In this short note, we give a localized version of the basic triangle theorem, first published in 2011 (see [4]) in order to prove the independence of hyperlogarithms over various function fields. This version provides direct access to rings of scalars and avoids the recourse to fraction fields as that of meromorphic functions for instance

Duchamp, G.

Gargava, Nihar

English

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A localized version of the basic triangle theorem
G. Duchamp, Nihar Gargava
To cite this version:
G. Duchamp, Nihar Gargava. A localized version of the basic triangle theorem. 2019. ￿hal-02264912￿
A localized version of the basic triangle
theorem
G.H.E. Duchamp
University Paris 13, Sorbonne Paris City, 93430 Villetaneuse, France,
Nihar Gargava
École Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland
August 7, 2019 07-08-2019 17:06
Abstract
In this short note, we give a localized version of the basic triangle theorem, first published in
2011 (see [4]) in order to prove the independence of hyperlogarithms over various function
fields. This version provides direct access to rings of scalars and avoids the recourse to
fraction fields as that of meromorphic functions for instance.
1 Original theorem
Non commutative differential equations with left multiplier can be expressed in the
context of general differential algebras. Notations about alphabets and (noncom-
mutative) series are standard and can be found in [1].
Theorem 1.1 (Th 1 in [4]) Let (A ,d) be a k-commutative associative differential
algebra with unit (ker(d) = k, a field) and C be a differential subfield of A (i.e.
d(C ) ⊂ C and k ⊂ C ). Let X be some alphabet (i.e. some set) and we define
d : A 〈〈X〉〉→A 〈〈X〉〉 to be the map given by 〈d(S) | w〉= d(〈S | w〉). We suppose
that S ∈A 〈〈X〉〉 is a solution of the differential equation
d(S) =MS ; 〈S | 1X∗〉= 1A (1)
where the multiplier M is a homogeneous series (a polynomial in the case of finite
X) of degree 1, i.e.
M = ∑
x∈X
uxx ∈ C 〈〈X〉〉 . (2)
The following conditions are equivalent :
i) The family (〈S | w〉)w∈X∗ of coefficients of S is free over C .
ii) The family of coefficients (〈S | y〉)y∈X∪{1X∗} is free over C .
iii) The family (ux)x∈X is such that, for f ∈ C and α ∈ k(X) (i.e. supp(α) is finite)
d( f ) = ∑
x∈X
αxux =⇒ (∀x ∈ X)(αx = 0) . (3)
iv) The family (ux)x∈X is free over k and
d(C )∩ spank
(
(ux)x∈X
)
= {0} . (4)
Proof. For convenience of the reader, we enclose here the demonstration given in
Th 1 [4].
(i)=⇒(ii) Obvious.
(ii)=⇒(iii)
Suppose that the family (〈S | y〉)y∈X∪{1X∗} (coefficients taken at letters and the
empty word) of coefficients of S were free over C and let us consider the relation
as in (3)
d( f ) = ∑
x∈X
αxux . (5)
We form the polynomial P=− f1X∗ +∑x∈X αxx. One has d(P) =−d( f )1X∗ and
d(〈S |P〉)= 〈d(S) |P〉+〈S | d(P)〉= 〈MS |P〉−d( f )〈S | 1X∗〉=(∑
x∈X
αxux)−d( f )= 0
(6)
whence 〈S | P〉 must be a constant, say λ ∈ k. For Q= P−λ .1X∗ , we have
supp(Q)⊂ X ∪{1X∗} and 〈S | Q〉= 〈S | P〉−λ 〈S | 1X∗〉= 〈S | P〉−λ = 0 .
This, in view of (ii), implies that Q= 0 and, as Q=−( f +λ )1X∗ +∑x∈X αxx, one
has, in particular, supp(α) = /0 (and, as a byproduct, f = −λ which is indeed the
only possibility for the L.H.S. of (3) to occur).
(iii)⇐⇒(iv)
Obvious, (iv) being a geometric reformulation of (iii).
(iii)=⇒(i)
2
Let K be the kernel of P 7→ 〈S | P〉 (linear C 〈X〉 →A ) i.e.
K = {P ∈ C 〈X〉|〈S | P〉= 0} . (7)
If K = {0}, we are done. Otherwise, let us adopt the following strategy.
First, we order X by some well-ordering < ([2] III.2.1) and X∗ by the graded lexi-
cographic ordering ≺ defined as follows
u≺ v⇐⇒ |u|< |v| or (u= pxs1 , v= pys2 and x< y). (8)
