# Hadamard Product and Resurgence Theory

Y. Li111Chern Institute of Mathematics and LPMC, Nankai university, Tianjin
300071, China D. Sauzin222CNRS – Observatoire de Paris, PSL Research
University, 75014 Paris, France 333Department of Mathematics, Capital Normal
University, Beijing 100048, China S. Sun††footnotemark:  444Academy for
Multidisciplinary Studies, Capital Normal University, Beijing 100048, China

(December 2020)

###### Abstract

We discuss the analytic continuation of the Hadamard product of two
holomorphic functions under assumptions pertaining to Écalle’s Resurgence
Theory, proving that if both factors are endlessly continuable with prescribed
sets of singular points $A$ and $B$, then so is their Hadamard product with
respect to the set $\\{0\\}\cup A\cdot B$. This is a generalization of the
classical Hadamard Theorem in which all the branches of the multivalued
analytic continuation of the Hadamard product are considered.

##  1 Introduction

1.1 The Hadamard product of two power series
$f(\xi),g(\xi)\in\mathbb{C}[[\xi]]$ is defined by the formula

$f(\xi)=\sum_{n\geq 0}a_{n}\xi^{n}\quad\text{and}\quad g(\xi)=\sum_{n\geq
0}b_{n}\xi^{n}\quad\Longrightarrow\quad f\odot g(\xi)\coloneqq\sum_{n\geq
0}a_{n}b_{n}\xi^{n}.$ (1.1)

If the radii of convergence of $f$ and $g$ are positive, _i.e._
$f(\xi),g(\xi)\in\mathbb{C}\\{\xi\\}$, then so is the radius of convergence of
$f\odot g$. From now on, we identify a convergent power series with the
holomorphic germ at $0$ that it defines. J. Hadamard published in 1898 the
proof of his now classical theorem [Ha99], to the effect that:

> _Given finite subsets $A,B\subset\mathbb{C}^{*}$ and $K,L>0$, if $f$ extends
> analytically to $\mathbb{D}_{K}\setminus\bigcup\limits_{\alpha\in
> A}R_{\alpha}$ and $g$ to $\mathbb{D}_{L}\setminus\bigcup\limits_{\beta\in
> B}R_{\beta}$, then $f\odot g$ extends analytically to
> $\mathbb{D}_{KL}\setminus\bigcup\limits_{\gamma\in A\cdot B}R_{\gamma}$._

Here, we have denoted by $\mathbb{D}_{M}$ the open disc centred at the origin
of radius $M$, by $R_{\omega}$ the portion of the ray
$\omega\,\mathbb{R}_{\geq 0}$ from $\omega$ onward, and by $A\cdot B$ the
“product set” of $A$ and $B$:

$\mathbb{D}_{M}\coloneqq D(0,M),\qquad
R_{\omega}\coloneqq\omega\,[1,+\infty),\qquad A\cdot
B\coloneqq\\{\alpha\beta\mid(\alpha,\beta)\in A\times B\\}.$ (1.2)

So, this is a statement about the “principal branches” of $f$, $g$ and $f\odot
g$: we are considering their analytic continuation to a part of their Mittag-
Leffler stars, assuming that any singular point that appears for $f$ or $g$
when following a ray emanating from the origin is isolated (in fact, J.
Hadamard also proves a more general version, in which the various
$R_{\omega}$’s are interpreted as portion of logarithmic spirals of the same
polar slope).

In [Bo98], É. Borel studied the analytic continuation of $f\odot g$, also
giving a proof of Hadamard’s theorem, with a view to giving more information
on the nature of the singularities of $f\odot g$ in terms of the nature of the
singularities of $f$ and $g$, at least in the case of “uniform singularities”
(_i.e._ with trivial monodromy: poles or essential singularities). We refer
the reader to [PM20a], [PM20b] for a modern study and a generalization of this
result.

Interestingly, Borel also mentions something about the analytic continuation
of $f$, $g$ and $f\odot g$ to other sheets than the principal one,555In this
article, when we speak of “principal” sheet or “principal” branch, we mean
that we refer to the analytic continuation along straight line segments
starting at the origin when possible, _i.e._ we may identify the principal
sheet with the Mittag-Leffler star.  and he gives a valuable hint in a
footnote which has been successfully used in the context of Écalle’s
Resurgence Theory to study the convolution product $f*g$—see §A.4. It is our
aim to consider other branches than the principal one for the analytic
continuation of $f\odot g$, under appropriate assumptions on $f$ and $g$.

1.2 In this article, we are concerned with the case when $f$ and $g$ satisfy a
property pertaining to Écalle’s Resurgence Theory [Eca81], namely a restricted
form of “endless continuability”:

######  Definition 1.1.

Given a non-empty closed discrete subset $\Omega$ of $\mathbb{C}$, what we
call “$\Omega$-continuable germ” is a holomorphic germ $f$ at the origin that
admits analytic continuation along any path
$\gamma\colon\thinspace[0,1]\to\mathbb{C}\setminus\Omega$ that has its initial
point $\gamma(0)$ in the disc of convergence of $f$.

