Carlos Amendola

Viet Son Pham

Carlos Améndola

Autocovariance Varieties of Moving Average Random Fields

Keywords: 14Q15, 60G60, 62F10, 62M10, 62R01

We study the autocovariance functions of moving average random fields over the integer lattice Z d from an algebraic perspective. These autocovariances are parametrized polynomially by the moving average coefficients, hence tracing out algebraic varieties. We derive dimension and degree of these varieties and we use their algebraic properties to obtain statistical consequences such as identifiability of model parameters. We connect the problem of parameter estimation to the algebraic invariants known as euclidean distance degree and maximum likelihood degree. Throughout, we illustrate the results with concrete examples. In our computations we use tools from commutative algebra and numerical algebraic geometry.

Introduction

Moving average random fields indexed by the integer lattice Z d generalize the class of discretetime moving average processes and constitute an important statistical spatial model. They are used to model texture images (cf. [START_REF] Francos | Maximum likelihood parameter estimation of textures using a wold decomposition based model[END_REF]), as well as in image segmentation and restoration (cf. [START_REF] Krishnamurthy | Adaptive restoration of textured images with mixed spectra using a generalized Wiener filter[END_REF]). Furthermore, they are connected to ARMA (autoregressive moving average) random fields (cf. [START_REF] Drapatz | Strictly stationary solutions of spatial ARMA equations[END_REF] and the references therein) and the sampling problem of CARMA (continuous autoregressive moving average) random fields, in which the autocovariance functions of moving average random fields play a crucial role (cf. [START_REF] Pham | Lévy-driven causal CARMA random fields[END_REF]Section 4.3]).

A moving average random field (Y t ) t∈Z d of order q = (q 1 , q 2 , . . . , q d ) ∈ N d is defined by the equation

Y t = q 1 k 1 =0 ⋯ q d k d =0 a k Z t-k , t ∈ Z d ,
where k = (k 1 , . . . , k d ), a k are real coefficients and (Z t ) t∈Z d is a real-valued zero-mean white noise (see Definition 2.1). The autocovariance function

γ(t) = Cov[Y 0 , Y t ], t ∈ Z d ,
for this type of random field is compactly supported, i.e. only finitely many values are nonzero. More precisely, we have γ(t) = 0 for every t = (t 1 , . . . , t d ) ∈ Z d with entries satisfying t i > q i for at least one i ∈ {1, . . . , d}.

We study the autocovariance functions of moving average random fields from an algebraic perspective. Our motivation stems from the field of algebraic statistics [START_REF] Sullivant | Algebraic Statistics[END_REF]. Specifically, inspired by the concept of moment varieties [START_REF] Améndola | Moment varieties of Gaussian mixtures[END_REF], here we introduce autocovariance varieties. The moving average variety MA q ⊆ P N (see Definition 3.1) is parametrized by (q 1 +1)⋯(q d +1) moving average coefficients a k where the indices k satisfy 0 ≤ k i ≤ q i for i = 1, . . . , d. These coefficients induce (2q 1 + 1)⋯(2q d + 1) nonzero autocovariance values γ(t). However, we only consider half of them since the relation γ(-v) = γ(v) holds for all v ∈ Z d .

Example 1.1. Let d = 2 and q = (1, 1). Then the Q = (1 + 1)(1 + 1) = 4 parameters a 00 , a 01 , a 10 , a 11 define the autocovariances γ(0, 0) = a 2 00 + a 2 10 + a 2 01 + a 2 11 , γ(1, 0) = a 00 a 10 + a 11 a 01 , γ(0, 1) = a 00 a 01 + a 11 a 10 , γ(1, -1) = a 10 a 01 , γ(1, 1) = a 00 a 11 .

The moving average variety MA (1,1) ⊆ P 4 is expected to be 3-dimensional. We characterize it in Theorem 3.4.

This paper is organized as follows. In Section 2, we give the main definition of a moving average random field and its autocovariance function. We define our main object of study, namely autocovariance varieties, in Section 3. We contrast the properties between moving average processes (one-dimensional) from the higher dimensional moving average random fields. In Theorem 3.9 we establish the dimension and degree of these varieties. In Section 4 we investigate identifiability of the associated models and prove that they are algebraically identifiable. In contrast to the d = 1 case where the degree of the fiber grows with q, we show that for d > 1 there are generically only two sets of parameters that yield the same autocovariance function. Next, we study two different approaches to estimate model parameters from given samples in Section 5. First, we fit the empirical autocovariance function to the theoretical counterpart using a least squares method. Second, we consider maximum likelihood estimation. Both approaches connect nicely to concepts from algebraic statistics: respectively the ED degree and the ML degree. In Example 5.8, we conduct a simulation study comparing classical local optimization methods to numerical homotopy continuation, where we find that the numerical algebraic geometry (NAG) method performs slightly better.