It is easy to check that X∗ is also a well-ordered by≺. For each nonzero polynomial
P, we denote by lead(P) its leading monomial; i.e. the greatest element of its
support supp(P) (for ≺).
Now, as R =K r{0} is not empty, let w0 be the minimal element of lead(R) and
choose a P ∈R such that lead(P) = w0. We write
P= f w0+ ∑
u≺w0
〈P | u〉u ; f ∈ Cr{0} . (9)
The polynomial Q = 1f P is also in R with the same leading monomial, but the
leading coefficient is now 1; and so Q is given by
Q= w0+ ∑
u≺w0
〈Q | u〉u . (10)
Differentiating 〈S | Q〉= 0, one gets
0= 〈d(S) | Q〉+ 〈S | d(Q)〉= 〈MS | Q〉+ 〈S | d(Q)〉=
〈S |M†Q〉+ 〈S | d(Q)〉= 〈S |M†Q+d(Q)〉 (11)
with
M†Q+d(Q) = ∑
x∈X
ux(x
†Q)+ ∑
u≺w0
d(〈Q | u〉)u ∈ C 〈X〉 . (12)
It is impossible that M†Q+d(Q) ∈ R because it would be of leading monomial
strictly less than w0, henceM†Q+d(Q) = 0. This is equivalent to the recursion
d(〈Q | u〉) =− ∑
x∈X
ux〈Q | xu〉 ; for x ∈ X , v ∈ X
∗
. (13)
From this last relation, we deduce that 〈Q | w〉 ∈ k for every w of length deg(Q)
and, because 〈S | 1X∗〉= 1A , one must have deg(Q)> 0. Then, we write w0 = x0v
and compute the coefficient at v
d(〈Q | v〉) =− ∑
x∈X
ux〈Q | xv〉= ∑
x∈X
αxux (14)
3
with coefficients αx =−〈Q | xv〉 ∈ k as |xv| = deg(Q) for all x ∈ X . Condition (3)
implies that all coefficients 〈Q | xv〉 are zero; in particular, as 〈Q | x0v〉= 1, we get
a contradiction. This proves that K = {0}.
✷
2 Localization
We will now establish the following extension of Theorem 1 in [4]. Let (A ,d) be
a k-commutative associative differential algebra with unit (ker(d) = k, a field). We
consider a solution of the differential equation
d(S) =MS ; 〈S | 1X∗〉= 1A (15)
where the multiplierM is a homogeneous series (a polynomial in the case of finite
X ) of degree 1, i.e.
M = ∑
x∈X
uxx ∈A 〈〈X〉〉 . (16)
Proposition 2.1 (Thm1 in [4], Localized form) Let (A ,d) be a commutative as-
sociative differential ring (ker(d) = k being a field) and C be a differential subring
(i.e. d(C )⊂C ) of A which is an integral domain containing the field of constants.
We suppose that, for all x ∈ X, ux ∈ C and that S ∈ A 〈〈X〉〉 is a solution of the
differential equation (15) and that (ux)x∈X ∈ C X .
The following conditions are equivalent :
i) The family (〈S | w〉)w∈X∗ of coefficients of S is free over C .
ii) The family of coefficients (〈S | y〉)y∈X∪{1X∗} is free over C .
iii’) For all f1, f2 ∈ C , f2 6= 0 and α ∈ k(X), we have the property
W ( f1, f2) = f
2
2 (∑
x∈X
αxux) =⇒ (∀x ∈ X)(αx = 0) . (17)
where W ( f1, f2), the wronskian, stands for d( f1) f2− f1d( f2).
Proof. (i.=⇒ ii.) being trivial, remains to prove (ii.=⇒ iii’.) and (iii’.=⇒ i.). To
this end, we localize the situation w.r.t. the multiplicative subset C× := Cr{0} as
can be seen in the following commutative cube
4
C Fr(C )
A A [(C×)−1]
C Fr(C )
A A [(C×)−1]
ϕC
j
d
d f rac
j f rac
ϕA
d f rac
ϕC
j j f rac
ϕA
d
(18)
We give here a detailed demonstration of the commutation which provides, in pass-
ing, the labelling of the arrows.
Left face. — Comes from the fact that d(C ) ⊂ C , j being the canonical embed-
ding.
Upper and lower faces. — We first construct the localization
ϕA : A −→A [(C
×)−1] w.r.t. the multiplicative subset C× ⊂Ar{0} (recall that
C has no zero divisor). Now, from standard theorems (see [3], ch2 §2 remark 3
after Def. 2, for instance), we have
ker(ϕA ) = {u ∈A |(∃v ∈ C
×)(uv= 0)} (19)
For every intermediate ring C ⊂B ⊂A , we remark that the composittion
B →֒A
ϕA−→A [(C×)−1]
realises the ring of fractions B[(C×)−1] which can be identified with the subal-
gebra generated by ϕA (B) and the set of inverses ϕA (C
×)−1. Applying this to
C , and remarking that C [(C×)−1] ≃ Fr(C ), we get the embedding j f rac and the
