Linxiao Chen
email: linxiao.chen@math.ethz.ch

A transfer theorem for multivariate ∆-analytic functions with a power-law singularity

This paper presents a multivariate generalization of Flajolet and Odlyzko's transfer theorem. Similarly to the univariate version, the theorem assumes ∆-analyticity (defined coordinate-wise) of a function A(z 1 , . . . , z d ) at a unique dominant singularity (ρ 1 , . . . , ρ d ) ∈ (C * ) d , and allows one to translate, on a term-by-term basis, an asymptotic expansion of A(z 1 , . . . , z d ) around (ρ 1 , . . . , ρ d ) into a corresponding asymptotic expansion of its Taylor coefficients a n1,...,n d . We treat the case where the asymptotic expansion of A(z 1 , . . . , z d ) contains only power-law type terms, and where the indices n 1 , . . . , n d tend to infinity in some polynomially stretched diagonal limit. The resulting asymptotic expansion of a n1,...,n d is a sum of terms of the form

where (λ 1 , . . . , λ d ) ∈ (0, ∞) d is the direction vector of the stretched diagonal limit for (n 1 , . . . , n d ), the parameter n 0 tends to ∞ at similar speed as n 1 , . . . , n d , while Θ ∈ R and I : (0, ∞) d → C are determined by the asymptotic expansion of A.

Introduction

Univariate transfer theorems. In their pioneering work [START_REF] Flajolet | Singularity analysis of generating functions[END_REF], Flajolet and Odlyzko developped a method, known as the transfer theorems, that allows one to compute a precise asymptotic expansion of a sequence a n as n → ∞, from an asymptotic expansion of its generating function A(z) = n≥0 a n z n around some singularity ρ ∈ C * ≡ C\{0}. We use the notation [z n ]A(z) for the coefficient a n = 1 n! A (n) (0). One distinctive feature of the transfer theorem is that it applies to generating functions that are ∆analytic, that is, analytic on a ∆-domain of the form ρ • ∆ δ ≡ {ρz | z ∈ ∆ δ } for some δ > 0, where

∆ δ := z ∈ C : |z| < 1 + δ, z = 1 and arg(1 -z) ∈ (-π 2 -δ, π 2 + δ) . (1)
See Figure 1(a). (Simple extensions of the transfer theorem also apply to functions analytic in a finite intersection of ∆-domains of the form (ρ

1 •∆ δ )∩ • • • ∩(ρ n •∆ δ
), but we shall not discuss that further here.) Under the ∆-analyticity assumption, ρ is the unique dominant singularity of the function A(z). By the change of variable z = z/ρ, we can bring this singularity to z = 1. In the following, we will assume without loss of generality that ρ = 1.

In its most general form, Flajolet and Odlyzko's transfer theorem applies to functions whose asymptotic expansion is composed of any regular varying functions taken from a large class of "standard scale functions", such as f (z) = (1 -z) α (log(1 -z)) β for α, β ∈ R. Here we focus on functions whose asymptotic expansion contains only power-law terms of the form (1 -z) α . In this simple case, the transfer theorem can be formulated as follows.

Theorem A (Univariate transfer theorem). Let α ∈ R and let A(z) be an analytic function on ∆ δ for some δ > 0.

(Analytic terms have exponentially small contribution)

If A(z) is also analytic at 1, then there exists r ∈ (0, 1) such that [z n ]A(z) = O(r n ) when n → ∞. (In this case, z = 1 is actually not the dominant singularity of A(z).)

(Coefficient asymptotics of power functions)

When A(z) = (1 -z) α with α ∈ R, we have the following asymptotic expansion as n → ∞,

[z n ](1 -z) α = n -α-1 Γ(-α) ∞ k=0 e k (α) n k ≡ n -α-1 Γ(-α) 1 + e 1 (α) n + e 2 (α) n 2 + • • • (2)
where e k (α) ∈ Q[α] is a polynomial of degree 2k in α. More precisely, e k (α) = k l=0 g k,l

Γ(-α) Γ(-α-k-l) , where g k,l is defined by the Taylor expansion G(x, y) := exp -y -

y x + 1 log(1 -x) = ∞ k=0 k l=0
g k,l y l x k .

(3)

(Big-O transfer and little-o transfer)

If

A(z) = O((1 -z) α ) when z → 1 in ∆ δ , then [z n ]A(z) = O(n -α-1 ) when n → ∞. If A(z) = o ((1 -z) α ) when z → 1 in ∆ δ , then [z n ]A(z) = o (n -α-1 ) when n → ∞.
In practice, the transfer theorem is usually applied to functions A(z) which are linear combinations of the three cases above, as in the following statement.

Theorem A (Univariate transfer theorem, integrated form). Let A(z) be an analytic function on ∆ δ for some δ > 0 such that when z → 1 in ∆ δ , we have

A(z) = A reg (z) + h 0 (1 -z) α 0 + • • • h m (1 -z) αm + o ((1 -z) αm ) , (4)
where A reg (z) is defined and analytic in a neighborhood of 1 in C, and h j ∈ C, α j ∈ R for all 0 ≤ j ≤ m.

Then, the coefficients of A(z) has the following expansion when n → ∞

[z n ]A(z) = h 0 • [z n ](1 -z) α 0 + • • • + h m • [z n ](1 -z) αm + o(n -αm-1 ) (5)
where each term [z n ](1 -z) α j has the asymptotic expansion given in Theorem A [START_REF] Casas-Alvero | Singularities of plane curves[END_REF].

The same result holds if the little-o estimates in both (4) and (5) are replaced by big-O estimates.

Remark.

1. Theorem A and Theorem A are not exactly equivalent, for the following reasons: When A(z) is analytic at 1, Theorem A asserts that [z n ]A(z) decays exponentially as n → ∞, while Theorem A only implies a super-polynomial decay. On the other hand, one cannot prove Theorem A by simply applying Theorem A to each term of the expansion (4), because the terms A reg (z) and o((1 -α) αm ) in ( 4) have a priori no analytic continuations on the whole domain ∆ δ .

2. For a sequence of functions s k (n), the asymptotic expansion S(n) = ∞ k=0 s k (n) ususally means that for all m ≥ 0, we have S(n) = s 0 (n)+• • •+s m-1 (n)+O(s m (n)) as n → ∞. Theorem A(2) uses a slightly modified notion of asymptotic expansion: we choose implicitly the family s β (n) = n -β (β ∈ R) as the reference asymptotic scale, and view each asymptotic expansion as a weighted sum of the form S(n) = ∞ k=0 c k • n -β k with some β 0 < β 1 < • • • . The difference is that now the prefactors c k may vanish for some or even all k ≥ 0, and the expansion should be read as: for all m ≥ 0, we have

S(n) = c 0 n -β 0 + • • • + c m-1 n -β m-1 + O(n -βm ) as n → ∞.
3. It is also possible to include α ∈ C \ R in Theorem A and theorem A . The asymptotic expansion

(2) would hold unmodified. But complex exponents α rarely appear in applications and they complicate the asymptotic scale s β (n) = n -β (since C is not totally ordered). We restrict ourselves to α ∈ R for the sake of simplicity.

4. Theorem A(1)-( 2) together imply that the prefactor 1 Γ(-α) in (2) must vanish when α ∈ N. Indeed, α → 1 Γ(-α) is an entire function and we have 1 Γ(-α) = 0 if and only if α ∈ N. The transfer theorems apply to a large class of naturally occuring generating functions. For example, it is well known that all D-finite functions analytic at 0 are linear combinations of ∆-analytic functions. Among them, the algebraic functions always have singularities of power-law type, to which Theorem A applies. Compared to alternative methods such as Darboux's method or the Tauberian theorems, the transfer theorem has the advantage of giving a transparent correspondance between the asymptotic expansion of the generating function and that of its coefficients. See [START_REF] Flajolet | Analytic combinatorics[END_REF]Sec. 5] for a detailed discussion about this comparison.

In this paper, we present a generalization of the transfer theorem to the multivariate setting. Below, we start by recalling some basic notations about multivariate generating functions, and then define the regimes of coefficient asymptotics with which our transfer theorem will be concerned.

Multivariate generating functions. In many practical problems, the relevant information is captured naturally by a multidimensional infinite array (a n ) n∈N d ∈ C N d . Such multidimensional arrays and the multivariate generating functions which encode them will be this paper's central objects. To make the formulas compact, we will use the following multi-index notations: For any formal or complex vectors z = (z 1 , . . . , z d ) and θ = (θ 1 , . . . , θ d ), we denote

θz = (θ 1 z 1 , . . . , θ d z d ) , θ • z = θ 1 z 1 + • • • + θ d z d , z θ = z θ 1 1 • • • z θ d d , dz = dz 1 • • • dz d . (6)
And, for any scalar σ and integer vector m ∈ N d , let

σ θ = (σ θ 1 , . . . , σ θ d ) and ∂ m = ∂ m 1 1 • • • ∂ m d d ≡ ∂ m 1 ∂z 1 m 1 • • • ∂ m d ∂z d m d . (7)
With these notations, the multivariate generating function of the array (a n ) n∈N d can be written as

A(z) = n∈N d a n z n ≡ ∞ n 1 =0 • • • ∞ n d =0 a n 1 ,...,n d z n 1 1 • • • z n d d . (8)
Similarly to the univariate case, we denote the coefficient a n by [z n ]A(z). We assume that every generating function in this paper is absolutely convergent in an open neighborhood of 0 ≡ (0, . . . , 0) ∈ C d , so that it defines an analytic function there. We refer to [START_REF] Hörmander | The analysis of linear partial differential operators. I. Springer Study Edition[END_REF] for the general theory on power series and analytic functions in several variables. One particular fact that we will use without further mention is the uniqueness of analytic continuation: if A and B are analytic functions on an open connected domain Ω ⊆ C d and A(z) = B(z) on any open subset of Ω, then A(z) = B(z) for all z ∈ Ω.

Stretched diagonal limits. One central problem of analytic combinatorics in several variables is to understand the asymptotics of the coefficients [z n ]A(z) when the components n 1 , . . . , n d of the multi-index tend to ∞ simultaneously. In general, one needs to put some constraint on the relative speeds at which n 1 , . . . , n d grow in order to get a useful asymptotic formula, and the interesting regimes are to a large extent dictated by the structure of the singularities of A(z).

In this work, we will be interested in the stretched diagonal limits, where n 1 , . . . , n d grow at polynomial speeds relative to each other. In other words, we require that for each j = 2, . . . , d, there exists a constant θ j > 0 such that the ratio n j /n θ j 1 remains in some compact interval I ⊂ R >0 ≡ (0, ∞) when n 1 → ∞. A symmetrized definition of this limit regime goes as follows:

Definition (Stretched diagonal limit). We say that the multi-index n ∈ N d tends to ∞ in the stretched diagonal regime if there exist an exponent vector θ ∈ R d >0 and an auxiliary variable n 0 > 0, such that

n = λn θ 0 ≡ λ 1 n θ 1 0 , . . . , λ d n θ d 0 ( 9
)
for some prefactor λ = λ(n 0 ) that remains in a compact set K ⊂ R d >0 when n 0 → ∞. In this case, we will also say that n → ∞ in the θ-diagonal limit.

Remark. If we require in addition that λ(n 0 ) → λ * ∈ R d >0 as n 0 → ∞, then the point n = λn θ 0 would tend to ∞ roughly along the curve D λ * ,θ = (λ * t θ , t > 0). But we only require λ(n 0 ) to stay in some compact set. Intuitively this means that n can jump between the curves D λ * ,θ for different values of λ * . Consequently, the asymptotics that we write in the stretched diagonal limit should be understood as uniform with respect to λ * on every compact subset of R d >0 . Notice that for any τ > 0, the (τ θ)-diagonal limit is the same as the θ-diagonal limit: it suffices to replace the variable n 0 by n τ 0 or n 1/τ 0 to pass from one to the other. The classical notion of diagonal limit, e.g. as defined in [START_REF] Pemantle | Analytic combinatorics in several variables[END_REF], corresponds to the (1, • • • , 1)-diagonal limit in our terminology. Apart from being the relevant limit regime for multivariate ∆-analytic function with power-law singularity (to be defined below), the stretch diagonal limits also arise naturally from the study of critical phenomena in probability and mathematical physics.