We use the following notation and terminology in this paper: The components of a vector u ∈ R d are given by u 1 , ..., u

d if not stated otherwise. If u, v ∈ Z d , then we set [u, v] ∶= {s ∈ Z d u i ≤ s i ≤ v i , 1 ≤ i ≤ d},
which may be an empty set. The symbol ⪯ denotes the lexicographic order and for x ∈ R d and t ∈ Z d we define x t ∶= x t 1 1 ⋯x t d d . If g∶ A → B is a mapping and y ∈ g(A), then g -1 (y) is called fiber of y and each point inside the fiber is called preimage. A statement that holds generically, or for a generic point, holds outside an algebraic hypersurface in x ∈ C d , hence in particular for almost all x ∈ C d with respect to Lebesgue measure.

Moving Average Random Fields

Throughout this article, all stochastic objects are defined on a fixed complete probability space (Ω, F, P).

Definition 2.1. (1) A random field (Y t ) t∈Z d is called weakly stationary if Y t ∈ L 2 (Ω, F, P) for every t ∈ Z d and γ(t) ∶= Cov[Y 0 , Y t ] = Cov[Y s , Y s+t ] for every t, s ∈ Z d .
It is called a white noise if γ(0) > 0 and γ(t) = 0 for every 0 ≠ t ∈ Z d . In this case, σ 2 ∶= γ(0) is called the white noise variance.

(2) Let q 1 , ..., q d be positive integers and (Z t ) t∈Z d be a real-valued zero-mean white noise on Z d . A random field (Y t ) t∈Z d is called a moving average random field of order q = (q 1 , ..., q d ), denoted M A(q), if it satisfies the equation

Y t = k∈[0,q] a k Z t-k , t ∈ Z d , (2.1)
where a k ∈ R such that for each i = 1, ..., d there exist at least two index vectors l, m ∈ [0, q] satisfying a l ≠ 0, a m ≠ 0, l i = q i and m i = 0.

The last condition on the two index vectors l, m guarantees that the M A(q) random field has indeed order q and not a smaller order. We associate to each M A(q) random field the moving average polynomial

θ(x) = k∈[0,q] a k x k ,
and further, we define the formal backshift operators B 1 , ..., B d which act on any random field (X t ) t∈Z d in the following way:

(B i X) t = X (t 1 ,...,t i-1 ,t i -1,t i+1 ,...,t d ) , t ∈ Z d , i = 1, ..., d.
Since the operators B i and B j commute for any i, j, equation (2.1) can be written as

Y t = θ(B)Z t , t ∈ Z d , where B = (B 1 , ..., B d ).
The following proposition establishes the link between the moving average polynomial θ and the autocovariance function γ of a M A(q) random field. Proposition 2.2. Suppose that (Y t ) t∈Z d is a M A(q) random field driven by a white noise (Z t ) t∈Z d with variance σ 2 . Then Y is weakly stationary, its autocovariance function γ is compactly supported and we have

σ 2 θ(x)θ(x -1 ) = t∈Z d γ(t)x t .
(2.2)

Proof. The facts that Y is weakly stationary and γ is compactly supported are straightforward. Let S = {k ∈ Z d ∶ a k ≠ 0} denote the set of indices with non-vanishing coefficient a k . Then we have that

σ 2 θ(x)θ(x -1 ) = σ 2 ⎛ ⎝ k∈[0,q] a k x k ⎞ ⎠ ⎛ ⎝ k∈[0,q] a k x -k ⎞ ⎠ = σ 2 t∈Z d ⎛ ⎝ k,k+t∈S a k a k+t ⎞ ⎠ x t = t∈Z d Cov ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ k∈[0,q] a k Z 0-k , k+t∈[0,q] a k+t Z 0-k ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ x t = t∈Z d Cov[θ(B)Z 0 , θ(B)Z t ]x t = t∈Z d Cov[Y 0 , Y t ]x t = t∈Z d γ(t)x t .