This is the definition that has been employed in [Eca81] with
$\Omega=2\pi{\mathrm{i}}\mathbb{Z}$, or in [Sau13], [Sau15], [MS16]. It
already covers a range of interesting examples, e.g. in complex dynamics. The
simplest examples of $\Omega$-continuable germs are provided by meromorphic
functions and algebraic functions; however, the definition puts no restriction
on the nature of the singularities of the analytic continuation of an
$\Omega$-continuable germ, only on their location. More general is the
definition of “endless continuability” [CNP93], [KS20], and even more general
that of “continuability without a cut” [Eca85].

Our main result is

######  Theorem A.

Given non-empty closed discrete subsets $A,B\subset\mathbb{C}$, if $f$ is an
$A$-continuable germ and $g$ a $B$-continuable germ, then

$\Omega\coloneqq\\{0\\}\cup(A\cdot B)$ (1.3)

is closed and discrete and $f\odot g$ is an $\Omega$-continuable germ.

The proof is given in Sections 2–3.

Note that in Definition 1.1 we may have $0$ belonging to $\Omega$ or not, but
in all cases does an $\Omega$-continuable germ have a principal branch regular
at $0$. Simple examples with $\Omega=\\{0,1\\}$ are
$f_{1}(\xi)\coloneqq-\frac{1}{\xi}\log(1-\xi)$ and
$\operatorname{Li}_{2}(\xi)\coloneqq\int_{0}^{\xi}f_{1}(\xi_{1})\,{\mathrm{d}}\xi_{1}$.
So, Theorem A is compatible with the Hadamard Theorem, and J. Hadamard is
right not to include $0$ among the possibly singular points of $f\odot g$ as
far as only the principal branches are concerned.

The necessity of including $0$ among the possibly singular points of $f\odot
g$ when we consider its analytic continuation on other sheets was already
noted by Borel, who gives credit to E. Lindelöf for that point [Bo98, ftn 2].
A simple example that illustrates it is $f_{0}(\xi)\coloneqq-\log(1-\xi)$: the
germ $f_{0}$ is $\\{1\\}$-continuable, but the germ $f_{0}\odot f_{0}$ is not;
in fact, $f_{0}\odot f_{0}=\operatorname{Li}_{2}$ is $\\{0,1\\}$-continuable.

1.3 The formal Borel transform $\mathcal{B}\colon\thinspace
t\,\mathbb{C}[[t]]\to\mathbb{C}[[\xi]]$ is defined by:

$\widetilde{f}(t)=\sum_{n\geq
0}c_{n}t^{n+1}\quad\Longrightarrow\quad\mathcal{B}\widetilde{f}(\xi)=\sum_{n\geq
0}\frac{c_{n}}{n!}\xi^{n}.$ (1.4)

In other words,
$\mathcal{B}\widetilde{f}(\xi)=\mathrm{e}^{\xi}\odot\frac{\widetilde{f}(\xi)}{\xi}$.
A formal power series $\widetilde{f}(t)\in t\,\mathbb{C}[[t]]$ is said to be
an _$\Omega$ -resurgent series_ when its Borel transform
$\mathcal{B}\widetilde{f}$ is an $\Omega$-continuable germ [Eca81], [Sau13].
The singularities of $\mathcal{B}\widetilde{f}$ then give us information on
the divergence of $\widetilde{f}(t)$, inasmuch as the Borel transform of a
convergent power series has no singularity at all.

Now,

$f=\mathcal{B}\widetilde{f}\quad\text{and}\quad
g=\mathcal{B}\,\widetilde{g}\quad\Longrightarrow\quad f\odot
g=\mathcal{B}(e_{1}\odot\widetilde{f}\odot\widetilde{g}),$ (1.5)

where $e_{1}(t)\coloneqq t\,\mathrm{e}^{t}$, hence our result can be rephrased
as

> _Given non-empty closed discrete subsets $A,B\subset\mathbb{C}$, if
> $\widetilde{f}(t)$ is an $A$-resurgent series and $\widetilde{g}(t)$ a
> $B$-resurgent series, then $e_{1}\odot\widetilde{f}(t)\odot\widetilde{g}(t)$
> is an $\Omega$-resurgent series with $\Omega$ as in (1.3). _

This is to be compared with Theorem 6.27 of [MS16], which can be rephrased as:

> _Given non-empty closed discrete subsets $A,B\subset\mathbb{C}$ such that
> $\Omega\coloneqq A\cup B\cup(A+B)$ is closed and discrete, if
> $\widetilde{f}(t)$ is an $A$-resurgent series and $\widetilde{g}(t)$ a
> $B$-resurgent series, then their product
> $\widetilde{f}(t)\cdot\widetilde{g}(t)$ is an $\Omega$-resurgent series. _

The latter statement is a variation on a result on the multiplication of
$\Omega$-resurgent series proved in [Sau13]. More generally, the stability
under multiplication of the space of resurgent series (defined as Borel
preimages of endlessly continuable germs or of germs continuable without a
cut) is a key fact of Écalle’s Resurgence Theory.