Preliminary definitions. Let us define some domains and function classes needed for stating the main theorems. Let C * = C \ {0} and R >0 = (0, ∞). For δ > 0, we write

K δ = {u ∈ C * : | arg(u)| < δ} and denote Ω δ = K π/2+δ
. The multivariate versions of these cones are denoted

K δ = K d δ and Ω δ = Ω d δ . For > 0, w ∈ C and w ∈ C d , let B (w) = {z ∈ C : |z -w| < } and B (w) = B (w 1 ) × • • • × B (w d ). Notice that Ω δ is related to the ∆-domain ∆ δ by Ω δ = {µ(1 -z) | µ > 0 and z ∈ ∆ δ }. Definition (Multivariate ∆-analytic functions). Let ρ ∈ C d * .
We say that a multivariate function A(z) is ∆-analytic at ρ if it has an analytic continuation on the product domain

ρ∆ δ := (ρ 1 ∆ δ ) × • • • × (ρ d ∆ δ )
for some δ > 0, where ∆ δ is the univariate ∆-domain defined in [START_REF] Bochnak | Real algebraic geometry[END_REF].

Like in the univariate case, one can make the change of variable z = z/ρ ≡ (z 1 /ρ 1 , . . . , z d /ρ d ) to bring the point ρ to 1 ≡ (1, . . . , 1). In the following, we will focus without loss of generality on functions which are ∆-analytic at 1.

Definition (Demi-analytic functions). Let

Ω = Ω 1 × • • • × Ω d be an open product domain in C d and let ρ ∈ (∂Ω 1 ) × • • • × (∂Ω d ). For j ∈ {1, . . . , d}, denote Ω ĵ = Ω 1 × • • • × Ω j-1 × C × Ω j+1 × • • • × Ω d .
We say that a function A : Ω → C is demi-analytic at ρ ∈ Ω if A is analytic on Ω and there exist > 0 and a decomposition

A = A 1 + • • • + A d such that A j is analytic in B (ρ) ∩ Ω ĵ for each j ∈ {1, . . . , d}.
If each term A j in the above decomposition is analytic on Ω ĵ , then we say that A is demi-entire.

Definition (Generalized homogeneous functions). Let K be a cone in C d , i.e. a subset of C d such that {σz | z ∈ K} = K for all σ > 0. For θ 0 ∈ R and θ ∈ R d >0 , we say that a function H :

K → C is (θ 0 , θ)-homogeneous if H(σ θ u) ≡ H(σ θ 1 u 1 , . . . , σ θ d u d ) = σ θ 0 H(u) (10)
for all u ∈ K and σ > 0.

It is clear that a (θ 0 , θ)-homogeneous function is also (τ θ 0 , τ θ)-homogeneous for all τ > 0. The classical notion of homogeneous functions of degree D becomes (D, 1)-homogeneous in our terminology.

The three definitions above generalize respectively the notions of ∆-analytic functions, locally analytic functions (for the term A reg in Theorem A ), and power functions (for the term h • (1 -z) α ) used in the univariate transfer theorems. To state the multivariate transfer theorem, we need one more definition whose counterpart does not appear explicitly in the univariate setting: Definition (Functions of polynomial type). For any δ > 0, we say that a function F is of polynomial type (globally) on K δ if there exist C, M > 0 such that

∀u ∈ K δ , |F (u)| ≤ C • |u 1 | -M + |u 1 | M + • • • + |u d | -M + |u d | M (11)
We say that F is of polynomial type locally at 0 ∈ K δ if the above bound only holds in a neighborhood of 0 ∈ K δ . Equivalently, F is of polynomial type locally at 0 ∈ K δ if there exist , C, M > 0 such that

∀u ∈ K δ ∩ B 0, , |F (u)| ≤ C • |u 1 | -M + • • • + |u d | -M . (12)
Similarly, we say that a function A is of polynomial type on ∆ δ if there exist C, M > 0 such that

∀z ∈ ∆ δ , |A(z)| ≤ C • |z 1 -1| -M + • • • + |z d -1| -M (13)
(here we do not need the terms |z j -1| M on the right hand side because ∆ δ is bounded), and we say that A is of polynomial type locally at 1 ∈ ∆ δ if it satisfies the above bound in ∆ δ ∩ B 1, for some > 0.

For S = K δ , Ω δ or ∆ δ , we write P(S) = {f : S → C | f is analytic and of polynomial type on S}.

Up to decreasing δ, a continuous function on ∆ δ is always bounded by a constant on (∆ δ \ U ) d for any neighborhood U of 1 ∈ C. Thus, the condition that A is of polynomial type (globally) on ∆ δ is essentially an upper bound for |A(z)| when z j → 1 in ∆ δ for some j ∈ {1, . . . , d}. In contrast, the condition of being of polynomial type locally at 1 ∈ ∆ δ only gives a bound for |A(z)| when z j → 1 for all j ∈ {1, . . . , d}. When d = 1, the two conditions are essentially the same. They do not appear explicitly in the statement of the univariarte transfer theorem because a function having an asymptotic expansion of the form (4) is automatically of polynomial type on ∆ δ .

Main results. We are now ready to state the multivariate transfer theorem. Like in the univariate case, we formulate it in two ways, one discussing the building blocks of the coefficient asymptotics piece by piece, and the other showing how the theorem would be applied in practice.

Theorem 1 (Multivariate transfer theorem). Let θ 0 ∈ R, θ ∈ R d >0 and A ∈ P(∆ δ ) for some δ > 0. We consider the asymptotics of the coefficients [z n ]A(z) when n → ∞ in the θ-diagonal regime.

(Demi-analytic terms have exponentially small contribution)

If A is demi-analytic at 1 ∈ ∆ δ , then there exist r ∈ (0, 1) and σ > 0 such that [z n ]A(z) = O(r n σ 0 ).

(Coefficient asymptotics of generalized homogeneous functions)

When A(z) = H(1 -z) and H is (θ 0 , θ)-homogeneous (θ 0 ∈ R), we have the asymptotic expansion

[z n ]H(1 -z) = 1 n Θ 0 k∈N d D k I(λ) n k•θ 0 ≡ 1 n Θ 0   I(λ) + d j=1 ∂ j I(λ) + 1 2 λ j ∂ 2 j I(λ) n θ j 0 + • • •   . ( 14
)
where

Θ = θ 0 + θ 1 + • • • + θ d , D k = l≤k g k,l • λ l ∂ k+l is a partial differential operator of order 2(k 1 + • • • + k d )
, and I : R d >0 → C is the inverse Laplace transform of H defined by

I(λ) = 1 2πi d ˆVδ H(u)e λ•u du . ( 15
)
In the above formulas, we denote

g k,l = g k 1 ,l 1 • • • g k d ,l d , with the numbers (g k,l
) k,l≥0 defined by (3).

The sum l≤k runs over l ∈ N d l j ≤ k j for all 1 ≤ j ≤ d , while the integral is over V δ = V d δ , where V δ ⊂ Ω δ is any piecewise smooth curve which coincides with the rays ∂Ω δ outside a bounded region for some δ ∈ (0, δ), e.g. as in Figure 1

(b). (Recall that Ω δ = u ∈ C * : | arg(u)| < π 2 + δ .) 3. (Big-O transfer and little-o transfer) If A(z) = O( H(1-z)) for some (θ 0 , θ)-homogeneous H as z → 1 in ∆ δ , then [z n ]A(z) = O(n -Θ 0 ). If A(z) = o ( H(1-z)) for some (θ 0 , θ)-homogeneous H as z → 1 in ∆ δ , then [z n ]A(z) = o (n -Θ 0 ).
Theorem 1 (Multivariate transfer theorem). Let θ ∈ R d >0 and A ∈ P(∆ δ ) for some δ > 0. Assume that when z → 1 in ∆ δ , the function A has an asymptotic expansion of the form

A(z) = A reg (z) + H 0 (1 -z) + • • • + H m-1 (1 -z) + o (H m (1 -z)) , (16)
where A reg is demi-analytic at 1 ∈ ∆ δ , and each H k is (θ (k) 0 , θ)-homogeneous and of polynomial type locally at 0 ∈ Ω δ , with θ

(0) 0 > • • • > θ (m-1) 0 ≥ θ (m)
0 . Then as n → ∞ in the θ-diagonal limit, we have

[z n ]A(z) = [z n ]H 0 (1 -z) + • • • + [z n ]H m-1 (1 -z) + o(n -Θm 0 ) ( 17
)
where each [z n ]H k (1-z) has the asymptotic expansion given in Theorem 1 [START_REF] Casas-Alvero | Singularities of plane curves[END_REF], Θ m = θ Remark. The following remarks are analoguous to their univariate counterparts below Theorem A .

1. Theorems 1 and 1 are not exactly equivalent, for the same reason as in the univariate case.

2. The asymptotic expansion (14) in Theorem 1(2) uses the same modified notion of asymptotic expansion as in Theorem A(2), with the asymptotic scale

s β (n 0 ) = n -β 0 (β ∈ R). 3.
It is also possible to include θ 0 ∈ C \ R in Theorems 1 and 1 . The expansion (14) would still hold with a complex value for Θ. But we restrict ourselves to θ 0 ∈ R for simplicity.

4. Theorem 1( 1)-( 2) imply that for a (θ 0 , θ)-homogeneous function

H ∈ P(Ω δ ), if H is demi-analytic at 0 ∈ Ω δ , then its inverse Laplace transform I(λ) vanishes for all λ ∈ R d >0 .
We will see in Corollary 3 below that the converse is also true.

Let us also make some remarks about the expansion (16) in Theorem 1 . These will be discussed in more details in Section 4. 16) is in general not unique. The reason is that a demi-analytic function can also contain (θ 0 , θ)-homogeneous components for any θ 0 ∈ R. In principle, one could fix A reg = 0 in Theorem 1 without reducing significantly the class of functions A(z) covered by the theorem. But having the flexibility of choosing any demi-analytic function A reg makes the theorem easier to apply.

The expansion (

6. Once A reg is chosen, one can write A(z) = A reg (z) + A sing (1 -z). If the expansion (16) exists, then its terms H k can be obtained as the coefficients in the univariate asymptotic expansion

A sing (ε θ u) = H 0 (u) • ε θ (0) 0 + • • • + H m-1 (u) • ε θ (m-1) 0 + o(ε θ (m) 0 ) with respect to ε → 0 + . 7. Although not assumed in Theorem 1 , the functions z → H k (1 -z) (0 ≤ k < m) are necessarily analytic on ∆ δ . See Lemma 4 below. In particular, the coefficients [z n ]H k (1 -z) are well-defined.
8. Unlike the univariate case, the generalized homogenous function H m used in the little-o estimate is in general different from the previous term H m-1 of the asymptotic expansion. The reason is that the class of (θ 0 , θ)-homogeneous functions is one-dimensional (generated by H(u) = u θ 0 /θ ) in the univariate case, but infinite-dimensional in the multivariate case.

In many applications, one is only interested in the dominant asymptotics of the coefficients. For this, we can simplfy Theorems 1 and 1 to the following statement: for any generating function A ∈ P(∆ δ ),

A(z) = A reg (z) + H(1 -z) + o( H(1 -z)) ⇒ [z n ]A(z) ∼ I(λ) • n -Θ 0 ( 18
)
where A reg is demi-analytic and of polynomial type locally at 1 ∈ ∆ δ , H and H are (θ 0 , θ)-homogeneous and of polynomial type locally at 0 ∈ ∆ δ , and Θ ∈ R and I : R >0 → C are defined as in Theorem 1 [START_REF] Casas-Alvero | Singularities of plane curves[END_REF]. As in the theorems, the asymptotics of the coefficients is taken in the θ-diagonal regime.

The functional form of the prefactor I(λ) is often of practical importance (see Section 4 for more discussions). In Theorem 1, I(λ) was expressed as an integral transform of H(u). The next theorem will provide some basic properties of this transform and its inverse. We define the inverse Laplace transform (also called Borel transform) of a function H by

B[H](λ) = 1 2πi d ˆVδ e λ•u H(u) du (19)
where δ ∈ (0, δ) and V δ is a contour of the form specified below Equation (15). On the other hand, for any given c ∈ R d >0 , we define the Laplace tranform (truncated at c) of a function I by

L c [I](u) = ˆ∞ c e -λ•u I(λ) dλ ≡ ˆ[c 1 ,∞)ו••×[c d ,∞) e -λ•u I(λ) dλ. ( 20
)
Theorem 2 (Properties of the Borel-Laplace transforms). Fix δ ∈ (0, π/2) and c ∈ R d >0 . 1. For all H ∈ P(Ω δ ), B[H] defines an analytic function on K δ independent of the value of δ ∈ (0, δ).

Moreover, B[H] ∈ P(K δ • ) for all δ • ∈ (0, δ).