Autocovariance Varieties

We have seen that for q = (q 1 , q 2 , . . . , q d ) ∈ N d , the autocovariance function of a moving average random field is only dependent on the coefficients a k of the moving average polynomial θ(x) = ∑ k∈[0,q] a k x k and the white noise variance σ 2 . In order to avoid redundancies in model specification, one can assume without loss of generality that σ 2 = 1 and we will do so for the rest of this paper. There are

Q + 1 ∶= ∏ d i=1 (q i + 1) coefficients a k and 2N + 1 ∶= ∏ d i=1 (2q i + 1) non-zero autocovariances γ(t) for t ∈ [-q, q].
Ordering them in two vectors a and γ a , we can think of this correspondence as a polynomial map Γ q ∶ R Q+1 ↦ R 2N +1 given by a ↦ γ a .

Since γ(-t) = γ(t) for all t ∈ Z d , we can drop half of the autocovariances and only consider γ(t) with t ∈ [-q, q] and t ⪰ 0, where ⪰ denotes the lexicographic order. In this way, we have a map R Q+1 ↦ R N +1 which we still denote by Γ q .

The points in the image represent the set of autocovariance functions of moving average random fields. Geometrically, this is a semialgebraic set, defined by polynomial equalities and inequalities. The closure of the image of this parametrization will give a real affine algebraic variety. However, as is standard in algebraic statistics, we will first change the underlying field to be algebraically closed (so the map becomes Γ q ∶ C Q+1 ↦ C N +1 over the complex numbers C) and then we pass to projective space arriving at Γ q ∶ P Q ⇢ P N . This last step requires the polynomials γ(t) to be homogeneous, and indeed they are in our case. Definition 3.1. Let q = (q 1 , q 2 , . . . , q d ) ∈ N d and define

Q ∶= ∏ d i=1 (q i + 1) -1 and N ∶= (∏ d i=1 (2q i + 1) -1) 2.
The autocovariance variety MA q is the image of the autocovariance map Γ q ∶ P Q ⇢ P N .

Moving average processes

If d = 1, moving average random fields are also called moving average processes. These processes are well-studied and belong to the important class of ARMA processes (cf. [START_REF] Brockwell | Time Series: Theory and Methods[END_REF]Chapter 3]). Suppose that (Y t ) t∈Z is a M A(q) process given by the equation

Y t = q k=0 a k Z t-k , t ∈ Z.
Then, the autocovariance function γ of Y has the simple expression

γ(t) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∑ q-t k=0 a k a k+ t , if t ≤ q, 0, if t > q.
For the class of moving average processes we have that Q = N = q. Thus, in this special case the autocovariance map takes the form Γ q ∶ P q ⇢ P q .

In the next subsection we will see that the map is actually defined in all of P q (there are no base points), so we conclude the following.

Proposition 3.2. If d = 1, then MA q = P q .
While MA q is not particularly interesting when d = 1, the parametrization coming from Γ q ∶ P q → P q has interesting fibers and computing them is important for statistical applications. This issue of identifiability will be explored in Section 4.

Remark 3.3. Going back for a moment to the real picture (over R), the equality MA q = P q is analogous to the statement that when d = 1, any autocovariance function with support [-q, q] is an autocovariance function of a M A(q) process [6, Prop. 3.2.1].

Moving average random fields

We start by carefully analyzing the case MA (1,1) mentioned in the introduction. We denote polynomial ring variables by g t = γ(t).

Theorem 3.4. The autocovariance variety MA (1,1) ⊆ P 4 is a threefold of degree 4. It is the hypersurface defined by the quartic g 2 10 g 2 01 -g 00 g 10 g 01 g 11 + g 2 10 g 2 11 + g 2 01 g 2 11 -g 00 g 10 g 01 g

1-1 + g 2 00 g 11 g 1-1 -2g 2 10 g 11 g 1-1 -2g 2 01 g 11 g 1-1 -4g 3 11 g 1-1 + g 2 10 g 2 1-1 + g 2 01 g 2 1-1 + 8g 2 11 g 2 1-1 -4g 11 g 3 1-1 = 0. (3.1)
Its singular locus is a quadratic surface, which is the union of the three irreducible components corresponding to the prime ideals

⟨g 10 -g 01 , g 00 -2g 11 -2g 1-1 , 4g 11 g 1-1 -g 2 01 ⟩ , (3.2)
⟨g 10 + g 01 , g 00 + 2g 11 + 2g 1-1 , 4g 11 g 1-1 -g 2 01 ⟩ , (3.3)
and ⟨g 11 -g 1-1 , g 00 g 1-1 -g 10 g 01 ⟩ .