1.4 The shift of indices in the definition of $\mathcal{B}$ (mapping $t^{n+1}$
to $\xi^{n}/n!$) is a convenient normalization which results in the following
relation between the ordinary multiplication in $\mathbb{C}[[t]]$ and the
convolution $*$ in $\mathbb{C}[[\xi]]$:

$\mathcal{B}(\widetilde{f}\cdot\widetilde{g})=f*g(\xi)\coloneqq\int_{0}^{\xi}f(\xi_{1})g(\xi-\xi_{1})\,{\mathrm{d}}\xi_{1}.$
(1.6)

This is the usual “additive” convolution (associated with Laplace transform
and Borel-Laplace summation), whereas the Hadamard product may be considered
as a “multiplicative” convolution. Indeed, as was well-known already in
Hadamard’s time,

$0<\rho<R_{f}\enspace\text{and}\enspace\lvert\xi\rvert<\rho
R_{g}\quad\Longrightarrow\quad f\odot
g(\xi)=\frac{1}{2\pi{\mathrm{i}}}\oint_{C_{\rho}}f({\zeta})g\Big{(}\frac{\xi}{{\zeta}}\Big{)}\frac{{\mathrm{d}}{\zeta}}{{\zeta}},$
(1.7)

where $C_{\rho}$ is the parametrized circle:
$s\in[0,1]\mapsto{\zeta}=\rho\,\mathrm{e}^{2\pi{\mathrm{i}}s}$ (we have
denoted by $R_{f}$ and $R_{g}$ the radii of convergence of $f$ and $g$).

Actually, Formula (1.7) is the basis of Hadamard’s and Borel’s arguments in
[Ha99] and [Bo98], and it will be the starting point of our analysis as well.
The idea, translated from Borel’s own words, is that “this expression of
$f\odot g$ stays valid _if one deforms the integration contour without letting
it cross any singular point of the integrand_ ” and, “the contour having been
fixed in an arbitrary manner, one obtains the analytic continuation of $f\odot
g$ _by moving $\xi$ in the plane, provided the singular points of the
integrand do not cross the integration contour”_.

It seems that, throughout [Bo98], Borel keeps in mind the possibility of going
to the non-principal sheets and dealing with multivalued analytic
continuation, yet reluctantly so, since when he explicitly mentions that
possibility (in the aforementioned footnote 2 or later in Sec. III when he
writes “if the functions $f$ and $g$ were multivalued…”) he tends to recommend
to discard it (see the last paragraph of his Sec. III: “It seems to us useless
to insist on the latter point… one would be led to complicated statements…”).
However, he puts in a footnote (footnote 3) an idea that has been successfully
adapted to study the analytic continuation of the _additive_ convolution (1.6)
of endlessly continuable germs in the context of Resurgence [Eca81], [CNP93].

We reproduce here the content of footnote 3 of [Bo98], with minor notational
changes. Borel writes (1.7) as $f\odot
g(\xi)=\frac{1}{2\pi{\mathrm{i}}}\oint_{C}f\big{(}\frac{1}{x}\big{)}g(\xi
x)\frac{{\mathrm{d}}x}{x}$, with $C=C_{\rho^{-1}}$.

> “Let us conceive the closed contour $C$ as a flexible extensible thread, the
> singular points of $f\big{(}\frac{1}{x}\big{)}$ as pins stuck into the
> plane, the singular points of $g(\xi x)$ as pins that travel as $\xi$ moves.
> It is necessary and sufficient that the thread always part the two systems
> of pins. Now, this will always be possible, by means of a suitable
> deformation, if, while travelling, the second pins never come to _hit_ the
> first ones (…); the thread may acquire a very complicated form, but this is
> harmless.” [Bo98, ftn 3, p. 240]

In the case of the additive convolution product (1.6), instead of the closed
contour $C=C_{\rho^{-1}}$, it is the line segment $[0,\xi]$ that must be
deformed. In [Sau13] and [MS16], it was shown how to contruct explicitly this
deformation when $\xi$ moves in the complex plane without meeting $A\cup
B\cup(A+B)$; this is done by applying to the initial integration contour a
homeomorphism that is a deformation of the identity obtained as the flow of an
explicit non-autonomous vector field tailored to the situation.

A benefit of such a detailed rigorous proof with respect to the arguments
given in [Eca81] or [CNP93] is that it allows for quantitative estimates.
These estimates, in turn, can be used to show the stability of the space of
resurgent series under nonlinear operations and not only multiplication
[Sau15], [MS16], [KS20].