For all

I ∈ P(K δ ), L c [I] defines an analytic function on Ω δ . Moreover, for all δ • ∈ (0, δ), there exists M > 0 such that B • L c [I] is well-defined and analytic on {λ ∈ K δ • : ∀j, |λ j | > M }. 3. (L c is a right inverse of B). For all I ∈ P(K δ ), we have B • L c [I] = I.
For all H ∈ P(Ω δ ), there exists a demi-entire function

E c : Ω δ → C such that L c • B[H] = H + E c .
Corollary 3. The scaling function I(λ) in Theorem 1( 2) is (-Θ, θ)-homogeneous and in

P(K δ • ) for all δ • ∈ (0, δ). It is identically zero if and only if H is demi-entire on Ω δ .
Outline of the rest of the paper. Sections 2 and 3 give the proofs of Theorems 1 and 2, respectively. In Section 4, I first provide some additional results on the classes of multivariate functions mentioned above Theorem 1, and then discuss the background of this paper and its relations to previous works.

Acknowledgement. The author is grateful for the support of the ETH Foundation. This work has also been supported by the Swiss National Science Foundation (SNF) Grant 175505 and the Agence Nationale de la Recherche project ProGraM (Projet-ANR-19-CE40-0025).

Proof of the multivariate transfer theorems

Contours and domains. We start by defining some contours and domains useful in the proof. Recall the definition (1) of the ∆-domain ∆ δ . For given values of n 0 1 and θ j > 0, we define the contours C j and C O j by Figure 1(a): C j is a contour inside ∆ δ and close to its boundary, while C O j is a portion of C j close to the point z = 1. We denote by = (δ) the maximal distance from a point of

C O j to 1. Let V O j be the image of C O j under the mapping z → n 0 θ j (1 -z), and V be the limit of V O j as n 0 → ∞. We have V O j ⊂ V ⊂ Ω δ , see Figure 1(b).
The multivariate versions of these objects are defined by

C = C 1 × • • • × C d C O = C O 1 × • • • × C O d V = V × • • • × V V O = V O 1 × • • • × V O d .
Uniform growth bounds. Now let us derive some uniform bounds of functions on the above contours and domains. Recall that in the θ-diagonal limit, the vector

λ = n -θ 0 n ≡ (n 0 -θ 1 n 1 , . . . , n 0 -θ d n d ) remains
in some compact subset of (0, ∞) d as n 0 → ∞. Let λ min , λ max > 0 be such that λ j ∈ [λ min , λ max ] for all 1 ≤ j ≤ d. Let θ min = min{θ 1 , . . . , θ d } and θ max = max{θ 1 , . . . , θ d }. We assume n 0 ≥ 2.

We assume without loss of generality that δ, and therefore , is small enough so that all the local assumptions near z = 1 in Theorem 1 hold globally in B 1, := z ∈ C d ∀1 ≤ j ≤ d, |z j -1| < . In other words, the functions A reg and z → H k (1 -z) are well-defined on ∆ δ ∩ B 1, and satisfy

∀z ∈ ∆ δ ∩ B 1, , |F (z)| ≤ C • |z 1 -1| -M + • • • + |z d -1| -M , (21)
C j 1 n θ j 0 1+ δ 2 V Ω δ 1 ∆ δ π 2 -δ 2 π 2 -δ 2 (a) (b) C O j V O j n θ j 0 Figure 1: (a)
The contour C j is composed of two circle arcs and two line segments. The two arcs have their centers at 0 and 1, and their radii 1 + δ/2 and 1/n 0 θj , respectively. The line segments connect the two arcs and form an angle π 2 -δ 2 with the positive real axis at 1. The contour C O j is C j with the large arc removed. (b) The contour V O j is the image of C O j under the mapping z → n 0 θj (1 -z), and V is the limit of V O j as n 0 → ∞ (which no longer depends on j). V is a special case of the curve V δ appeared in the definition (15) of I(λ) in Theorem 1.

and there exist functions

A reg,j analytic in (∆ j-1 δ ×C×∆ d-j δ )∩B 1, such that A reg = A reg,1 +• • •+A reg,d . Notice that C O ⊂ ∆ δ ∩ B 1, and |z j -1| ≥ n 0 -θ j for all z ∈ C. So the bound (21) implies that ∀z ∈ C O , |F (z)| ≤ C • n M θ 1 0 + • • • + n M θ d 0 ≤ dC • n M θmax 0 . ( 22
)
Since A is of polynomial type globally on ∆ δ , it satisfies (21) for all z ∈ ∆ δ and ( 22) for all z ∈ C. We leave the reader to check the elementary fact that there exist c, µ > 0 depending only on δ, θ min and λ min , λ max , such that

∀z ∈ C O j , z n j +1 ≥ c • exp µ • n θ j 0 Re(z -1) . ( 23
)
And since Re(z-1) ≥ n 0 -θ j for z ∈ C O j and |z| > 1 for z ∈ C j \C O j , there exists c = c(δ, θ min , λ min , λ max ) > 0 such that ∀z ∈ C j , z n j +1 ≥ c .

Under the change of variable u = n 0 θ j (1 -z), the bound (23) becomes

∀u ∈ V O j , (1 -n -θ j 0 u) n j +1 ≥ c • e -µ•Re(u) . (25)
In Section 4, we will discuss several equivalent formulations of the polynomial-type condition, and their consequences on homogeneous functions. In particular, we will see in Lemma 7 that a homogeneous function of polynomial type locally at 0 ∈ Ω δ is always bounded by a Laurent polynomial globally on Ω δ . An easy consequence of Lemma 7 is the following global polynomial bound for the functions H 0 , . . . , H m : there exists C, M > 0 such that

∀u ∈ V , |H k (u)| ≤ C • |u 1 • • • u d | M (26)
Analyticity of the homogeneous components. Before starting the proof of the main theorems, let us show the analyticity of the functions H k (0 ≤ k < m) mentioned in the 7-th remark below Theorem 1 .

Actually, we show that they are analytic on the larger domain Ω δ , thanks to homogeneity.

Lemma 4. The functions H 0 , . . . , H m-1 in Theorem 1 have analytic continuations on Ω δ .

Proof. Recall that by our assumption on the smallness of , the function A reg is analytic on

∆ δ ∩ B 1, . Let B 0, = {1 -z | z ∈ B 1, } ≡ {z ∈ C d | ∀1 ≤ j ≤ d, |z j | < } and A sing (1 -z) = A(z) -A reg (z) and f σ (u) = σ -θ (0) 0 A sing (σ θ u).
Then f σ is analytic on Ω δ ∩ B 0, for all 0 < σ ≤ 1, and the expansion (16) implies that H 0 (u) = lim σ→0 + f σ (u) for all u ∈ Ω δ ∩ B 0, . Moreover, thanks to the polynomial-type bound, H 0 , . . . , H m are bounded on all compact subsets of Ω δ ∩ B 0, . It follows that the family (f σ ) σ∈(0,1] is uniformly bounded there. By Vitali's theorem, the convergence

f σ (u) ---→ σ→0 H 0 (u) is uniform on compact subsets of Ω δ ∩ B 0, . Therefore H 0 is analytic on Ω δ ∩ B 0, .
The same argument applied to A sing -H 0 , A sing -H 0 -H 1 , etc. shows that H 1 , . . . , H m-1 are also analytic on Ω δ ∩B 0, . The homogeneity property

H k (σ θ u) = σ θ (k) 0 H k (u) extends this to all u ∈ Ω δ .
Proof of Theorem 1 . Now let us prove Theorem 1 . The proof will also imply Theorem 1 by considering cases in which only one term on the right hand side of ( 16) is nonzero. We follow the same general steps as the proof of the univariate transfer theorem in [START_REF] Flajolet | Singularity analysis of generating functions[END_REF].

Step 1. Contour deformation and localization. The coefficient [z n ]A(z) are related to A(z) by the Cauchy integral formula

[z n ]A(z) = 1 2πi d ˛Tr A(z) z n+1 dz ( 27
)
where T r = z ∈ C d ∀j, |z j | = r is the polytorus of radius r for some r > 0 small enough. Thanks to the analyticity of A(z) in ∆ δ , we can deform the contour in (27) from T r to C. By spliting the integral according to the partition

C = C O ∪ (C \ C O ), we obtain [z n ]A(z) = I loc + 1 2πi d ˆC\C O A(z) z n+1 dz where I loc = 1 2πi d ˆCO A(z) z n+1 dz . (28)
By the definition of C O , for each z ∈ C \C O , there is at least one j * ∈ {1, . . . , d} such that

|z j * | = 1+δ/2.
Together with the growth bounds ( 22) and ( 24), this implies

∀z ∈ C \ C O , A(z) z n+1 ≤ dC • n M θmax 0 c d-1 • (1 + δ/2) n j * +1 ≤ C • n M θmax 0 (1 + δ/2) λmin•n θ min 0 ≤ C • r n θ min 0 ( 29
)
for some constants r < 1 and C independent of n 0 . It follows that

[z n ]A(z) = I loc + O r n θ min 0 (30)
for some constant r ∈ (0, 1) as n 0 → ∞.

Step 2. Removing the demi-analytic term. Let A sing (1 -z) = A(z) -A reg (z). Then we have

I loc = I reg + I sing := 1 2πi d ˆCO A reg (z) z n+1 dz + 1 2πi d ˆCO A sing (1 -z) z n+1 dz (31)
Recall that the demi-analytic term A reg admits a decomposition

A reg = A reg,1 + • • • + A reg,d , where each A reg,j is analytic in (∆ j-1 δ × C × ∆ d-j δ ) ∩ B 1, .
Within this domain, we can deform the j-th component the contour

C O ≡ C O 1 × • • • × C O d
to the arc CO j on the circle {z ∈ C : |z| = 1 + δ/2} with the same endpoints as C O j . Then, with the same reasoning as in Step 1, we obtain ˆCO A reg,j (z)

z n+1 dz = ˆCO 1 ו••× CO j ו••×C O d A reg,j (z) z n+1 dz = O(r n θ min 0 ) (32)
for some r ∈ (0, 1). Summing over j gives I reg = O(r n θ min 0

). Together with Step 1, this implies

[z n ]A(z) = I sing + O(r n θ min 0 ) (33)
for some constant r ∈ (0, 1) as n 0 → ∞. In particular, when A sing = 0, we obtain Case 1 of Theorem 1.

Step 3. Transfer. By linearity, the expansion (16) implies I sing = I 0 + • • • + I m , where

I k = 1 2πi d ˆCO H k (1 -z) z n+1 dz for 0 ≤ k < m and I m = 1 2πi d ˆCO o(H m (1 -z)) z n+1 dz (34)
Thanks to the analyticity of H 0 , . . . , H m-1 in Lemma 4, we can apply Step 1 to them to obtain

[z n ]H k (1 -z) = I k + O(r n θ min 0 ) for all 0 ≤ k < m. (35
C o j = {z ∈ C O j : |z -1| < ε(n 0 )} and C o = C o 1 × • • • × C o d . (36)
Contrary to C O , which has a small but fixed size , the contour C o shrinks to the point 1 when n 0 → ∞. According to the above definition, for all z ∈ C O \ C o , there exists at least one j * ∈ {1, . . . , d} such that |z j * | ≥ 1 + sin(δ/2) • ε(n 0 ). Then, similarly to (29), we have

∀C O \ C o , H m (1 -z) z n+1 ≤ Cn M θmax 0 1 + sin(δ/2) • ε(n 0 ) λmin•n θ min 0 ≤ C • r ε(n 0 )•n θ min 0 ( 37
)
for some r < 1 and C > 0. We choose a function ε(x) such that ε(x)

x -θmin log(x) when x → ∞. (For example, ε(x) = x θmin/2 .) Then the right hand side of the above display decays faster than any negative power of n 0 when n 0 → ∞. In particular, this implies

ˆCO \C o o(H m (1 -z)) z n+1 dz = o n -Θm 0 . ( 38
)
as n 0 → ∞. On the other hand, since the contour C o shrinks to z = 1 when n 0 → ∞, for any τ > 0, there exists n * 0 such that

∀z ∈ C o , |o(H m (1 -z))| ≤ τ |H m (1 -z)| (39)
for all n 0 ≥ n * 0 . By plugging this inequality into the integral over C o and making the change of variable

u = n θ 0 (1 -z) ≡ (n 0 θ 1 (1 -z 1 ), . . . , n 0 θ d (1 -z d )), we obtain ˆCo o(H m (1 -z)) z n+1 dz ≤ τ ˆV o H m (n -θ 0 u) 1 -n -θ 0 u n+1 |du| n θ 1 +•••+θ d 0 = τ n Θm 0 ˆV o H m (u) 1 -n -θ 0 u n+1 |du| (40)
for n 0 ≥ n * 0 , where V o is the image of C o under the change of variable, and the last equality used the (θ (m) 0 , θ)-homogeneity of H m . The bounds (25) and ( 26) imply that the integrand on the right hand side is bounded by