(3.4)

Proof. The proof is computational. One way to obtain the quartic (3.1) is through the following Macaulay2 [START_REF] Grayson | Macaulay2, a software system for research in algebraic geometry[END_REF] commands: R = QQ[a00,a01,a10,a11] S = QQ[g00,g01,gm11,g10,g11] h = map(R,S,{ a00^2 + a10^2 + a01^2 + a11^2, a00*a01 + a10*a11, a10*a01, a00*a10 + a01*a11, a00*a11} ) I = kernel h For the singular locus, we compute the radical ideal of the quartic along with its vanishing gradient, and then compute its prime decomposition. The complexity of MA q increases rapidly when d > 1. It is computationally challenging to obtain generators for its prime ideal even for small values of q and d. Beyond q = (1, 1), we were also able to do this for q = (1, 2) and q = (1, 1, 1). Proposition 3.6. The autocovariance variety MA (1,2) ⊆ P 7 is 5-dimensional of degree 16. Its prime ideal is cut out by 7 sextics. One of those is 1-2 g 1-1 g 01 g 00 + g 12 g 2 02 g 1-1 g 01 g 00 + g 1-2 g 2 02 g 1-1 g 01 g 00 -4g 12 g 1-2 g 11 g 10 g 01 g 00 + g 11 g 2 02 g 10 g 01 g 00 -4g 12 g 1-2 g 1-1 g 10 g 01 g 00 + g 2 02 g 1-1 g 10 g 01 g 00 -4g 12 g 1-2 g 02 g 2 01 g 00 + g 2 1-2 g 2 11 g 2 00 + 2g 12 g 1-2 g 11 g 1-1 g 2 00 + g 2 12 g 2 1-1 g 2 00 -g 1-2 g 11 g 02 g 01 g 2 00 -g 12 g 02 g 1-1 g 01 g 2 00 + g 12 g 1-2 g 2 01 g 2 00

4g 3 12 g 1-2 g 2 11 -28g 2 12 g 2 1-2 g 2 11 + 28g 12 g 3 1-2 g 2 11 -4g 4 1-2 g 2 11 -4g 2 1-2 g 4 11 -g 2 12 g 2 11 g 2 02 + 6g 12 g 1-2 g 2 11 g 2 02 -5g 2 1-2 g 2 11 g 2 02 -32g 3 12 g 1-2 g 11 g 1-1 + 64g 2 12 g 2 1-2 g 11 g 1-1 -32g 12 g 3 1-2 g 11 g 1-1 -8g 12 g 1-2 g 3 11 g 1-1 + 8g 2 1-2 g 3 11 g 1-1 + 6g 2 12 g 11 g 2 02 g 1-1 -12g 12 g 1-2 g 11 g 2 02 g 1-1 + 6g 2 1-2 g 11 g 2 02 g 1-1 -4g 4 12 g 2 1-1 + 28g 3 12 g 1-2 g 2 1-1 -28g 2 12 g 2 1-2 g 2 1-1 + 4g 12 g 3 1-2 g 2 1-1 -4g 2 12 g 2 11 g 2 1-1 + 16g 12 g 1-2 g 2 11 g 2 1-1 -4g 2 1-2 g 2 11 g 2 1-1 -5g 2 12 g 2 02 g 2 1-1 +6g 12 g 1-2 g 2 02 g 2 1-1 -g 2 1-2 g 2 02 g 2 1-1 +8g 2 12 g 11 g 3 1-1 -8g 12 g 1-2 g 11 g 3 1-1 -4g 2 12 g 4 1-1 +8g 2 12 g 1-2 g 2 11 g 10 -16g 12 g 2 1-2 g 2 11 g 10 + 8g 3 1-2 g 2 11 g 10 -2g 12 g 2 11 g 2 02 g 10 + 6g 1-2 g 2 11 g 2 02 g 10 + 4g 12 g 11 g 2 02 g 1-1 g 10 + 4g 1-2 g 11 g 2 02 g 1-1 g 10 + 8g 3 12 g 2 1-1 g 10 -16g 2 12 g 1-2 g 2 1-1 g 10 + 8g 12 g 2 1-2 g 2 1-1 g 10 + 6g 12 g 2 02 g 2 1-1 g 10 -2g 1-2 g 2 02 g 2 1-1 g 10 + 4g 12 g 1-2 g 2 11 g 2 10 -4g 2 1-2 g 2 11 g 2 10 -g 2 11 g 2 02 g 2 10 -2g 11 g 2 02 g 1-1 g 2 10 -4g 2 12 g 2 1-1 g 2 10 + 4g 12 g 1-2 g 2 1-1 g 2 10 -g 2 02 g 2 1-1 g 2 10 + 12g
The autocovariance variety MA (1,1,1) ⊆ P 13 is 7-dimensional of degree 64. Its prime ideal is cut out by 56 quartics, 90 quintics and 50 sextics.