1.5 Our proof of Theorem A is spread over

  1. –

Section 2, which gives details on the kind of “deformation of the identity”
that we need to deform the integration contour in (1.7),

  2. –

Section 3, which explains how to obtain this deformation of the identity out
of the flow of an explicit non-autonomous vector field.

This proof bears some resemblance with the proof of the theorem on convolution
presented in [Sau13], however a number of modifications were necessary,
notably a novel method to construct a non-autonomous vector field adapted to
Formula (1.7) (Section 3).

We were led to Theorem A while studying the Borel transform of the Moyal star
product of two resurgent series $\widetilde{f}(t,q,p)$ and
$\widetilde{g}(t,q,p)$ [LSS20]. Indeed, the Borel counterpart of the Moyal
star product can be written in terms of the Borel transforms of the factors,
$f(\xi,q,p)$ and $g(\xi,q,p)$, and the formula appears as a mixture of
additive convolution with respect to $\xi$ and Hadamard product; more
specifically, it involves the Hadamard product with respect to ${\zeta}$ of
the Taylor expansions $f(\xi_{1},q,p+{\zeta})\odot g(\xi_{2},q+{\zeta},p)$ and
then a convolution-like integration with respect to $\xi_{1}$ and $\xi_{2}$.
However, in such a many-variable context, the restrictions one needs to put on
the singular locus of $f$ and $g$ are more stringent than in Definition 1.1,
namely one requires “algebro-resurgence” [GGS14], [LSS20]:
$f,g\in\mathbb{C}\\{\xi,q,p\\}$ have analytic continuation away from a proper
algebraic subvariety of $\mathbb{C}^{3}$ (or $\mathbb{C}^{2N+1}$ if one is
interested in deformation quantization of $\mathbb{C}^{2N}$ with coordinates
$q_{1},\ldots,q_{N},p_{1},\ldots,p_{N}$). As a result, the functions of
${\zeta}$ involved in the Hadamard product part of the formula have at most
finitely many singularities.

Our work appears as complementary to the one of [PM20a] and [PM20b], which is
devoted to the nature of the singularities of the principal branch of $f\odot
g$ and contains new formulas for the monodromy at a given point
$\omega=\alpha\beta$. Considering the monodromy operator $\Delta_{\omega}$
forces to go around $\omega$ and visit other sheets of the Riemann surface of
$f\odot g$, but only the nearby sheets (the “next-to-principal” one for
$\Delta_{\omega}(f\odot g)$, or the ones reached by circling several times
around $\omega$ if one wants to iterate $\Delta_{\omega}$). Dealing with these
nearby branches of the analytic continuation of the Hadamard product requires
a deformation of the integration contour in (1.7) that is tractable by means
of relatively elementary geometry; the deformation becomes much more intricate
if the variable $\xi$ is allowed to travel to arbitrary sheets, but it can
still be mathematically described and studied by following the method of
Sections 2–3. Therefore, it would be interesting to see whether this can be
used to study the nature of the singularities of the non-principal branches of
$f\odot g$ in the spirit of [PM20a] and [PM20b]. Moreover, as noted in
[PM20a], one might try to explore more deeply the connection with Écalle’s
Resurgence Theory; the so-called “alien operators”, designed to study the
singularities of endlessly continuable functions, behave particularly well in
relation with the additive convolution (1.6) and they lead to contour
deformations reminiscent of those employed in [PM20a]—see e.g. pp. 236–237 of
[MS16].

Another avenue of research that naturally suggests itself would be to consider
the Hadamard product of endlessly continuable germs $f$ and $g$ which are not
$\Omega$-continuable in the sense of Definition 1.1 for any discrete closed
set $\Omega$. One may conjecture that $f\odot g$ is still endlessly
continuable because, during the process of analytic continuation by means of
contour deformation, one “feels” the presence of only finitely many
singularities of $f$ and $g$ at the same time, which is why (additive)
convolution (and even iterated convolutions) could be handled in [KS20]; one
might try to apply similar techniques to the Hadamard product.

##  2 Deformations of the identity adapted to a path $\gamma$

We now begin the proof of Theorem A. We thus give ourselves $A$ and $B$ non-
empty closed discrete subsets of $\mathbb{C}$, and we define $\Omega$ by
(1.3). We also introduce

$A^{\prime}\coloneqq A\setminus\\{0\\},\quad
a\coloneqq\min\\{\lvert\alpha\rvert\mid\alpha\in A^{\prime}\\},\qquad
B^{\prime}\coloneqq B\setminus\\{0\\},\quad
b\coloneqq\min\\{\lvert\beta\rvert\mid\beta\in B^{\prime}\\},$ (2.1)

regardless of $0$ belonging to $A\cup B$ or not. Note that

$\Omega=\\{0\\}\cup(A^{\prime}\cdot B^{\prime}).$ (2.2)

2.1 The first claim to be proved is

######  Lemma 2.1.

The set $\Omega$ is closed and discrete.