C|u 1 • • • u d | M c d e -µ•Re(u 1 +•••+u d ) = c -d C d j=1 |u j | M e µRe(u j ) , (41)
which is integrable on V ⊃ V o . Hence the inegral on V o is bounded by a constant independent of n 0 . It follows that ´Co Step 4. Coefficient asymptotics of homogeneous functions. It remains to prove the asymptotic expansion (14) for a general (θ 0 , θ)-homogeneous function H satisfying the assumptions of Theorem 1. With the same argument as in Step 3, there exists r ∈ (0, 1) such that

[z n ]H(1 -z) = J + O(r n θ min 0 ) with J := 1 2πi d ˆCO H(1 -z) z n+1 dz (42)
We perform the same change of variable as in Step 3, which gives

J = 1 2πi d ˆV O H(n -θ 0 u) (1 -n -θ 0 u) n+1 du n θ 1 +•••+θ d 0 = 1 n Θ 0 • 1 2πi d ˆV O H(u) (1 -n -θ 0 u) n+1 du ( 43
)
where Θ = θ 0 + • • • + θ d as defined in Theorem 1. According to the definition (3) of the coefficients g k,l , we have

1 -n -θ 0 u -λn θ 0 -1 = e λu G n -θ 0 u, λu = e λu ∞ k=0 k l=0 g k,l λ l u k+l 1 n k•θ 0 . ( 44
)
Applying the above formula to each factor of the product d j=1 1 -n 0 -θ j u j -n j -1 gives

1 (1 -n -θ 0 u) n+1 = e λ•u k∈N d l≤k g k,l λ l u k+l 1 n k•θ 0 . ( 45
)
If we treat the right hand side as a formal sum over k, and approximate the contour V O by its limit V , then (43) would imply heuristically that:

J ≈ 1 n Θ k∈N d 1 n k•θ 0 • 1 2πi d ˆV l≤k g k,l λ l u k+l e λ•u H(u)du = 1 n Θ k∈N d 1 n k•θ 0 • l≤k g k,l λ l ∂ k+l λ 1 2πi d ˆV e λ•u H(u)du = 1 n Θ k∈N d 1 n k•θ 0 • D k I(λ) .
where the function I(λ) and the differential operator D k are defined as in Theorem 1. Now let us show that the last line is indeed an asymptotic expansion of J. In other words, for any N > 0, we have

n Θ 0 • J = k•θ<N D k I(λ) n k•θ 0 + O 1 n N 0 ( 46
)
when n 0 → ∞. For this, let us go back to (44). It can be seen as the Taylor series expansion of the function f y (x) = (1 -x) -y/x-1 around x = 0 evaluated at x = n -θ 0 u and y = λu. The corresponding Taylor expansion with remainder term writes

f y (x) = e y m k=0 k l=0 g k,l y l x k + R m (x, y) with R m (x, y) = 1 m! ˆx 0 (x -ξ) m f (m+1) y (ξ)dξ . ( 47
)
It is not hard to show by induction that there are functions ϕ k,m continuous on the unit disk, such that

f (m) y (x) = f y (x) • m k=0 ϕ k,m (x)y k , (48)
It follows that for each m, there exists a constant C m > 0 such that f

(m) y (x) ≤ C m (1 + |y| m ) • f y (x) for all |x| ≤ 1/2 and y ∈ C. Plugging this into the definition of R m (x, y) gives that ∀|x| ≤ 1/2, ∀y ∈ C, |R m (x, y)| ≤ C m (m + 1)! (1 + |y| m ) • |x| m+1 sup ξ∈[0,x] f y (ξ) (49)
Consider the case where (x, y) = (n 0 -θ j u, λu) with n 0 > 0, λ ∈ [λ min , λ max ] and u ∈ V O j . In this case the condition |x| ≤ 1/2 is satisfied because |u| ≤ n 0 θ j on V O j and = (δ) is assumed to be small enough. Since |u| ≥ 1 on V O j , the term (1 + |y| m ) is bounded by a constant times |u| m uniformly for u ∈ V O j . Moreover, for all ξ ∈ [0, x], there exists ñ0 > 0 such that ξ = ñ0 -θ j u. Thus we can apply the bound (25)

to see that |f y (ξ)| = (1 -ñ0 -θ j u) -λu-1 ≤ c -1 e µ•Re(u) for all ξ ∈ [0, x]. It follows that ∀u ∈ V O j , R m (n -θ j 0 u, λu) ≤ Cm |u| 2m+1 e µ•Re(u) n (m+1)θ j 0 (50)
for some constant Cm depending only on δ, λ min , λ max and m. Then, the Taylor expansion (47) becomes

(1 -n -θ j 0 u) -λn θ j 0 -1 = e λu m k=0 k l=0 g k,l λ l u k+l 1 n k•θ j 0 + O |u| 2m+1 e µ•Re(u) n (m+1)θ j 0 , (51)
where the big-O estimate is uniform with respect to u ∈ V O j as n 0 → ∞. Now take m > N/θ min and replace each factor in d j=1 1 -n 0 -θ j u j -n j -1 by the right hand side of the above formula. After expanding the resulting product, we obtain a finite sum. By collecting all the terms of order O(1/n N 0 ) together, we obtain an expansion of the form

1 (1 -n -θ 0 u) n+1 = e λ•u k•θ<N l≤k g k,l λ l u k+l 1 n k•θ 0 + O R(u) n N 0 , (52)
where the big-O estimate is uniform on u ∈ V O . By following the above calculation more closely, it is not hard to see that we can choose

R(u) = |u 1 • • • u d | M e µ•Re(u 1 +•••+u d ) for some µ, M > 0. With the bound (26) for H(u), it follows that H(u)R(u) is integrable on V ⊃ V O .
So we can integrable the previous display term by term with respect to u ∈ V O to obtain

ˆV O H(u) (1 -n -θ 0 u) n+1 du = k•θ<N ˆV O H(u)e λ•u l≤k g k,l λ l u k+l du • 1 n k•θ 0 + O 1 n N 0 . (53)
Like in Step 1, replacing the contour V O by V in each term of the above equation only produces an error of order O(r n θ min 0 ) as n 0 → ∞. The resulting equation divided by (2πi) d gives exactly (46). The expansion (14) follows readily from (42) and (46). This concludes the proof of Theorem 1 .

The Borel-Laplace transforms

In this section, we prove the three statements in Theorem 2.

Proof of Theorem 2(1). Fix δ ∈ (0, π

2 ) and H ∈ P(Ω δ ). Recall that B[H] is defined as the integral

I V δ (λ) := 1 2πi d ˆVδ e λ•u H(u)du, (54)
on a contour V δ chosen from the class of contours

V δ := V 1 × • • • × V d ∀1 ≤ j ≤ d, V j is a piecewise smooth curve in Ω δ
which coincide with ∂Ω δ outside a bounded set, as in Figure 2(a) .

Thanks to the analyticity of H on the domain Ω δ = (Ω δ ) d , it is clear that for each fixed δ , the value of the integral I V δ (λ) does not depend on the choice of the contour V δ within the class V δ .

First, let us fix δ ∈ (0, δ) and V δ ∈ V δ and show that the integral I V δ (λ) is absolutely convergent and analytic with respect to λ ∈ K δ . Let U δ ,r be the closure of (C \ Ω δ ) ∪ B 0,r , where B 0,r ⊂ C is the disk of radius r around the origin. It is a simple exercise to show that for δ • ∈ (0, δ ) and r > 0, we have

∀u ∈ U δ ,r , ∀λ ∈ K δ • , e λu ≤ C r|λ| * e -σ * |λ|•|u| , (55)
with σ * = sin(δ -δ • ) > 0 and C * = e 1+σ * . Since each component of the contour V δ ∈ V δ is bounded away from the origin and H is of polynomial type, there exist C, M > 0 such that

∀u ∈ V δ , |H(u)| ≤ C • |u 1 | M + • • • + |u d | M . ( 56
)
On the other hand, there exists r > 0 such that V δ ⊂ U δ ,r . So the bound (55) implies that for any bounded set S ⊆ K δ • , there exists C, σ > 0, such that

∀u ∈ V δ , ∀λ ∈ S, e λ•u ≤ C • e -σ(|u 1 |+•••+|u d |) . ( 57
)
Up to increasing the value of C and decreasing σ, the polynomial bound on the right hand side of (56) can be absorbed by the exponential decay in the above display. Therefore we have

∀u ∈ V δ , ∀λ ∈ S, e λ•u H(u) ≤ C • e -σ(|u 1 |+•••+|u d |) (58)
for some C, σ > 0. Since the right hand side is independent of λ ∈ S and integrable on V δ , it follows that the integral I V δ (λ) is absolutely convergent and analytic with respect to λ ∈ S. And since this is true for all δ • ∈ (0, δ ) and bounded set S ⊂ K δ • , the integral I V δ defines an analytic function on K δ . Now let us show that I V δ (λ) is independent of δ ∈ (0, δ) as well. For this, we fix δ , δ ∈ (0, δ) and λ ∈ K min(δ ,δ ) . For each N > 0, let V N δ be the contour obtained by deforming each component of V δ inside the disk B 0,N to coincide with the corresponding component of V δ there, while keeping it unchanged outside B 0,N . See Figure 2(a). By the triangular inequality, we have ˆVδ -

ˆVδ e λ•u H(u)du = ˆVδ - ˆV N δ e λ•u H(u)du ≤ ˆVδ ∆ V N δ e λ•u H(u) • |du| , (59)
where V δ ∆ V N δ is the symmetric difference between V δ and V N δ . Let

S n = u ∈ V δ ∆ V N δ n ≤ max j |u j | < n + 1 . (60)
It is not hard to see that S n = ∅ for n < N , and the d-dimensional Lebesgue measure of S n is bounded by cn d for some constant c = c(δ , δ , d). Moreover, since each point λ ∈ K min(δ ,δ ) belongs to K δ • for some δ • < min(δ , δ ), we can deduce from ( 58) that e λ•u H(u) ≤ Ce -σn for all u ∈ S n and all n.

(Here we also use the observation that (58) remains valid when V δ is replaced by

V N δ ). It follows that ˆVδ ∆V N δ e λ•u H(u) • |du| ≤ ∞ n=N Ce -σn • cn d ----→ N →∞ 0 . ( 61
)
Together with (59), this implies that I V δ (λ) = I V δ (λ) for all λ ∈ K min(δ ,δ ) . In this sense, B[H] = I V δ is independent of δ ∈ (0, δ) and thus defines an analytic function on K δ .

To show that B[H] ∈ P(K δ • ) for all δ • ∈ (0, δ), let us fix λ ∈ K δ • and δ ∈ (δ • , δ), and consider the particular contour

V δ = V 1 × • • • V d with V j = ∂U δ ,1/|λ j |
, where U δ ,r is defined above (55). By the bound (55), we have

∀u ∈ V δ , e λ•u ≤ C d * • e -σ * (|λ 1 u 1 |+•••+|λ d u d |) (62)
Since the contour V j is not bounded away from the origin when |λ j | → ∞, we need to use the complete bound (11) for the function of polynomial type H ∈ P(Ω δ ) here, which implies for some C, M > 0. We integrate the above bound on V δ and make the change of variables v j = λ j u j . Notice that the contour

∀u ∈ V δ , e λ•u H(u) ≤ C • |u 1 | M + |u 1 | -M + • • • + |u d | M + |u d | -M • e -σ * •(|λ 1 u 1 |+•••+|λ d u d |) (63) e -iϕ Ω -ψ (b) (a) δ U δ ,r ∩ Ω δ K δ • r V δ V N δ λ R c,ϕ c N R N c,ϕ K δ ϕ ψ u
λV δ ≡ (λ 1 V 1 ) × • • • × (λ d V d )
for the variable v after this change no longer depends on λ. This gives us

B[H](λ) = 1 2πi d ˆVδ e λ•u H(u)du ≤ C (2π) d ˆλV δ v 1 λ 1 M + v 1 λ 1 -M + • • • + v d λ d M + v d λ d -M e -σ * (|v 1 |+•••+|v d |) |dv| |λ 1 • • • λ d | ≤ max |λ 1 | M , |λ 1 | -M , . . . , |λ d | M , |λ d | -M |λ 1 • • • λ d | • J
where the integral

J = C (2π) d ˆλV δ |v 1 | M + |v 1 | -M + • • • + |v d | M + |v d | -M • e -σ * (|v 1 |+•••+|v d |) |dv| (64)
is finite and independent of λ. It follows that

∀λ ∈ K δ • , B[H](λ) ≤ J • |λ 1 | M + |λ 1 | -M + • • • + |λ d | M + |λ d | -M |λ 1 • • • λ d | . ( 65
)
We will see in Lemma 8 that the monomial function λ → λ α is in P(K δ • ), for all α ∈ R d and δ • ∈ (0, π). This allows us to bound the right hand side of the above display by a polynomial-type bound of the form (11). Hence we have

B[H] ∈ P(K δ • ), for all δ • ∈ (0, δ). Proof of Theorem 2(2). Fix c ∈ R d >0 , δ ∈ (0, π 2 ) and I ∈ P(K δ ). For each ϕ ∈ (-δ, δ) d , we define R ϕ = R c 1 ,ϕ 1 × • • • × R c d ,ϕ d ⊂ K δ ,
where R c,ϕ ⊂ C is the contour which starts at c ∈ R >0 , then follows an arc on the circle ∂B 0,|c| , and then goes to ∞ along the ray {λ ∈ C * | arg(λ) = ϕ}, as in Figure 2(b).