Table 1 presents the basic properties of the first autocovariance varieties MA (q 1 ,q 2 ) , that is, for d = 2. The dimension appears to be the expected one, while the degree follows a clear pattern as a power of two. We will prove that this actually holds for any MA q . To that end, we use the next two lemmas. Lemma 3.7. The map Γ q ∶ P Q -⇢ P N has no base points.

q 1 q 2 dim(MA q ) N deg(MA q ) generators 1 1 3 4 4
Proof. Assume that Γ q (a) = 0. We know from (2.2) that

θ(x)θ(x -1 ) = t∈Z d γ(t)x t = 0.
Multiplying both sides by the monomial x q = x q 1 1 ⋯x q d d , we obtain the product of two polynomials θ(x) ⋅ x q θ(x -1 ) that equals the zero polynomial. Since the polynomial ring K[x] is an integral domain when K is a field, we must have that either θ(x) = 0 or x q θ(x -1 ) = 0. In particular, all the coefficients a k = 0, that is, a = 0 is the zero vector.

Lemma 3.8. The autocovariance variety MA q is a linear projection of the Veronese variety. Furthermore, γ(t) is the sum of exactly (q 1 -t 1 + 1)⋯(q d -t d + 1) quadratic monomials for every t ∈ [-q, q] with t ⪰ 0, and each monomial appears exactly once.

Proof. The quadratic Veronese embedding precisely consists of all quadratic monomials. The parametrization of MA q consists of quadrics, each one is a sum of quadratic monomials. Moreover, Proposition 2.2 implies that

γ(t) = k,k+t∈[0,q] a k a k+t , (3.8)
for every t ∈ [-q, q], which shows the second part of the assertions. Now we state our main theorem concerning the varieties MA q .

Theorem 3.9. Let q ∈ N d . Then

dim(MA q ) = Q = d i=1 (q i + 1) -1 and if d > 1, then deg(MA q ) = 2 Q-1 = 2 ∏ d i=1 (q i +1)-2 .
Proof. Let D ∶= dim(MA q ) denote the dimension of MA q and consider the regular map Γ q ∶ P Q → P N . Since the domain is Q-dimensional, the inequality D ≤ Q has to hold. However, Γ q is not a constant map and has no base points by Lemma 3.7. As there are no nonconstant regular maps from a projective space to a variety of smaller dimension, we must have equality, that is, D = Q. Now consider the map Γ q as a composition of the quadratic Veronese embedding ν 2 of P Q followed by the linear projection to P N , as in Lemma 3.8. By [15, Proposition 5.5], we have the following equality of degrees:

deg(ν 2 (P Q )) = deg(MA q ) deg(Γ -1 (γ))
where γ ∈ MA q is generic. Here we know that the linear space we project from does not intersect the Veronese variety V Q,2 = ν 2 (P Q ) , by Lemma 3.7. Thus the left hand side equals the degree of the Veronese variety, which is 2 Q . In addition, the identifiability Theorem 4.4 below proves that deg(Γ -1 (γ)) = 2. Hence, we conclude that deg(MA q ) = 2 Q-1 .

Identifiability

We show that all models M A(q) with q ∈ N d are algebraically identifiable (in the sense of [START_REF] Améndola | Algebraic identifiability of Gaussian mixtures[END_REF]). This means that the map from the model parameters to the autocovariances is generically finite to one.

Moving Average Processes

The following result shows that the map Γ q is not injective and the moving average parameters a i are not identifiable from a second order point of view if d = 1. Proposition 4.1. If d = 1, the fibers of a generic point γ ∈ MA q consist of 2 q points. Proof. Let α 1 , ..., α q ∈ C be the q roots of the moving average polynomial

θ(x) = q k=0 a k x k = a q (x -α 1 )⋯(x -α q ), x ∈ C.
Using Proposition 2.2, we see that there are exactly 2 q+1 polynomials which generate γ as above, all of which have the form

±a q (x -α 1 ) ± ⋯(x -α q ) ± , where (x -α i ) + ∶= (x -α i ) and (x -α i ) -∶= (α i x -1). (4.1)
Hence, the fiber of any point in MA q under Γ q ∶ P q → P q consists in general of 2 q points. Proposition 4.1 shows that it is possible to deduce all preimage points from a single one by inverting the roots of θ as suggested in (4.1).