###### Proof.

Clearly, this is equivalent to saying that

$\text{For any
$R\in\mathbb{R}_{>0}$,}\quad\Omega\cap\overline{\mathbb{D}}_{R}\enspace\text{is
finite.}$ (2.3)

Let $R\in\mathbb{R}_{>0}$. Any pair $(\alpha,\beta)\in A^{\prime}\times
B^{\prime}$ such that $\alpha\beta\in\overline{\mathbb{D}}_{R}$ must satisfy
$\lvert\alpha\rvert\geq a>0$, $\lvert\beta\rvert\geq b>0$ and
$\lvert\alpha\beta\rvert\leq R$. Hence, $\alpha\in\overline{\mathbb{D}}_{R/b}$
and $\beta\in\overline{\mathbb{D}}_{R/a}$. But
$A\cap\overline{\mathbb{D}}_{R/b}$ and $B\cap\overline{\mathbb{D}}_{R/a}$ are
finite, by the rephrasing (2.3) of our assumption on $A$ and $B$, hence there
only finitely many such pairs $(\alpha,\beta)$ and
$\Omega\cap\overline{\mathbb{D}}_{R}$ is finite. ∎

2.2 Let $I\coloneqq[0,1]$. We now give ourselves an $A$-continuable germ $f$,
a $B$-continuable germ $g$, and a path $\gamma\colon\thinspace
I\to\mathbb{C}\setminus\Omega$ that has its initial point close enough to the
origin, specifically

$0<\lvert\gamma(0)\rvert<ab.$ (2.4)

Let us choose $\rho>0$ such that

$\frac{\lvert\gamma(0)\rvert}{b}<\rho<a.$ (2.5)

Note that the radius of convergence $R_{f}$ of $f$ is at least $a$, and the
radius of convergence $R_{g}$ of $g$ at least $b$, hence Formula (1.7) implies
that

$f\odot
g(\xi)=\frac{1}{2\pi{\mathrm{i}}}\oint_{C_{\rho}}f({\zeta})g\Big{(}\frac{\xi}{{\zeta}}\Big{)}\frac{{\mathrm{d}}{\zeta}}{{\zeta}}\quad\text{for}\enspace\xi\;\text{close
enough to}\;\gamma(0).$ (2.6)

To prove Theorem A, we just need to prove that $f\odot g$ admits analytic
continuation along $\gamma$. Without loss of generality, we may suppose that
$\gamma$ is $C^{1}$ with a derivative $\gamma^{\prime}$ that is Lipschitz.

2.3 We now make precise the notion of deformation of the identity that we have
alluded to in the introduction:

######  Definition 2.2.

We call “deformation of the identity adapted to $\gamma$” any family
$(\Psi_{t})_{t\in I}$ of Lipschitz homeomorphisms of $\mathbb{C}$ such that
$\Psi_{0}=\mathop{\hbox{{\rm Id}}}\nolimits_{\mathbb{C}}$ and

$\displaystyle\alpha\in\\{0\\}\cup A^{\prime}$
$\displaystyle\quad\Longrightarrow\quad\quad\enspace\Psi_{t}(\alpha)=\alpha\qquad\text{for
all}\enspace t\in I,$ (2.7) $\displaystyle\beta\in B^{\prime}\quad$
$\displaystyle\quad\Longrightarrow\quad\Psi_{t}\Big{(}\frac{\gamma(0)}{\beta}\Big{)}=\frac{\gamma(t)}{\beta}\quad\text{for
all}\enspace t\in I.$ (2.8)

######  Lemma 2.3.

Suppose we have found $(\Psi_{t})_{t\in I}$, a deformation of the identity
adapted to $\gamma$, then the analytic continuation of $f\odot g$ along
$\gamma$ exists and is given, at each $\gamma(t)$, $t\in I$, by the formula

$\operatorname{cont}_{\gamma\mid t}(f\odot
g)(\xi)=\frac{1}{2\pi{\mathrm{i}}}\oint_{\Psi_{t}(C_{\rho})}f({\zeta})g\Big{(}\frac{\xi}{{\zeta}}\Big{)}\frac{{\mathrm{d}}{\zeta}}{{\zeta}}\quad\text{for}\enspace\xi\;\text{close
enough to}\;\gamma(t).$ (2.9)

###### Proof.

Since each $\Psi_{t}$ is a homeomorphism of $\mathbb{C}$, condition (2.7)
entails $\Psi_{t}^{-1}(\alpha)=\alpha$ for $\alpha\in A$, whence

${\zeta}\in\mathbb{C}\setminus
A\quad\Longrightarrow\quad\Psi_{t}({\zeta})\in\mathbb{C}\setminus
A\quad\text{for all}\enspace t\in I.$ (2.10)