Let H ϕ (u) = ˆRϕ e -λ•u I(λ)dλ . ( 66
)
Recall that

Ω ψ = (Ω ψ ) d with Ω ψ ≡ K π 2 +ψ := u ∈ C * : | arg(u)| < π 2 + ψ , for any ψ ∈ (-π 2 , π 2
). First, let us show that the integral H ϕ (u) is absolutely convergent and analytic with respect to u ∈ e -iϕ Ω 0 := (e -iϕ 1 Ω 0 ) × • • • × (e -iϕ d Ω 0 ). The proof is similar to the one for the integral I V δ (λ): It is a simple exercise to show that for any ϕ ∈ (-δ, δ), ψ ∈ (0, π 2 ) and c ∈ {c 1 , . . . , c d } we have

∀λ ∈ R c,ϕ , ∀u ∈ e -iϕ Ω -ψ , e -λu ≤ C |u| * e -σ * |λ|•|u| (67)
with σ * = sin(ψ) > 0 and C * = e (1+σ * ) max j |c j | . Following the same steps as for the bounds ( 56)-( 58), we deduce that for any bounded subset S ⊂ e -iϕ Ω -ψ , where ψ ∈ (0, π 2 ) and ϕ ∈ (-δ, δ) d , there exists

C, σ > 0 such that ∀λ ∈ R ϕ , ∀u ∈ S, e λ•u I(λ) ≤ C • e -σ(|λ 1 |+•••+|λ d |) . ( 68
)
Since the right hand side is independent of u ∈ S and integrable on R ϕ , it follows that the integral H ϕ (u) is absolutely convergent and analytic with respect to u ∈ S. And since this is true for all bounded S ⊂ e -iϕ Ω -ψ with ψ ∈ (0, π 2 ), we conclude that H ϕ defines an analytic function on e -iϕ Ω 0 . Now let us show that H ϕ (u) = H 0 (u) for all ϕ ∈ (-δ, δ) d and u ∈ Ω -δ . Fix some ϕ ∈ (-δ, δ) d . For any

N > 0, let R N ϕ = R N c 1 ,ϕ 1 × • • • × R N c d ,ϕ d , where R N c,ϕ
is the curve which coincides with the interval [c, N ] inside the disk B 0,N , while staying on the ray {λ ∈ C * | arg(λ) = ϕ} outside B 0,N . See Figure 2(b). By construction, the contour R N ϕ coincides with R 0 in the polydisk (B 0,N ) d . Like in the previous proof for I V δ (λ), the sets

Sn = λ ∈ R N ϕ ∆ R 0 n ≤ max j |λ j | < n + 1 (69)
satisfy Sn = ∅ for all n < N and ´Sn |dλ| ≤ cn d for some c < ∞ independent of n. Moreover, since each point u ∈ Ω -δ belongs to Ω -ψ ∩ e -iϕ Ω -ψ for some ψ ∈ (0, π 2 ), we can deduce from ( 68) that e -λ•u I(λ) ≤ Ce -σn for all λ ∈ Sn and all n. (Here we also use the fact that (68) remains valid when R ϕ is replaced by R N ϕ ). It follows that

H ϕ (u) -H 0 (u) = ˆRN ϕ - ˆR0 e -λ•u I(λ)dλ ≤ ˆRN ϕ ∆R 0 e -λ•u I(λ) |dλ| ≤ ∞ n=N Ce -σn • cn d ----→ N →∞ 0
Therefore, H ϕ (u) = H 0 (u) for all ϕ ∈ (-δ, δ) d and u ∈ Ω -δ . And since each H ϕ is analytic on e -iϕ Ω 0 , the function L c [I] ≡ H 0 has an analytic continuation on the union Ω δ = ϕ∈(-δ,δ) d e -iϕ Ω 0 . Finally, let us fix δ • ∈ (0, δ) and λ ∈ K δ • , and show that B • L c [I] is well-defined and analytic at λ when the |λ j |'s are large enough. For this, let us first prove that for all δ ∈ (0, δ), there exists m > 0

such that ∀u ∈ Ω δ ∩ B, |L c [I](u)| ≤ e m•(|u 1 |+•••+|u d |) . ( 70
)
where

B := {u ∈ C d | ∀j, |u j | ≥ 1}. Indeed, for all ϕ ∈ (-δ, δ) d and ψ ∈ (0, π 2 ), (67) implies that ∀u ∈ e -iϕ Ω -ψ , ∀λ ∈ R ϕ , e -λ•u ≤ C |u 1 |+•••+|u d | * • e -σ * (|λ 1 u 1 |+•••+|λ d u d |) . (71)
In addition, since I ∈ P(K δ ) and R ϕ ⊂ K δ is bounded away from 0, there exist C, M > 0 such that

∀λ ∈ R ϕ , I(λ) ≤ C |λ 1 | M + • • • + |λ d | M . (72)
For u ∈ B, we can increase the value of C * and decrease that of σ * on the right hand side of (71) to absorbe the polynomial function on right hand side of (72). It follows that there exist constants C, σ > 0, which do not depend on ϕ, such that

∀u ∈ e -iϕ Ω -ψ ∩ B, ∀λ ∈ R ϕ , e -λ•u I(λ) ≤ C |u 1 |+•••+|u d | • e -σ(|λ 1 u 1 |+•••+|λ d u d |) . (73)
One can easily check by direct computation that

J := sup ϕ∈(-δ,δ) d sup u∈ B ˆRϕ e -σ(|λ 1 u 1 |+•••+|λ d u d |) |dλ| (74) is finite. It follows that |L c [I](u)| ≤ J • C |u 1 |+•••+|u d | for all u ∈ e -iϕ Ω -ψ ∩ B.
This implies (70), because J, C are independent of ϕ, and we have Ω δ = ϕ∈(-δ,δ) d e -iϕ Ω -ψ with the choice ψ := δ -δ . Now assume δ • < δ < δ and consider the contour V δ = (∂U δ ,1 ) d , where U δ ,r is defined above (55) (see also Figure 2(a)). Then (55) implies that

∀λ ∈ K δ • , ∀u ∈ V δ , e λ•u ≤ C |λ 1 |+•••+|λ d | * e -σ * (|λ 1 u 1 |+•••+|λ d u d |) (75)
Since we have V δ ⊂ Ω δ ∩ B, the last display and (70) imply that B[L c [I]] is well-defined and analytic on λ ∈ K δ

• : ∀j, |λ j | > m+1 σ *
.

Proof of Theorem 2(3). Notice that the Borel transform B and the Laplace transform L c can both be written as the product of d univariate integral transforms: we have

B = B (1) • • • • • B (d) and L c = L (1)
c 1 • • • • • L (1)
c d with

B (j) [H](u ĵ ) := 1 2πi ˆVδ e λu j H(u)du j and L (j) c [I](λ ĵ ) := ˆ∞ c e λ j u I(λ)dλ j , (76)
where u ĵ = (u 1 , . . . , u j-1 , λ, u j+1 , . . . u d ) and λ ĵ = (λ 1 , . . . , λ j-1 , u, λ j+1 , . . . λ d ). Moreover, for all j = k,

B (j) commute with L (k)
c k , because they operate on different variables. It follows that

B •L c = B (1) • L (1) c 1 •• • •• B (d) • L (d) c d and L c •B = L (1) c 1 • B (1) •• • •• L (d) c d • B (d) . ( 77
)
The above decompositions allow us to reduce the proof of Theorem 2(3) to the univariate case. In the following, we assume d = 1 and fix some δ ∈ (0, π 2 ) and c ∈ R >0 . Let I ∈ P(K δ ). Let us show that B • L c [I](λ) = I(λ) for all λ > c. For this, fix some δ ∈ (0, δ) and let V δ = ∂Ω δ with the domain Ω δ defined above (55). Let V + δ (resp. V - δ ) be the part of V δ in the upper half-plane (resp. lower half-plane). Recall from the proof of Theorem 2(2) that L c [I] can be expressed as an integral over R c,ϕ for any ϕ ∈ (-δ, δ), where R c,ϕ is the contour shown in Figure 2(3). Fix some ϕ ∈ (δ , δ). Then we can write

B • L c [I](λ) = 1 2πi ˆVδ e λu L c [I](u)du = 1 2πi ˆV + δ e λu ˆRc,-ϕ e -τ u I(τ )dτ du + ˆV - δ e λu
ˆRc,ϕ e -τ u I(τ )dτ du .

Thanks to the bounds (55) and (67), it is not hard to see that (u, τ ) → e λu e -τ u I(τ ) is integrable on both V + δ × R c,-ϕ and V - δ × R c,ϕ . Thus we have by Fubini's theorem

B • L c [I](λ) = 1 2πi ˆRc,-ϕ I(τ ) ˆV + δ e λu-τ u du dτ + ˆRc,ϕ I(τ ) ˆV - δ e λu-τ u du dτ = 1 2πi    ˆRc,-ϕ I(τ ) • e (λ-τ )u λ -τ e i( π 2 +δ ) ∞ 1 dτ + ˆRc,ϕ I(τ ) • e (λ-τ )u λ -τ 1 e -i( π 2 +δ ) ∞ dτ    = 1 2πi ˆRc,-ϕ I(τ ) • e λ-τ τ -λ dτ + ˆRc,ϕ I(τ ) • e λ-τ λ -τ dτ = 1 2πi ˆR I(τ ) • e λ-τ τ -λ dτ
where R = R c,-ϕ ∪ Rc,ϕ and Rc,ϕ is the contour R c,ϕ oriented in the opposite direction. Notice that R ⊂ K δ is a bi-infinite contour whose two ends extend to ∞. Moreover, the integrand τ → I(τ ) e λ-τ τ -λ is meromorphic on K δ , has a unique simple pole at τ = λ (which is on the right of the contour R), and decays exponentially when Re(τ ) → ∞. It follows from the residue theorem that

1 2πi ˆR I(τ ) • e λ-τ τ -λ dτ = Res τ →λ I(τ ) e λ-τ τ -λ = I(λ) . (78)
This proves B • L c [I](λ) = I(λ) for all λ > c, and thus for all λ ∈ K δ by analytic continuation. In other words, B • L c is the identity on P(K δ ). By the decomposition (77), the same is true in any dimension d. Now fix some H ∈ P(Ω δ ) and u > 0. Consider the contour V δ = ∂U δ ,r for some δ ∈ (0, δ) and r ∈ (0, u). It is not hard to check that the function (λ, v) → e -λu e λv H(v) is integrable on [c, ∞) × V δ . By Fubini's theorem, we have

L c • B[H](u) = 1 2πi ˆ∞ c e -λu ˆVδ e λv H(v)dv) dλ (79) = 1 2πi ˆVδ H(v) ˆ∞ c e -(u-v)λ dλ dv = 1 2πi ˆVδ H(v) e -(u-v)c u -v dv . (80)
For each u ∈ C, let

E c (u) = - 1 2πi lim R→∞ ˆ∂U δ ,R H(v) e -(u-v)c u -v dv . (81)
It is not hard to see that the above limit stablizes when R > |u|, and defines an entire function of u.

Actually, by the residue theorem, we have for all R > |u|,

1 2πi ˆ∂U δ ,R H(v) e -(u-v)c u -v dv = 1 2πi ˆVδ H(v) e -(u-v)c u -v dv + Res v→u H(v) e -(u-v)c u -v = L c • B[H](u) -H(u) . It follows that L c • B[H](u) = H(u) + E c (u)
for all u > 0. By analytic continuation, the same is true for all u ∈ Ω δ . To recover the case of general dimension d, we apply this formula of L c • B[H] to each factor in the decomposition (77). To simplify notation, we write

I j = L (j) c j • B (j) , then L c • B[H] = (I 1 • • • • • I d-1 • I d ) [H] = (I 1 • • • • • I d-1 ) [H] + (I 1 • • • • • I d-1 ) [E (d) c ] = • • • • • • • • • = H + E (1)
c + I 1 [E (2)
c ] + • • • + (I 1 • • • • • I d-1 ) [E (d) c ] .
where each E

c is an analytic function on Ω δ which is entire with respect to the variable u j . The same is true for

I 1 • • • • I j-1 [E (j) c ], because the integral transform I 1 • • • • I j-1 does not affect the variable u j . It follows that E (1) c + I 1 [E (2) c ] + • • • + (I 1 • • • • • I d-1 ) [E (d)
c ] is a demi-entire function on Ω δ . This proves the decomposition L c • B[H] = H + E c in any dimension d, and concludes the proof of Theorem 2.