commutation of upper and lower faces.
Front and rear faces. — From standard constructions (see e.g. the book [5]), there
exists a unique d f rac ∈Der(A [C×)−1]) such that these faces commute.
Right face. — Commutation comes from the fact that d f rac j f rac and j f racd f rac co-
incide on ϕC (C ) hence on ϕC (C
×) and on their inverses. Therefore on all Fr(C ).
From the constructions it follows that the arrows (derivations, morphisms) are ar-
rows of k-algebras.
5
Now, we set
(i) S¯= ∑w∈X∗ ϕA (〈S | w〉)w ∈A [C
×)−1]〈〈X〉〉
(ii) M¯ = ∑x∈X ϕC (ux)x ∈A [C
×)−1]〈〈X〉〉
it is clear, from the commutations, that (A [C×)−1]〈〈X〉〉,d f rac) where d f rac is the
extension of d f rac to the series, is a differential algebra and that
d f rac(S¯) = M¯S¯ ; 〈S¯ | 1〉= 1 (20)
we are now in the position to resume the proofs of (ii.=⇒ iii’.) and (iii’.=⇒ i.).
ii.=⇒ iii’.) Supposing (ii), we remark that the family of coefficients
(〈S¯ | y〉)y∈X∪{1X∗}
is free over C 1 . Indeed, let us suppose a relation
∑
y∈X∪{1X∗}
gy 〈S¯ | y〉= 0 with (gy)y∈X∪{1X∗} ∈ C
(X∪{1X∗}) (21)
this relation is equivalent to
ϕA ( ∑
y∈X∪{1X∗}
gy 〈S | y〉) = 0 (22)
which, in view of (19), amounts to the existence of v ∈ C× such that
0= v( ∑
y∈X∪{1X∗}
gy 〈S | y〉) = ∑
y∈X∪{1X∗}
vgy 〈S | y〉 (23)
which implies (∀y ∈ X ∪{1X∗})(vgy = 0) but, C being without zero divisor, one
gets
(∀y ∈ X ∪{1X∗})(gy = 0) (24)
which proves the claim. This implies in particular, by chasing denominators, that
the family of coefficients
(〈S¯ | y〉)y∈X∪{1X∗}
is free over Fr(C ). This also implies 2 that ϕA is injective on
spanC (〈S | y〉)y∈X∪{1X∗} (25)
1 As ϕC is injective on C we identify ϕC (C ) and C , this can be unfolded on request, of course.
2 And indeed is equivalent under the assumption of (ii).
6
To finish the proof that (ii.=⇒ iii’.), let us choose f1, f2 ∈ C with f2 6= 0 and set
some relation which reads
W ( f1, f2) = f
2
2 (∑
x∈X
αxux) (26)
with α ∈ k(X), then
(∑
x∈X
αxux) =
W ( f1, f2)
f 22
= d f rac(
f1
f2
) (27)
but, in view of Th1 in [4] applied to the differential field Fr(C ), we get α ≡ 0.
(iii’.=⇒ i.) The series S¯ satisfies
d(S¯) = M¯S¯ ; 〈S¯ | 1X∗〉= 1A [C×)−1] = 1Fr(C ) (28)
and remarking that
(i) all f in the differential field Fr(C ) can be expressed as f = f1f2
(ii) condition (iii’) for (S,A ,C ,d,X) implies condition (iii) for
(S¯,A [C×)−1],Fr(C ),d f rac,X) 3 which, in turn, implies the Fr(C )-
freeness of (〈S¯ | w〉)w∈X∗ hence its C -freeness and, by inverse image 4 the
C -freeness of (〈S | w〉)w∈X∗.
✷
Remark 2.2 It seems reasonable to think that the whole commutation of the cube
could be understood by natural transformations within an appropriate category. If
yes, this will be inserted in a forthcoming version.
References
[1] .– J. Berstel, C. Reutenauer.– Rational series and their languages, Springer Verlag,
1988.
[2] N. Bourbaki.– Theory of Sets, Springer-Verlag Berlin and Heidelberg GmbH & Co. K;
(2nd printing 2004)
[3] N. Bourbaki.– Commutative Algebra: Chapters 1-7, Springer (1998)
3 Once again we identify, with no loss, k ⊂ C , the latter being idetified with its image through
ϕ
A [C×)−1].
4 If the image (through a A-linear arrow) of a family is A-free then the family itself is A-free.
7
[4] .– M. Deneufchâtel, G.H.E. Duchamp, Hoang Ngoc Minh, A.I. Solomon.–
Independence of hyperlogarithms over function fields via algebraic combinatorics,
in Lecture Notes in Computer Science (2011), Volume 6742/2011, 127-139.
[5] .– M. van der Put, M. F. Singer.– Galois Theory of Linear Differential Equations,
Springer (2003)
8


A localized version of the basic triangle theorem

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