We also have $\Psi_{t}^{-1}(0)=0$ and, by condition (2.8),
$\Psi_{t}^{-1}\big{(}\frac{\gamma(t)}{\beta}\big{)}=\frac{\gamma(0)}{\beta}$
for all $\beta\in B^{\prime}$, whence

${\zeta}\neq
0\enspace\text{and}\enspace\frac{\gamma(0)}{{\zeta}}\in\mathbb{C}\setminus
B^{\prime}\quad\Longrightarrow\quad{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Psi_{t}({\zeta})\neq
0\enspace\text{and}\enspace}\frac{\gamma(t)}{\Psi_{t}({\zeta})}\in\mathbb{C}\setminus
B^{\prime}\quad\text{for all}\enspace t\in I$ (2.11)

(indeed: $\frac{\gamma(t)}{\Psi_{t}({\zeta})}=\beta\in B^{\prime}$ would
require
${\zeta}=\Psi_{t}^{-1}\big{(}\frac{\gamma(t)}{\beta}\big{)}=\frac{\gamma(0)}{\beta}$).
Therefore, since by (2.5) all the points of $C_{\rho}$ satisfy the
requirements indicated in the left-hand sides of (2.10)–(2.11), the integral
in the right-hand side of (2.9) is a well-defined function $h_{t}(\xi)$ of
$\xi$, which represents a holomorphic germ at $\gamma(t)$.

The Cauchy theorem entails that, for $t<t^{\prime}$ close enough, the
functions $h_{t}$ and $h_{t^{\prime}}$ coincide in a neighbourhood of
$\gamma_{\mid[t,t^{\prime}]}$, hence $h_{t}$ is the analytic continuation
along $\gamma$ at $\gamma(t)$ of $h_{0}$, and $h_{0}=f\odot g$ by (2.6). ∎

##  3 Construction of a non-autonomous vector field

3.1 To conclude the proof of Theorem A, it is thus sufficient to find a
deformation of the identity in the sense of Definition 2.2. Our tool will be
the flow map induced by a non-autonomous vector field.

######  Lemma 3.1.

Suppose $X\colon\thinspace I\times\mathbb{C}\to\mathbb{C}$ is locally
Lipschitz (identifying $\mathbb{C}$ and $\mathbb{R}^{2}$) and satisfies

$\lvert X(t,{\zeta})\rvert\leq K\lvert{\zeta}\rvert$ (3.1)

with some $K>0$. Then, for each $t^{*}\in I$ and each initial condition
${\zeta}_{0}\in\mathbb{C}$, the non-autonomous vector field

$\frac{{\mathrm{d}}{\zeta}}{{\mathrm{d}}t}=X(t,{\zeta})$ (3.2)

has a unique maximal solution $t\mapsto\Phi^{t^{*},t}({\zeta}_{0})$ such that
$\Phi^{t^{*},t^{*}}({\zeta}_{0})={\zeta}_{0}$. This maximal solution is
defined for all $t\in I$.

Moreover, the map $(t^{*},t,{\zeta})\in I\times
I\times\mathbb{C}\mapsto\Phi^{t^{*},t}({\zeta})\in\mathbb{C}$ thus defined is
locally Lipschitz, and $(\Phi^{t^{*},t})_{t^{*},t\in I}$ is a family of
locally Lipschitz homeomorphisms of $\mathbb{C}$ satisfying

$\Phi^{t,t^{**}}\circ\Phi^{t^{*},t}=\Phi^{t^{*},t^{**}},\qquad\Phi^{t^{*},t^{*}}=\mathop{\hbox{{\rm
Id}}}\nolimits_{\mathbb{C}}.$ (3.3)

###### Proof.

Apply the classical Cauchy-Lipschitz theorem about the existence and
uniqueness of solutions to systems of ordinary differential equations and
their regular dependence upon the initial condition. The maximal solutions are
defined for all times because of the bound (3.1). ∎

######  Corollary 3.2.

Suppose that we have found $c\colon\thinspace I\times\mathbb{C}\to\mathbb{C}$
locally Lipschitz, bounded by $K\lvert{\zeta}\rvert$ with some $K>0$, such
that

$\displaystyle\alpha\in\\{0\\}\cup A^{\prime}$
$\displaystyle\quad\Longrightarrow\quad\quad\enspace
c(t,\alpha)=0\qquad\text{for all}\enspace t\in I,$ (3.4)
$\displaystyle\beta\in B^{\prime}\quad$
$\displaystyle\quad\Longrightarrow\quad
c\Big{(}t,\frac{\gamma(t)}{\beta}\Big{)}=\frac{1}{\beta}\quad\text{for
all}\enspace t\in I.$ (3.5)

Then the flow map $\Psi_{t}\coloneqq\Phi^{0,t}$ of the non-autonomous vector
field

$X(t,z)\coloneqq c(t,z)\,\gamma^{\prime}(t)$ (3.6)

gives rise to a deformation of the identity adapted to $\gamma$.