Discussions

In this section, we discuss some additional properties of the classes of multivariate functions used in the statement of the multivariate transfer theorems.

∆-analytic functions. Like in the univariate case, the multivariate transfer theorems relies fundamentally on the ∆-analyticity of the generating function. But unlike the univariate case, ∆-analyticity is a rather restrictive condition in the multivariate setting. In particular, it implies that the dominant singularity (for any reasonable definition of the term) is unique and independent of the exponent θ and the direction λ of the θ-diagonal limit taken. This is in stark contrast with the case of rational functions, where the dominant singularities (a.k.a. contributing critical points) generically depend on the direction of the diagonal limit taken. In fact, a multivariate rational function is never ∆-analytic at any of its poles, unless its denominator has a univariate linear factor which vanishes at this pole.

Proposition 5 (Genuinely multivariate rational functions are never ∆-analytic). If a rational function is ∆-analytic at 1 and has a pole at 1, then its denominator is divisible by z j -1 for some 1 ≤ j ≤ d.

Remark. The proposition as well as its proof given below can be easily generalized to meromorphic functions. We shall not enter into the details here and refer to [START_REF] Melczer | Algorithmic and symbolic combinatorics-an invitation to analytic combinatorics in several variables[END_REF]Section 3.1.1] for an introduction to meromorphic functions in several variable.

Proof. Consider a rational function A = F G , where F, G are coprime polynomials. The proposition follows directly from Lemma 6 below about the local geometry of the zero set of a polynomial: it suffices to apply Lemma 6 to the polynomial

H(u) = G(1 -u). For δ, > 0, let Ω δ, = (Ω δ, ) d with Ω δ, := Ω δ ∩ B 0, ≡ u ∈ C : 0 < |u| < and | arg(u)| < π 2 + δ . Lemma 6 (Local version of Proposition 5). If a polynomial H ∈ C[u]
has a zero at 0, but no zero on Ω δ, for some δ, > 0, then H(u) is divisible by u j for some 1 ≤ j ≤ d.

Proof. When d = 1, the lemma is trivial: a univariate polynomial H(u) has a zero at 0 if and only if it is divisible by u.

When d = 2, consider a polynomial H ∈ C[u, v] not divisible by u nor v, such that H(0, 0) = 0. According to the Newton-Puiseux theorem (see e.g. [2, Corollary 1.5.5, Theorem 1.7.2]), there exist an integer n ≥ 1 and a nonzero analytic function s defined on a neighborhood of 0, such that for every determination of the n-th root, the mapping ϕ(u) = s(u 1/n ) satisfies H(u, ϕ(u)) = 0 for all u in some neighborhood of 0. Up to swapping the variables u and v, we can assume without loss of generality that ϕ(u) = O(u) as u → 0. Since s is analytic at 0 and is not identically zero, there exist c = 0 and α ≥ 1 such that ϕ(u) ∼ c • u α as u → 0. Geometrically, this means that the mapping ϕ multiplies the angles at u = 0 by α. In particular, for any δ, > 0 with small enough, the image ϕ(Ω δ, ) conains an angle of α(π + 2δ) at 0. Since α ≥ 1, we have α(π + 2δ) + (π + 2δ) > 2π. This implies that ϕ(Ω δ, ) ∩ Ω δ, = ∅. In other words, the graph of the mapping ϕ intersects (Ω δ, ) 2 . It follows that the polynomial H has zeros on (Ω δ, ) 2 for any δ, > 0. By contraposition, this proves Lemma 6 when d = 2.

For d ≥ 3, we give a proof by contradiction based on the result of the case d = 2: Let H ∈ C[u] be a polynomial with a zero at 0, no zero on Ω δ, for some δ, > 0, and such that H(u) is not divisible by u j for any 1 ≤ j ≤ d. For 1 ≤ j ≤ d and m ≥ 0, define E

(j) m = n ∈ N d (n 1 + • • • + n d ) -n j = m .
Let us prove the following statement by induction on m:

∀j ∈ {1, . . . , d}, ∀n ∈ E (j) m , [u n ]H(u) = 0 . (H m ) For x ∈ R d-1 >0 , let H x (u, v) = H(u, vx 1 , . . . , vx d-1
). One can check that for all k ∈ N, we have

[v k ]H x (u, v) = n∈E (1) k [u n ]H(u) • u n 1 • x n 2 1 • • • x n d d-1 . (82)
Now fix some m ≥ 0 and assume that (H k ) is true for all 0 ≤ k < m. By (82), we have

[v k ]H x (u, v) = 0 for all k < m. Hence H (m) x (u, v) := v -m H x (u, v) is a polynomial in (u, v). Moreover, H (m)
x (0, 0) = 0 : For m = 0, we have H (0)

x (0, 0) = H x (0, 0) = H(0) = 0 by assumption. When m ≥ 1, one can check that n ∈ E (1) m n 1 = 0

⊆ k<m d j=1 E (j) k . ( 83
)
Hence the hypotheses (H k ) k<m and (82) imply that H (m)

x (0, 0) ≡ [u 0 v m ]H x (u, v) = 0 as well. On the other hand, since H(u) = 0 for all u ∈ Ω δ, , we have H x (u, v) is either divisible by u or divisible by v. In particular, we have

H (m) x (0, 1) • H (m) x (u, 0) ≡ H(0, x) • [v m ]H x (u, v) = 0 (84)
set. Since H is (θ 0 , 1)-homogeneous, we have

H(u) = -θ 0 |u i | θ 0 H u 1 |u i | , . . . , u d |u i |
, and therefore

|H(u)| ≤ -θ 0 |u i | θ 0 C • -M u 1 u i -M + • • • + -M u d u i -M = C -θ 0 -M u θ 0 +M i u -M 2 + • • • + u θ 0 +M i u -M d
Summing the right hand side over i gives a bound of H(u) of the form (86) for all u ∈ K.

Lemma 7 was briefly used in the proof of Theorem 1 to obtain the bound (26). It implies (26) because the variable u ∈ V in (26) satisfies |u j | ≥ 1 for all 1 ≤ j ≤ d.

In practical examples, it is usually not hard to check that a function is of polynomial type, in particular thanks to the following lemma.

Lemma 8 (Closure properties of P(K δ ) and P(∆ δ )). For any δ ∈ (0, π), the space P(K δ ) forms a C-algebra with respect to pointwise addition and multiplication of functions. Moreover, if f is any function such that |f (x)| is bounded by a polynomial of |x| (e.g. f (x) = x β , β ∈ C), then for all h ∈ P(K δ ) such that f • h is well-defined and analytic on K δ , we have f • h ∈ P(K δ ).

The same is true for P(∆ δ ) with δ ∈ (0, π/2).

Proof. Fix δ ∈ (0, π) and let g, h ∈ P(K δ ). By definition there exist C, M > 0 such that

|g(u)| ≤ C • |u 1 | M + |u 1 | -M + • • • + |u d | M + |u d | -M (88) and |h(u)| ≤ C • |u 1 | M + |u 1 | -M + • • • + |u d | M + |u d | -M (89)
for all u ∈ K δ . It is clear that any linear combination of g and h satisfies a bound of the same form. So P(K δ ) is a C-vector space. In addition, since ab ≤ 1 2 (a 2 + b 2 ) for all a, b ∈ R, we have

|g(u)h(u)| ≤ C 2 d i,j=1 |u i | M |u j | M + |u i | M |u j | -M + |u i | -M |u j | M + |u i | -M |u j | -M ≤ C 2 d i,j=1 |u i | 2M + |u j | 2M + |u i | -2M + |u j | -2M .
So P(K δ ) is also closed under multiplication, and therefore forms a C-algebra. Now let f be a function such that |f (x)| is bounded by a polynomial of |x|. Then there exist c > 0 and m ∈ Z ≥0 such that |f (x)| ≤ c • (1 + |x| m ). If h ∈ P(K δ ) and f • h is well-defined and analytic on K δ , then we have h m ∈ P(K δ ) by the closure of P(K δ ) under multiplication, and therefore

|f • h(u)| ≤ c • (1 + |h(u) m |) ≤ c • 1 + C • |u 1 | M + |u 1 | -M + • • • + |u d | M + |u d | -M (90)
for some C, M > 0. It follows that f • h ∈ P(K δ ). The same proof works for P(∆ δ ).

Let

Ω ⊆ C d . A function A : Ω → C is algebraic if there exists a polynomial E ∈ C[a, z], such that E(A(z), z) = 0 for all z ∈ Ω.
Lemma 9 (Algebraic functions are of polynomial type). If A is an algebraic function analytic on ∆ δ , then A ∈ P(∆ δ ) for all δ ∈ (0, δ).

Proof. Let A be an algebraic function which is analytic on ∆ δ . For z ∈ ∆ δ , let

h(z) = 1 dist(z, C d \ ∆ δ ) ≡ inf w∈C d \∆ δ z -w -1 , (91)
where zw is the Euclidean distance between z and w, viewed as points in R 2d . Let us show that there exist C, M > 0 such that

∀z ∈ ∆ δ , A(z) ≤ C • h(z) M . (92)
This is a variant of the Łojasiewicz inequality in semi-algebraic geometry. See e.g. [1, Chapter 2] for an introduction. Recall that a set S ⊆ R n is semi-algebraic if and only if it can be defined by a first order logic formula involving only polynomial conditions (i.e. equations or inequalities) and quantifiers ∀/∃ over R.

And a function

f : S → R m is semi-algebraic if and only if Graph(f ) := {(x, f (x)) ∈ R n+m | x ∈ S} is a semi-algebraic set.
It is not hard to write a first order formula that describes the set ∆ δ (viewed as a subset of R 2d ). Therefore ∆ δ is a semi-algebraic set. It is well-known that a continuous algebraic function defined on an open semi-algebraic set is always semi-algebraic (see e.g. [START_REF] Wakabayashi | Remarks on semi-algebraic functions[END_REF]Theorem 11]). Hence the function A : ∆ δ → C ≡ R 2 is semi-algebraic. Thanks to general closure properties of the class of semi-algebraic functions (c.f. [1, Section 2.2]), we deduce from the above facts that the functions h : ∆ δ → R and |A| : ∆ δ → R are also semi-algebraic.

For each t > 0, let G t = {z ∈ ∆ δ : h(z) = t} and g(t) = sup z∈Gt |A(z)|, with the convention that sup ∅ = 0. The graph of the function g : R >0 → R can be described by a first order formula as follows:

Graph(g) = (t, y) ∈ R 2 t > 0 and (∀z ∈ G t , |A(z)| ≤ y) and (G t = ∅ and y = 0) or (∃z ∈ G t , |A(z)| = y) where ∀z ∈ G t , |A(z)| ≤ y ⇔ ∀z ∈ R 2d , z ∈ ∆ δ and (z, t) ∈ Graph(h) ⇒ ∃w ∈ R, (z, w) ∈ Graph(|A|) and w ≤ y , G t = ∅ and y = 0 ⇔ ∀z ∈ R 2d , z ∈ ∆ δ ⇒ (z, t) ∈ Graph(h) , ∃z ∈ G t , |A(z)| = y ⇔ ∃z ∈ R 2d , z ∈ ∆ δ and (z, t) ∈ Graph(h) and (z, y) ∈ Graph(|A|) .
Since h and |A| are semi-algebraic functions, the conditions (z, t) ∈ Graph(h) and (z, y) ∈ Graph(|A|) can be further expanded into first order logic formulas involving only polynomial conditions and quantifiers over R. It follows that g is a semi-algebraic function. A classical result on the growth rate of univariate semi-algebraic functions [1, Proposition 2.6.1] states that there exist constants C, M, t 0 > 0 such that g(t) ≤ C • t M for all t > t 0 . The definition of g implies that it is bounded on (0, t 0 ]. Hence the bound g(t) ≤ C • t M extends to all t > 0. This proves (92).