In order to obtain injectivity of Γ q , one usually imposes the condition that all roots α i of the polynomial θ lie strictly outside the unit disk (and a 0 > 0). It follows from the proof above that this is always possible as long as all roots don't lie on the boundary of the unit disk. Compare to [6, Proposition 3.5.1]. This property is also called invertibility in the literature [6, Definition 3.1.4] since it holds if and only if there exists coefficients π 0 , π 1 , ... with ∑ ∞ k=0 π k < ∞ such that the white noise sequence (Z t ) t∈Z can be expressed as

Z t = ∞ k=0 π k Y t-k , t ∈ Z.
Example 4.2. Let q = 1. This is the simplest moving average model M A(1). We have that θ(x) = a 0 + a 1 x and Γ 1 ∶ P 1 → P 1 is given by

Γ 1 (a 0 , a 1 ) = (a 2 0 + a 2 1 , a 0 a 1 ).
The fiber of a generic point γ = (γ 0 , γ 1 ) consists of 2 = 2 1 points in P 1 . They are (ã 0 , ã1 ) and (ã 1 , ã0 ) where

ã0 = γ 0 + γ 2 0 -4γ 2 1 2 , ã1 = γ 0 -γ 2 0 -4γ 2 1 2 . (4.2)
The invertibility condition is equivalent to a 0 > a 1 .

The observed symmetry of the two points (ã 0 , ã1 ) and (ã 1 , ã0 ) above extends to higher q. In fact, it holds that Γ q (a 0 , a 1 , . . . , a q-1 , a q ) = Γ q (a q , a q-1 , . . . , a 1 , a 0 ).

This can be seen from (2.2), where the reversal occurs by inverting all the roots in (4.1).

For general q > 1, there exist algorithms to numerically approximate the invertible solution with a 0 = 1. A basic one is the innovations algorithm, which recursively converges to the moving average parameters a k given the autocovariance values γ(t) under the invertibility condition (we refer to Section 2 in [START_REF] Brockwell | Simple consistent estimation of the coefficients of a linear filter[END_REF] for details). Other approaches use spectral factorization methods [START_REF] Sayed | A survey of spectral factorization methods[END_REF]. While we do not pursue this in this paper, the fact remains that the desired parameters are solutions to a polynomial system of equations, so it would be interesting to compare these with state-of-the-art algorithms in numerical algebraic geometry. See Example 5.8 for an illustration of such techniques.

Furthermore, the symmetry in the polynomial system means that one does not necessarily need to find a root of a polynomial of degree 2 q even when there are 2 q solutions. We illustrate this with q = 2.

Example 4.3. For q = 2 we have θ(x) = a 0 + a 1 x + a 2 x 2 and Γ 2 ∶ P 2 → P 2 given by Γ 2 (a 0 , a 1 , a 2 ) = (a 2 0 + a 2 1 + a 2 2 , a 0 a 1 + a 1 a 2 , a 0 a 2 ).

The fiber of a generic point γ = (γ 0 , γ 1 , γ 2 ) ∈ MA 2 consists of 2 2 = 4 points. A Gröbner basis elimination from the system Γ 2 (a 0 , a 1 , a 2 ) = (γ 0 , γ 1 , γ 2 ) with order a 2 , a 0 , a 1 reveals a triangular system with a quadric in a 2 1 :

a 4 1 -(γ 0 + 2γ 2 )a 2 1 + γ 2 1 = 0 a 2 0 a 1 -γ 1 a 0 + a 1 γ 2 = 0 a 2 a 1 + a 0 a 1 -γ 1 = 0.
And hence the solutions for (a 0 , a 1 , a 2 ) in terms of (γ 0 , γ 1 , γ 2 ) can be obtained as

a 1 = γ 0 + 2γ 2 ± (γ 0 + 2γ 2 ) 2 -4γ 2 1 2 , a 0 = γ 1 ± γ 2 1 -4a 2 1 γ 2 2a 1 , a 2 = γ 1 -a 0 a 1 a 1 .

Moving Average Random Fields

The following result demonstrates a fundamental difference between d = 1 and d > 1 in terms of identifiability. On the other hand, it shows how the symmetry in (4.3) generalizes to higher dimensions.

Theorem 4.4. Suppose that the moving average polynomial θ is generic. Then for d > 1, the fibers of a point γ ∈ MA q are only two points a and a ′ in P Q . One is obtained from the other by a ′ k = a q-k for any k ∈ [0, q].

Proof. Let γ be the image of the coefficients a k of a moving average polynomial θ under the mapping Γ q and assume that θ ′ is another polynomial which also generates γ and has coefficients a ′ k . Due to Proposition 2.2, the polynomial equation

θ(x)(x q θ(x -1 )) = θ ′ (x)(x q θ ′ (x -1 )) (4.4)
has to hold. Since generically θ is irreducible, we either have θ ′ = θ or θ ′ = x q θ(x -1 ), which proves the assertion.