###### Proof.

We have $\Psi_{t}(\alpha)=\Phi^{0,t}(\alpha)=\alpha$ for all $\alpha\in A$ by
(3.4), and (3.5) shows that $t\mapsto\frac{\gamma(t)}{\beta}$ is a particular
solution of $X$ for each $\beta\in B^{\prime}$, hence
$\frac{\gamma(t)}{\beta}=\Phi^{0,t}\big{(}\frac{\gamma(0)}{\beta}\big{)}=\Psi_{t}\big{(}\frac{\gamma(0)}{\beta}\big{)}$.
∎

3.2 Thus, we are just left with the question of finding a function $c$ meeting
the requirements of Corollary 3.2. Since $\Omega$ is closed and $\gamma(I)$ is
compact, we can pick $\delta,M>0$ so that, for each $t\in I$,

$\delta\leq\lvert\gamma(t)\rvert\leq
M,\qquad\lvert\gamma(t)-\alpha\beta\rvert\geq\delta\quad\text{for
all}\enspace(\alpha,\beta)\in A\times B.$ (3.7)

######  Lemma 3.3.

If ${\varepsilon}>0$ is chosen small enough so that

${\varepsilon}<\frac{a}{2},\qquad\frac{a\delta}{{\varepsilon}}>M+\frac{2M}{a}{\varepsilon},$
(3.8)

then

$\beta\in
B^{\prime}\quad\Longrightarrow\quad\operatorname{dist}\Big{(}\frac{\gamma(t)}{\beta},A^{\prime}\Big{)}\geq{\varepsilon}\quad\text{for
all}\enspace t\in I.$ (3.9)

###### Proof.

Suppose $\lvert\frac{\gamma(t)}{\beta}-\alpha\rvert<{\varepsilon}$ with
$\alpha\in A^{\prime}$. We would have

$\delta\leq\lvert\gamma(t)-\alpha\beta\rvert<{\varepsilon}\lvert\beta\rvert$
(3.10)

by (3.7b). We also have $\lvert\alpha\beta\rvert\geq a\lvert\beta\rvert$ by
(2.1) and $\lvert\gamma(t)\rvert\leq M$ by (3.7a), whence

$a\lvert\beta\rvert\leq{\varepsilon}\lvert\beta\rvert+M$ (3.11)

by the triangle inquality. Now, either $\lvert\beta\rvert\geq\frac{2M}{a}$,
but then
$a\lvert\beta\rvert\leq{\varepsilon}\lvert\beta\rvert+M\leq({\varepsilon}+\frac{a}{2})\lvert\beta\rvert$
leads to a contradiction with (3.8a); or $\lvert\beta\rvert<\frac{2M}{a}$, but
then $\frac{a\delta}{{\varepsilon}}<a\lvert\beta\rvert$ by (3.10), which is
$\leq M+\lvert\beta\rvert{\varepsilon}$ by (3.11), whence the inequality
$\frac{a\delta}{{\varepsilon}}<M+\frac{2M}{a}{\varepsilon}$, which contradicts
(3.8b). ∎

Let us choose a Lipschitz function
$\eta=\eta_{{\varepsilon},A^{\prime}}\colon\thinspace\mathbb{C}\to[0,1]$ such
that

$\alpha\in
A^{\prime}\enspace\Longrightarrow\enspace\eta(\alpha)=0,\qquad\operatorname{dist}({\zeta},A^{\prime})\geq{\varepsilon}\enspace\Longrightarrow\enspace\eta({\zeta})=1.$
(3.12)

For instance, we may pick a Lipschitz function
$\chi_{\varepsilon}\colon\thinspace\mathbb{R}_{\geq 0}\to[0,1]$ such that
$\chi_{\varepsilon}(0)=0$ and

$d\geq{\varepsilon}\enspace\Longrightarrow\enspace\chi_{\varepsilon}(d)=1,$

e.g.
$\chi_{\varepsilon}(d)\coloneqq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\min}\big{\\{}\frac{d}{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\varepsilon}}},1\big{\\}}$,
and define
$\eta({\zeta})\coloneqq\chi_{\varepsilon}\big{(}\operatorname{dist}({\zeta},A^{\prime})\big{)}$.

We conclude by checking that the function defined by

$c(t,{\zeta})\coloneqq\eta({\zeta})\frac{{\zeta}}{\gamma(t)}$ (3.13)

fulfils all the requirements of Corollary 3.2:

  1. –

The function $c$ is locally Lipschitz and bounded by
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{1}{\delta}}\lvert{\zeta}\rvert$,
by virtue of (3.7a), because $\lvert\eta\rvert\leq 1$.

  2. –

We have $c(t,0)=0$.