Finally, notice that for all δ ∈ (0, δ), there exists a constant c > 0 such that

∀z ∈ ∆ δ , c -1 • dist(z, C d \ ∆ δ ) ≤ min 1≤j≤d |z j -1| ≤ c • dist(z, C d \ ∆ δ ) . (93)
Moreover, we have

min 1≤j≤d |z j | -M ≤ |z 1 -1| -M + • • • + |z d -1| -M ≤ d • min 1≤j≤d |z j | -M . ( 94
) It follows that |A(z)| ≤ C d j=1 |z j -1| -M on ∆ δ for some C, M > 0. That is, A ∈ P(∆ δ ).
Background on ACSV and its general strategy. The results in paper fall under the topic of analytic combinatorics in several variables (ACSV). The remaining paragraphs provide some background on ACSV and where this work stands relative to the others.

In general, analytic combinatorics aims at understanding the enumerative properties of large combinatorial structures through the analytic properties of their generating functions. This is usually done in two steps: First, some (possibly implicit) expression of the generating function must be derived from the definition of the combinatorial structure that it encodes. Then, one studies the generating function as an complex analytic function to derive asymptotic formulas of its coefficients.

Compared to analytic combinatorics in one variable, to which the transfer theorems A and A belong, analytic combinatorics in several variables is a much less mature theory. The difference is especially stark when it comes to derving coefficient asymptotics from the generating functions (i.e. the second step outlined above). This process is often known as singularity analysis, since the asymptotic expansion of the coefficients of a generating function is mostly determined by the properties of the function near its singularities. For instance, if a function A(z) has a unique dominant singularity at ρ ∈ C * , then thanks to the analyticity of A everywhere else inside and on the circle of radius |ρ|, we can deform the contour of integration in the Cauchy integral formula (27) to a curve C that coincides with a circle of radius r > |ρ| everywhere except in a neighborhood N of ρ. By splitting the parts of the contour inside and outside N , we get [z n ]A(z) = I loc + I rem with

I loc = 1 2πi ˆC ∩ N
A(z) z n+1 dz and

I rem = 1 2πi ˆC \ N A(z) z n+1 dz . ( 95
)
Since |z| = r on C \ N , we have I rem = O(r -n ), which is exponentially small compared to |ρ| -n . On the other hand, lim sup n→∞ ([z n ]A(z)) 1/n = |ρ| -1 by the root test of radius of convergence. Hence the asymptotics of [z n ]A(z) is dominated by the term I loc . Since I loc only depends on A(z) in an arbitrarily small neighborhood of ρ, its asymptotics can be further studied via asymptotic expansions of the function A(z) near its singularity z = ρ. (This is basically the beginning of a proof of Theorem A .) At first glance, it is not clear how the above approach of singularity analysis could be generalized to multivariate functions. Indeed, the singularities of a multivariate complex function A(z) always form a continuous set with no isolated points (c.f. Hartog's extension theorem). The crucial remark here is that in general, not all singularities of A(z) contribute to the dominant asymptotics of its coeffcients. In nice cases, one can even expect to find a finite number of contributing singularities, so that the asymptotics of [z n ]A(z) is dominanted by the values of A(z) in arbitrarily small neighborhoods of these points. In practice, one would like deform the torus T r in the Cauchy integral formula (27) to a cycle (i.e. d-chain without boundary) C homologous to T r in the domain of analyticity of the integrant A(z) z n+1 , such that the denominator z n+1 attains its mimimum only at a finite number of points ρ (1) , . . . , ρ (m) on C. By Stokes' theorem (see e.g. [9, Appendix A.2]), such a deformation does not change the value of the integral. One can then take arbitrarily small neighborhoods N (1) , . . . N (m) of the points ρ (1) , . . . , ρ (m) , and split the integral inside and outside these neighborhoods as in the univariate setting. This gives the decomposition [z n ]A(z) = I loc , so that the latter dominates the the asymptotics of [z n ]A(z). After this, one still needs to expand the localized integrals I (s) loc to obtain a simple asymptotic expansion of [z n ]A(z). How this can be done depends on the form of the function A(z). But usually this is a simpler problem, since there are a lot more tools at our disposal for the local analysis of a function.

The above general strategy to the multivariate singularity analysis has been outlined by Pemantle and Wilson in [START_REF] Pemantle | Analytic combinatorics in several variables[END_REF]. Over the past twenty years, they and their collaborators developped an impressive theory that treats the singularity analysis of multivariate rational-type functions in an algorithmic way. The results of their project are collected at http://acsvproject.com/. More precisely, they consider general rational functions in several variables (and also some other functions whose singularities form algebraic varieties) and compute their coefficient asymptotics in the diagonal limit (i.e. θ-diagonal limit with θ = (1, . . . , 1)). In this broad setting, they carry out the strategy described in the previous paragraph in a systematic way, identifying the contributing singularities ρ (s) and simplifying the localized integrals I (s) loc , using powerful tools from algebraic topology and Morse theory. The reason that they need such advanced tools is that the singular set of a general rational function can have a quite complicated geometry. In particular, the location of the dominant singularities ρ (1) , . . . , ρ (m) (or contributing critical points, as they are called in [START_REF] Pemantle | Analytic combinatorics in several variables[END_REF]) depends on the direction λ of the diagonal limit. And in general, they cannot be reached by the easy-to-visualize cycles of product form

C = C 1 × • • • × C d .
This paper explores the case of ∆-analytic generating functions, following the same general strategy of singularity analysis. As shown by Proposition 5, this case is essentially disjoint from that of rational functions. ∆-analytic functions have a simpler singularity structure, in the sense that they always have a unique, fixed dominant singularity reachable by a cycle of product form. This allows us to study their coefficients in the θ-diagonal limits for general θ ∈ R d >0 , while using mostly elementary tools from univariate complex analysis. We will discuss more the relation between our case and the case of rational functions in Section 4. For more background on ACSV, we refer to the historical accounts in [9, Chapter 1] and [8, Section 1.2.2].

0

  +θ 1 +• • •+θ d , and the little-o estimate is uniform with respect to λ on all compact subsets of R d >0 . The same result holds if the little-o estimates in both (16) and (17) are replaced by big-O estimates.

)).

  Now let us show that I m = o(n -Θm 0 To make use of the little-o estimate in the integrand, we want to further localize the contour C O : For a function ε(x) ---→ x→∞ 0, we define the little-o versions of the contours C O j and C O by

o(Hm( 1 -

 1 z)) z n+1 dz = o(n -Θm ), Adding this to (38) gives I m = o(n -Θm 0 ). Together with the conclusions of Steps 2 and 3, this implies the asymptotic expansion (17) in Theorem 1 . When the little-o estimate in (16) is replaced by a big-O, the same proof (actually simpler, since one no longer needs to localize C O to C o ) shows the expansion (17) with o(n -Θm 0 ) replaced by O(n -Θm 0 ).

Figure 2 :

 2 Figure 2: (a) Some domains and contours used in the proof of Theorem 2(1). (b) Some domains and contours used in the proof of Theorem 2(2).

  (m) x (u, v) = v -m H(u, vx 1 , . . . , vx d-1 ) = 0 for all (u, v) ∈ (Ω δ,˜ ) 2 , where ˜ = min 1, x -1 1 , . . . , x -1 d-1 • > 0. So,according to the result of the case d = 2, the polynomial H (m)

( 1 )

 1 ∪•••∪C (m) ) A(z) z n+1 dz , (96) with C (s) := C ∩ N (s) . For a well chosen cycle C, we expect the non-local term I rem to be exponentially small compared to the local terms I (s)

for all x ∈ R d-1 >0 . By analytic continuation, the above identity is valid for all x ∈ C d-1 . Because H(u) is not divisible by u 1 , the polynomial x → H(0, x) is not identically zero. Since the ring of polynomials is an integral domain, this implies that [v m ]H x (u, v) = 0 as a polynomial in x. Comparing this to (82), we see that

m . The same argument works for E (j)

By induction, (H m ) holds for all m ≥ 0. But this implies that H(u) ≡ 0, which contradicts the assumption that H(u) is not divisible by u j . This completes the proof of Lemma 6 for general d ≥ 3.

Demi-analytic functions. A univariate function A is demi-analytic at ρ if and only if it is analytic in a neighborhood of ρ, and it is demi-entire if and only if it is an entire function. Thus the names "demi-analytic" and "demi-entire".

It is easy to check that a (θ 0 , θ)-homogeneous function is demi-entire with respect to a cone if and only if it is demi-analytic at 0 with respect to the same cone.

Given the form of the asymptotic expansion (17) and the analyticity of the scaling function I(λ), it is not hard to see that the coefficients [z n ]A(z) decays exponentially as n 0 → ∞ if and only if I is identically zero. By Corollary 3, this happens if and only if the homogeneous component H(u) is demi-analytic. In this sense, the demi-analyticity condition in Theorem 1(1) is optimal.

Generalized homogeneous functions. If H is a (θ 0 , θ)-homogeneous function, then for all u 1 > 0, we have

Inversely, for any function

In particular, when d = 1, the only (θ 0 , θ 1 )-homogeneous analytic functions are the power functions H(u) = C u θ 0 /θ 1 (C ∈ C).

For each α ∈ R d , the monomial u → u α is (θ 0 , θ)-homogeneous for all (θ 0 , θ) such that α • θ = θ 0 . Clearly, a linear combination of finitely many such monomials is also (θ 0 , θ)-homogeneous, as long as their multi-exponents α satisfy the above linear relation for the same (θ 0 , θ). Such polynomials are obviously not all the homogeneous functions, but they provide a good intuition for how a non-homogeneous function could be decomposed into homogeneous components.

Functions of polynomial type. In Theorem 1 , the remainder term in the asymptotic expansion of the generating function A(z) was expressed as a big-O of some (θ 0 , θ)-homogeneous function of polynomial type. The following lemma tells us that every such remainder term can also be bounded by finitely many monomials with the same homogeneity. Lemma 7. Let K be a cone in C d . If H : K → C is a (θ 0 , θ)-homogeneous function of polynomial type locally at 0, then there exist a constant C and finitely many vectors α

Proof. By considering H(s) = H(s θ ) instead of H, we can assume without loss of generality that θ = 1.

By assumption, there exist constants C, M, > 0 such that on {u ∈ K | ∀1 ≤ j ≤ d, |u j | < }, we have

Let i be such that |u i | = max 1≤j≤d |u j |. Then we can rescale the vector u by |u i | to place it in the above


Wenzhuo Liu
email: wenzhuo.liu@irt-systemx.fr

David Danan
email: david.danan@irt-systemx.fr

Marc Schoenauer
email: marc.schoenauer@inria.fr

Multi-Fidelity Transfer Learning for accurate data-based PDE approximation

Data-driven approaches to accelerate computation time on PDE-based physical problems have recently received growing interest. Deep Learning algorithms are applied to learn from samples of accurate approximations of the PDEs solutions computed by numerical solvers. However, generating a large-scale dataset with accurate solutions using these classical solvers remains challenging due to their high computational cost. In this work, we propose a multi-fidelity transfer learning approach that combines a large amount of low-cost data from poor approximations with a small but accurately computed dataset. Experiments on two physical problems (airfoil flow and wheel contact) show that by transferring prior-knowledge learned from the inaccurate dataset, our approach can predict well PDEs solutions, even when only a few samples of highly accurate solutions are available.

Introduction

In recent years, there has been a rapid growth in the use of machine learning algorithms to numerically solve problems from physical domains for which accurate numerical solvers are too computationally expensive. In such context, data-driven methods have been proposed [START_REF] Bhatnagar | Prediction of aerodynamic flow fields using CNNs[END_REF][START_REF] Tompson | Accelerating eulerian fluid simulation with convolutional networks[END_REF][START_REF] Thuerey | Deep learning methods for reynolds-averaged navier-stokes simulations of airfoil flows[END_REF][START_REF] Kashefi | A point-cloud deep learning framework for prediction of fluid flow fields on irregular geometries[END_REF][START_REF] De | Combining differentiable PDE solvers and GNNs for fluid flow prediction[END_REF][START_REF] Pfaff | Learning mesh-based simulation with graph networks[END_REF] to learn Deep Learning models (DL) that approximate the solutions of complex partial differential equations (PDEs) from large amounts of data created by classical numerical solvers. A widely used family of such classical numerical methods is the Finite Element Method (FEM), which uses a discretization of the physical domain into small polygonal elements, aka a mesh. The accuracy of the FEM solution depends on the size of the elements and hence on the size of the mesh, i.e., the number of elements. In particular, fine meshes (i.e., with a large number of elements) are needed for acceptable accuracy.