Example 4.5. We consider again the autocovariance variety MA (1,1) and assume that γ ∈ MA (1,1) is generated by a generic moving average polynomial θ as in the setting of Theorem 4.4, that is, θ is irreducible. Using Macaulay2 to compute a Gröbner basis, where equation (3.1) is added to the list of polynomials, we obtain the fiber of γ explicitly as

a 00 = γ 00 γ 11 -γ 01 γ 10 -(γ 01 γ 10 -γ 00 γ 11 ) 2 -4γ 2 11 (γ 11 -γ 1-1 ) 2 2(γ 11 -γ 1-1 )
,

a 10 = -γ 00 γ 1-1 + γ 01 γ 10 -(γ 01 γ 10 -γ 00 γ 1-1 ) 2 -4γ 2 1-1 (γ 11 -γ 1-1 ) 2 2(γ 11 -γ 1-1 )
,

a 01 = -γ 00 γ 1-1 + γ 01 γ 10 + (γ 01 γ 10 -γ 00 γ 1-1 ) 2 -4γ 2 1-1 (γ 11 -γ 1-1 ) 2 2(γ 11 -γ 1-1 )
,

a 11 = γ 00 γ 11 -γ 01 γ 10 + (γ 01 γ 10 -γ 00 γ 11 ) 2 -4γ 2 11 (γ 11 -γ 1-1 ) 2 2(γ 11 -γ 1-1 )
, and a ′ 00 = a 11 , a ′ 01 = a 10 , a ′ 10 = a 01 , a ′ 11 = a 00 . Substituting in the formulas from Example 1.1, we observe that the discriminants (a 01 a 10 -a 00 a 11 ) 2 , which are nonnegative in the real case (as they should be when the moving average parameters are real). If however θ is not irreducible, it has to be the product of two linear factors. Then identifiability from the above theorem fails and we have up to 4 preimages in the fiber of γ. We note that this explains the irreducible component (3.4) of the singular locus from Theorem 3.4, which is equivalent to (3.7). In order to see this, we first assume that Equation (3.7) holds. This implies that there exists a constant b * ∈ R such that we have a 01 = b * a 00 and a 11 = b * a 10 . Thus, the polynomial θ satisfies

(γ 01 γ 10 -γ 00 γ 11 ) 2 -4γ 2 11 (γ 11 -γ 1-1 ) 2 , (γ 01 γ 10 -γ 00 γ 1-1 ) 2 -4γ 2 1-1 (γ 11 -γ 1-1 )
θ(x) = (a 00 + a 10 x 1 )(1 + b * x 2 )
and is therefore reducible. On the other hand, assuming that θ(x) = a 00 + a 10 x 1 + a 01 x 2 + a 11 ).

Remark 4.6. If, in contrast to the setting of Theorem 4.4, θ is not irreducible, then the fiber of γ under Γ q consists of more than two preimages, as illustrated in the previous example. By (4.1), the maximum number of preimages is 2 q 1 +⋯+q d and occurs exactly when θ is completely separable, that is, a product of linear forms.

Parameter Estimation

In this section we go one step further and consider the problem of parameter estimation from observed sample points. We consider two methods: least squares estimation and maximum likelihood estimation. Both involve solving polynomial systems of equations. Algebraically, the computational complexity of the estimation problem is measured by the ED degree [START_REF] Draisma | The euclidean distance degree of an algebraic variety[END_REF] of the associated variety in the first case and by the ML degree [START_REF] Améndola | The maximum likelihood degree of toric varieties[END_REF][START_REF] Catanese | The maximum likelihood degree[END_REF] in the second.

Least squares estimation

Let (Y t ) t∈Z d be a M A(q) random field, which by Definition 2.1 has mean zero. If we are given observations of Y on a lattice L = {1, ..., n} d , we can estimate the autocovariance function γ(t) by the empirical autocovariance estimator

γn (t) ∶= 1 B n,t s∈Bn,t Y (t + s)Y (s), t ∈ [-q, q], t ⪰ 0,
where

B n,t ∶= {s ∈ Z d s, s + t ∈ L} and B n,t = d i=1 (n -t i )1 { t i ≤n} .
If γn (t) were exact values, we would be in the situation of the previous section. However, these are just numerical estimates which form a point γn that almost surely lies outside the model MA q . One approach is to project the estimated vector γn onto the autocovariance variety MA q , that is, obtaining γ * n ∈ MA q which has the smallest Euclidean distance to γn :

γ * n ∶= argmin γ∈MAq γ -γn . (5.1)
The number of critical points of this least squares optimization problem is counted by the Euclidean distance degree (ED degree).

