈ ÂS . We have
f(w0) = 0, D̂(f)(w0) = 0,
so we get (f.h)(w0) = 0 and
D̂(fh)(w0) = D̂(f)(w0).h(w0) + f(w0).D̂(h)(w0) = 0.
When w0 is an infinite ramification point, it is clear that ℵ∞ ⊂ mw0 and ℵ∞ 6= mw0 because
d ≥ 2. Taking preimages in ℵ∞ ⊂ mw0 we get
ÂS ⊂ D̂−1(mw0),
thus D̂−1(mw0) = ÂS .
When w0 ∈ S is a regular point, we have eP0◦k0 /∈ mw0 , so eP0◦k0 ∈ ℵ∞−mw0 . Also, there are
functions f ∈ mw0−ℵ∞. For example, one can choose f = F ◦k0 where F is a linear combination
of 1, F0, . . . , Fd−1 vanishing at z0 = k0(w0) (codimension 1 condition) and not a multiple of eP0
(another codimension 1 condition by the non-vanishing of the Ramificant determinant), then
not all asymptotic values of F can be 0 because otherwise F would be a multiple of eP0 by
Theorem 4.1. �
28 K. Biswas and R. Pérez-Marco
Acknowledgements
We are grateful to Y. Levagnini and the referees for their careful reading and corrections that
improved the article.
References
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https://arxiv.org/abs/1606.06449
https://arxiv.org/abs/1602.08219
https://doi.org/10.1090/conm/639/12826
https://arxiv.org/abs/1011.0535
https://doi.org/10.1090/conm/639/12827
https://arxiv.org/abs/1011.0812
https://arxiv.org/abs/1512.03776
https://arxiv.org/abs/1512.04035
https://doi.org/10.1515/crll.1882.92.181
https://doi.org/10.1007/978-3-642-56478-9_39
https://doi.org/10.1515/crll.1835.13.93
https://doi.org/10.1007/978-1-4612-0989-8
https://doi.org/10.1007/BF02547780
https://doi.org/10.1007/978-3-642-85590-0
https://doi.org/10.1007/978-3-642-85590-0
https://doi.org/10.1007/978-3-662-02770-7
https://doi.org/10.3792/pjaa.77.68
https://doi.org/10.3792/pjaa.77.68
https://doi.org/10.1090/conm/303/05238
1 Introduction
2 A ring of special functions
2.1 Definitions
2.2 Asymptotics at infinite
2.3 Linear independence
2.4 Algebraic independence
2.5 Computation of integrals
2.6 Differential ring structure
2.7 Picard–Vessiot extensions
2.8 Liouville classification
3 The Ramificant determinant
3.1 Definition of the Ramificant determinant
3.2 Formula for the Ramificant determinant
3.3 The universal polynomials d
4 Applications of the Ramificant determinant
4.1 Integrability and Abel-like theorem
4.2 The period mapping is étale
4.3 Separation of asymptotic directions
4.4 Transalgebraic symmetric formulas
4.5 Torelli-like theorem
5 Introduction to transalgebraic Dedekind–Weber theory
5.1 Transalgebraic curves of genus 0
5.2 The structural ring
5.3 Transcendence degree and number of infinite ramification points
5.4 Stolz limits and refined analytic estimates
5.5 Liouville theorem
5.6 Separation of points
5.7 Regular vs. infinite ramification points
References
|
| id | nasplib_isofts_kiev_ua-123456789-210302 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:25:04Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Biswas, K. Pérez-Marco, R. 2025-12-05T09:27:25Z 2019 The Ramificant Determinant / K. Biswas, R. Pérez-Marco // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 19 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 30F99; 30D99 arXiv: 1903.06770 https://nasplib.isofts.kiev.ua/handle/123456789/210302 https://doi.org/10.3842/SIGMA.2019.086 We give an introduction to the transalgebraic theory of simply connected log-Riemann surfaces with a finite number of infinite ramification points (transalgebraic curves of genus 0). We define the base vector space of transcendental functions and establish, by elementary methods, some transcendental properties. We introduce the Ramificant determinant constructed with transcendental periods, and we give a closed-form formula that gives the main applications to transalgebraic curves. We prove an Abel-like theorem and a Torelli-like theorem. Transposing to the transalgebraic curve, the base vector space of transcendental functions, they generate the structural ring from which the points of the transalgebraic curve can be recovered algebraically, including infinite ramification points. We are grateful to Y. Levagnini and the referees for their careful reading and corrections that improved the article. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Ramificant Determinant Article published earlier |
| spellingShingle | The Ramificant Determinant Biswas, K. Pérez-Marco, R. |
| title | The Ramificant Determinant |
| title_full | The Ramificant Determinant |
| title_fullStr | The Ramificant Determinant |
| title_full_unstemmed | The Ramificant Determinant |
| title_short | The Ramificant Determinant |
| title_sort | ramificant determinant |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210302 |
| work_keys_str_mv | AT biswask theramificantdeterminant AT perezmarcor theramificantdeterminant AT biswask ramificantdeterminant AT perezmarcor ramificantdeterminant |