However, beyond issues related to generalization, that will not be addressed here, a most critical issue in the data-based approach is the computational cost of the generation of training datasets: Complex phenomena can only be captured accurately using large enough deep networks that require large training datasets. Furthermore, the total error of the trained model is the sum of the training error of the network and the numerical error of the samples in the training set, that hence should be made of accurate enough solutions, i.e., in the case of FEM, solutions computed on very fine meshes. To tackle this challenge, we introduce MFT, a Multi-Fidelity Transfer learning approach that uses two meshes of different granularity: On the coarse mesh, FEM approximate solutions of the PDE at hand can be computed at low cost -but only poorly approximate the exact solution of the PDE. On the fine mesh, on the other hand, the FEM solutions are good approximations of the exact solution of the PDE, but are very costly to compute, making it unrealistic to generate enough samples to train an accurate-enough deep model. The goal of this work is to train such an accurate-enough deep model Figure 1: Diagram of the multi-fidelity transfer learning approach on Wheel Contact problem using only a small dataset of highly accurate solutions, but leveraging the result of the training on the many samples computed on the coarse mesh, using principles from Transfer Learning: the model trained on the large dataset generated on the coarse mesh is used as a starting point for the learning process on the small dataset of accurate solutions. We demonstrate the effectiveness of MFT on two PDEs describing respectively the air flow around an airfoil and the contact of a tire on the road.

The multi-fidelity transfer learning approach

In the following, we aim at learning a data-based model (see below) to solve a given PDE for different values of some input quantities x that govern the system on a given physical domain. We will use a given FEM numerical solver on two different meshes of that domain, a coarse mesh M c and a fine mesh M f . The solver can compute so-called low-fidelity solutions at low cost on the coarse mesh M c , and high-fidelity solutions on the fine mesh M f , but at a high cost. The goal is to train a deep model that can compute high-fidelity solutions of the PDE for any given input, using as few as possible high-fidelity samples but taking advantage of as many low-fidelity samples as needed.

Prediction on the Coarse Mesh

The first step consists in generating a dataset D c of low-fidelity solutions by applying the FEM solver on the coarse mesh M c using a representative set of inputs x. A deep model f c is then learned from D c (details follow). As many samples as needed can be computed efficiently at low cost, and the loss function is the MSE between the network output and the numerical solution in D c , considered as ground truth at this step. Even in the ideal case where the error of the learned network is zero, thanks to very large dataset D c data, the accuracy of the learned model w.r.t. the exact solution of the PDE is not satisfactory since the solutions in D c are poor approximations of these exact solutions.

Integration of low-fidelity prior knowledge on the Fine Mesh

The second step starts from this low-fidelity model f c , which was fully trained on the low-fidelity dataset D c and from a dataset D f of high-fidelity solutions assumed to be too small to allow direct training of an accurate enough deep model. Inspired by the ideas from Transfer Learning algorithms [START_REF] Zhuang | A comprehensive survey on transfer learning[END_REF], where the knowledge gained while solving a task is re-used for a different but related task, we propose to leverage some knowledge that has been encapsulated in f c to improve the learning of a deep model from the small dataset D f . Transfer Learning is widely used nowadays to solve textand image-related tasks [START_REF] Howard | Universal language model fine-tuning for text classification[END_REF][START_REF] Huang | Cross-language knowledge transfer using multilingual deep neural network with shared hidden layers[END_REF][START_REF] Oquab | Learning and transferring midlevel image representations using convolutional neural networks[END_REF], especially in similar situations when only a few data samples are available or training.

Latent Feature Extraction The basic assumption here is that the pre-trained model f c has learned a meaningful representation of the problem and can be viewed as a feature extractor for the high-fidelity task. For each input x, the vector h, outputs of all neurons before the last layer of f c is considered as a vector of latent features of the problem. It is first upsampled to the fine mesh M f using k-nearest neighbors interpolation and used as additional inputs for the high-fidelity learning, as illustrated in Fig. 1.

Transfering Using the high-fidelity dataset D f and additional input h, we train a transfer model f t using again the MSE loss. The final output is:

y = f t (x, h) = f t (x, UpSample[f c (x)])

Model Architecture

Mesh to Graph A mesh can be viewed as a graph whose nodes are the vertices of the polygonal elements, and whose edges are the edges of the elements. Formally, mesh data can then be expressed as M = (N, E, V, E), where N is the number of nodes of the mesh, E the set of edges, V ∈ R N ×F are the F -dimensional node features attached to each of the N nodes, and E ∈ R E×D the attached attributes on each edge.

Graph Neural Networks GNNs [START_REF] Bruna | Spectral networks and locally connected networks on graphs[END_REF][START_REF] Defferrard | CNNs on Graphs with Fast Localized Spectral Filtering[END_REF][START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF][START_REF] Atwood | Diffusion-convolutional neural networks[END_REF][START_REF] Veličković | Graph attention networks[END_REF] are designed to treat data from non-Euclidean space such as graphs, meshes or manifolds, and are now routinely used to handle mesh data [START_REF] Kashefi | A point-cloud deep learning framework for prediction of fluid flow fields on irregular geometries[END_REF][START_REF] De | Combining differentiable PDE solvers and GNNs for fluid flow prediction[END_REF][START_REF] Pfaff | Learning mesh-based simulation with graph networks[END_REF]. Needed here are a convolutional operator, invariant by permutation of the nodes, and a pooling operator, allowing to exchange data between meshes of different granularities. Among the different types of GNN, we have chosen the MoNet GNN [START_REF] Monti | Geometric deep learning on graphs and manifolds using mixture model CNNs[END_REF].

Convolution operator In MoNet, it is defined as follows. Consider a weighted graph G = (N, E, V, E). Let x i ∈ R F be the feature vector of node i, and x e ij ∈ R D be the feature of the edge i, j defining the set of neighbours N (i) of node i. The basic idea of MoNet is to define a trainable function w that computes an edge weight w ij from the edge feature x e ij . MoNet then defines the convolutional operator on node i as

x ′ i = 1 |N (i)| j∈N (i) 1 K K k=1 w k (x e ij ) ⊙ Θ k x j
,where K is the user-defined kernel size, Θ k ∈ R M ×N stands for the trainable matrix applying a linear transformation on the input data, ⊙ is the element-wise product, and w k , k = 1, . . . , K are trainable edge weights. Following [START_REF] Monti | Geometric deep learning on graphs and manifolds using mixture model CNNs[END_REF], we use Gaussian kernels:

w k (x e ij ) = exp(-1 2 (x e ij - µ k ) T Σ -1 k (x e ij -µ k ))
. Both µ k and Σ k are trainable variables. Pooling Operator We use simply the k-nearest interpolation proposed in PointNet++ [START_REF] Qi | Pointnet++: Deep hierarchical feature learning on point sets in a metric space[END_REF] when designing pooling operators. Let y be a node from M 1 , and assume its k nearest neighbors on M 2 are (x 1 , . . . , x k ). The interpolated feature for y is defined from those of the x i 's as:

f (y) = k i=1 w(x i )f (x i ) k i=1 w(x i ) , where w(x i ) = 1 ||y -x i || 2
MFT Architecture Both learning phases of MFT are performed using the Graph U-Net architecture that was proposed in [START_REF] Liu | Multi-resolution graph neural networks for pde approximation[END_REF]. As argued above, Graph Neural Networks (GNNs) can directly handle mesh data. On the other hand, the U-Net architecture allows to extract hierarchical spatial features. Modern mesh generators today easily allow us to build a set of hierarchical meshes for the same domain without sacrificing geometric information. We benefit from these algorithms and use such a hierarchy of meshes to construct the U-Net architecture. A set of graph layers is applied on the same mesh level while pooling operators transfer data between different mesh levels. The edge attribute x e ij ∈ E of edge (i, j) is x e ij = p j -p i , where p i and p j are the coordinates of nodes i and j respectively. The node attribute at node i is the vector of values of the unknown variables of the PDE at node i, the output of the network.

Experimental Results

The MFT approach is validated on two different PDEs:

Airfoil Flow : Incompressible fluid flow around an airfoil, modeled by the Navier Stokes PDEs. Flow fields on 80 distinct 2D NACA airfoils are simulated by the CFD solver OpenFOAM. Inputs are the Mach number, in [0.03, 0.3], and the Angle-of-Attack (AoA), in [-22.5, 22.5] (in degrees). The target variable are the velocity v = (v x , v y ) and the pressure p. The low-fidelity dataset is generated with meshes of approx. 500 nodes and the high-fidelity dataset uses meshes with around 3800 nodes.

Wheel Contact : The target is the deformation field u = (u x , u y ) of a tire on a fixed 2D wheel under some external force f . The FEM solver is GetFEM++. The input quantities are the mechanical carateristics of the rubber: Young modulus E ∈ [5, 7] × 10 6 and Poisson's ratio ν ∈ [0.38, 0.495]; the strength A ∈ [0.1, 0.5], and the angle α ∈ [-0.78, -2.35] of the external forces; and the friction coefficient of the road µ ∈ [0.5, 0.8]. The low-fidelity dataset is generated on a fixed coarse mesh with 504 nodes, while the high-fidelity dataset uses a fixed mesh with 3398 nodes. Note that the network is asked to predict the correction w.r.t. the projection of the prediction by the low-fidelity model f c rather than directly the deformation y.

Datasets For each problem, we generate a coarse mesh M c and a fine mesh M f . The low-fidelity datasets D c contains 2 000 samples, while the high-fidelity datasets D f is made of only 400 samples.

Baselines We compare MFT with three baselines. An interpolation model LF-Int predicts the outputs by simply up-sampling the output of model f c . Model HF-400 is directly trained on the 400 high-fidelity samples without any knowledge transfer. Finally, in order to be fair with the highfidelity-only approach, we add some high-fidelity samples to D f to compensate for the computational cost of creating D c . Model HF-ST is trained on this extended high-fidelity dataset.

Training Both the coarse model f c and the transfer model f t use the Graph U-Net architecture described above. Hyper-parameters setting and training process are detailed in Appendix ??. Standard 10-fold cross-validation is applied when training f t to assess the robustness of the approach, and we report the averages and standard deviations of test errors over the different folds. Models are trained on a single Nvidia A100 GPU, and each training takes from 3 to 6 hours.

Evaluation We create 800 new samples on M f for each task to form the ultimate test set and evaluate the results of all approaches using the rooted mean squared error (RMSE) metric.

Results Table 3 reminds the characteristics of the datasets (right side) and the error on the test set described above (left). The latter clearly demonstrates that the MFT model significantly outperforms the three baselines on both physical problems. The performance of the interpolation baseline LF-Int shows that the transfer model MFT largely improved the predictions based on the pre-trained model f c . In the meanwhile, the prior knowledge extracted from D c does help the prediction on fine meshes compared with HF-400, which has only access to the high-fidelity dataset D f . Moreover, the comparison between HF-ST and MFT indicates that the multi-fidelity datasets containing many more samples tend to be more informative than a pure high-fidelity dataset with the same computational budget. The MFT model combines the benefit of the low-fidelity dataset for the fast generation with the high-fidelity simulations to create accurate ground truth.

Conclusions

This work introduced MFT, a multi-fidelity transfer learning model, to use Machine Learning approaches (namely GNNs) to numerically solve PDEs on fine meshes efficiently. MFT benefits from a large amount of low-fidelity data to extract some prior-knowledge, and then transfers it to predict high-fidelity solutions. The training process on fine meshes can then be achieved by using only a small training set. The experimental results on two complex physical problems are a first proof of the concept that MFT can solve PDEs accurately when a small number of high-fidelity samples is available. On-going and future work is to evaluate the proposed approach on more complex physical systems such as three-dimensional problems.

Broader Impact

This paper investigates the ability to learn PDEs solutions when large training datasets are required to solve the problem accurately. Our proposed method can be widely used in data-driven methods for physical systems as the lack of a large-scale and high-fidelity dataset is a common issue in the physical domain.

Table 1 :

 1 Rooted mean squared error (RMSE) for both Airfoil Flow and Wheel Contact problem (left); dataset composition for each discussed model (right); and generation time for each sample in the last two rows.

		Tasks			Datasets
	Models	Airfoil Flow (e-2) Wheel Contact (e-4)	D c	D f
	LF-Int	9.81	27.82	2 000	\
	HF-400	2.98 ± 0.30	6.62 ± 0.25	\	400
	HF-ST	2.43 ± 0.39	4.80 ± 0.19	\	400 + 127/240 1
	MFT	1.95 ± 0.05	4.57 ± 0.10	2 000	400
	Generation	Airfoil Flow	∼ 2.33s	∼ 36.7s
	Time (each)	Wheel Contact	∼ 0.20s	∼ 1.64s

Machine Learning and the Physical Sciences workshop, NeurIPS 2022.

We generated 127 additional high-fidelity examples on fine meshes for the Airfoil Flow problem, and

for Wheel Contact problem.