  3. –

If $\alpha\in A^{\prime}$, then $c(t,\alpha)=0$ because (3.12a) says that
$\eta(\alpha)=0$.

  4. –

If $\beta\in B^{\prime}$, then (3.9) says that
$\operatorname{dist}\big{(}\frac{\gamma(t)}{\beta},A^{\prime}\big{)}\geq{\varepsilon}$,
hence
$c\big{(}t,\frac{\gamma(t)}{\beta}\big{)}=\eta\big{(}\frac{\gamma(t)}{\beta}\big{)}\frac{1}{\beta}=\frac{1}{\beta}$
by (3.12b).

Therefore, the proof of Theorem A is complete.

3.3 Remark. When $B$ is a finite set, the construction of § 3.2.2 can be
replaced by one closer to that of [Sau13], namely:

$c(t,{\zeta})\coloneqq\sum_{\beta_{1}\in
B^{\prime}}\,\,\frac{\eta_{\beta_{1}}({\zeta},t)}{\eta_{\beta_{1}}({\zeta},t)+\lvert{\zeta}-\frac{\gamma(t)}{\beta_{1}}\rvert}\cdot\frac{1}{\beta_{1}},\qquad\text{with}\quad\eta_{\beta_{1}}({\zeta},t)\coloneqq\operatorname{dist}\big{(}{\zeta},\\{0\\}\cup
A^{\prime}\cup B(\beta_{1},t)\big{)},$ (3.14)

where
$B(\beta_{1},t)\coloneqq\big{\\{}\frac{\gamma(t)}{\beta_{2}}\mid\beta_{2}\in
B^{\prime}\setminus\\{\beta_{1}\\}\big{\\}}$. Indeed, the denominators are all
always $>0$ precisely because $\gamma(I)$ and $\\{0\\}\cup(A^{\prime}\cdot
B^{\prime})$ do not intersect.

### Acknowledgements

The second author thanks Capital Normal University for their hospitality
during the period September 2019–February 2020, where this research was
started. The third author is partially supported by NSFC (No.s 11771303,
11911530092, 11871045).

This paper is partly a result of the ERC-SyG project, Recursive and Exact New
Quantum Theory (ReNewQuantum) which received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme under grant agreement No 810573.

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##  Appendix A Additional Figures

Figure 3 shows an overview of our method (Section 3), and Figure 4 illustrates
an example of our Argmin weighting (Section 3.2).

Figure 5 shows the average prediction of each expert, calculated across MNLI
ID dev (Section 4.3). We observe a significant variance between experts’
predictions, which indicates that different experts tend to make different
predictions.

Figure 3:  Overview of our method. We fit training data using a mixture model
consisting of $K$ expert networks $\\{E^{k}\\}_{k=1}^{K}$ and a router network
$\pi$ (Section 3.1). During inference, the model is used as is for ID data,
and $\pi$ is replaced with $\pi^{*}$ for OOD data (Section 3.2).  Figure 4:
An example of decision-making with argmin weighting, where $K=2$ and
$\mathcal{|Y|}=3$. After performing argmin weighting, label 2 achieves the
highest score (starred) and is thus chosen as the answer.  Figure 5:  The
average prediction of each expert $E^{k}$ across MNLI ID dev.

##  Appendix B Further Details of Setup

Table 5 specifies the URLs of the datasets, pre-trained models, and codes of
the previous methods we introduced in Section 4.1.

Following the fine-tuning hyperparameters of DeBERTa${}_{\text{v3-large}}$ (He
et al., 2023), we set the learning rate to 5e-6 and used gradient clipping
with the maximum gradient norm of 1.0 in the ablation study with
DeBERTa${}_{\text{v3-large}}$ (Section 4.3).

Datasets MNLI https://cims.nyu.edu/~sbowman/multinli/ HANS
https://github.com/tommccoy1/hans QQP and PAWS https://github.com/google-
research-datasets/paws FEVER and Symm. v1/v2
https://github.com/TalSchuster/FeverSymmetric Pre-Trained Models BERT
https://huggingface.co/bert-base-uncased DeBERTa${}_{\text{v3-large}}$
https://huggingface.co/microsoft/deberta-v3-large Codes Conf-reg
$\spadesuit~{}_{\text{self-debias}}$∗
https://github.com/UKPLab/emnlp2020-debiasing-unknown JTT∗
https://github.com/YyzHarry/SubpopBench RISK
https://github.com/CuteyThyme/RISK EIIL https://github.com/PluviophileYU/BAI
BAI https://github.com/PluviophileYU/BAI GroupDRO${}_{\text{label-group}}$∗
https://github.com/YyzHarry/SubpopBench ReWeightCRT∗
https://github.com/YyzHarry/SubpopBench

Table 5:  URLs of the datasets, pre-trained models, and codes of the previous
methods we used in the experiments. The methods with ∗ needed modification on
the codes to cover all the datasets we used.