Proposition 5.1. The ED degree of MA q is 1 if d = 1. The ED degree of MA (1,1) is 16. The vector u represents a generic choice of γ and the saturation is needed to remove the critical points that lie in the singular locus. After centering the sample, we compute the empirical autocovariances γn = (γ n (0, 0), γn (0, 1), γn (1, 0), γn (1, 1), γn (1, -1)).

By Proposition 5.

with q = (1, k), k = 1, 2, 3, 4
The computation of the ED degree for MA (1,k) with k > 1 is harder than for MA [START_REF] Améndola | The maximum likelihood degree of toric varieties[END_REF][START_REF] Améndola | The maximum likelihood degree of toric varieties[END_REF] . We therefore resort to numerical methods and obtain Table 2 above. The computations suggest the following pattern. for all k > 0.

Note that the optimization problem (5.1) gives a point in MA q and not a corresponding a * ∈ P Q . Theoretically, one could apply the identifiability results of the last section to obtain such a * by a * n ∶= Γ -1 q (γ * n ). However, since γ * n will most often be a numerical approximation, this is not feasible in practice. Instead, one should solve the optimization problem in parametrized form: a * n ∶= argmin a∈Θ γ a -γn , where Θ ⊆ R Q+1 is a compact parameter space. We note then that the ED degree gets multiplied by the algebraic identifiability degree of the model parametrization.

Maximum likelihood estimation

Suppose as before that (Y t ) t∈Z d is a M A(q) random field with mean zero. Furthermore, we assume that n observations Y (t 1 ), ..., Y (t n ) are given, where the vectors t 1 , ..., t n ∈ Z d are ordered according to the lexicographic order. If the driving white noise (Z t ) t∈Z d is Gaussian, then the vector Y ∶= (Y (t 1 ), ..., Y (t n )) ⊺ is Gaussian as well, and its likelihood is of the form

L(a) ∝ Σ -1 2 exp - 1 2 Y ⊺ Σ -1 Y ,
where Σ = Σ(a) is the covariance matrix of Y and Σ its determinant. The maximum likelihood estimator (MLE) is then defined as the value which maximizes the log-likelihood:

ân ∶= argmax a∈Θ - 1 2 log( Σ ) - 1 2 Y ⊺ Σ -1 Y, (5.2)
where Θ ⊆ R Q+1 is a compact parameter space. If Z is not Gaussian, then the latter estimator is called the quasi maximum likelihood estimator (QMLE).

Remark 5.4. In [START_REF] Yao | Gaussian maximum likelihood estimation for ARMA models II: Spatial processes[END_REF] it was shown that under mild assumptions including an invertibility condition, the QMLE ân is consistent as n tends to infinity. Furthermore, a slightly modified version of ân (to account for the edge effect) is shown to be asymptotically normal in [START_REF] Yao | Gaussian maximum likelihood estimation for ARMA models II: Spatial processes[END_REF]Theorem 2].

Conveniently, the optimization problem (5.2) is still algebraic, in the sense that the critical or score equations form a system of rational functions of a ∈ Θ. The number of critical points of the log-likelihood is invariant under generic data Y and this is known as the maximum likelihood degree (ML degree). In terms of likelihood, this is the same definition as ML degree for a discrete finite model. However, we avoid saying 'ML degree of the autocovariance variety' to avoid confusion with the discrete case, where the model is identified by a set of vectors in probability coordinates and hence one can also talk of the ML degree of the associated algebraic variety, as in [START_REF] Huh | The maximum likelihood degree of a very affine variety[END_REF].

We analyze the first nontrivial case, when q = 1 and n = 2. Even this simple model is interesting. It has been observed that the MLE can sometimes correspond to non-invertible models, which in this case is equivalent to a 0 = a 1 , and contrary to what was previously thought, this occurs with positive probability [START_REF] Cryer | Small-sample properties of the maximum likelihood estimator in the first-order moving average model[END_REF].

Proposition 5.5. Consider the M A(1) model with observed sample Y = (Y 1 , Y 2 ). The ML degree is 4, and these four critical points can be divided into three groups:

(1) The parameters a 0 and a 1 satisfy the two equations

a 0 a 1 = Y 1 Y 2 and a 2 0 + a 2 1 = Y 2 1 + Y 2 2 2 .
(2)

a 0 = a 1 = Y 2 1 + Y 2 2 -Y 1 Y 2 3
(3)

a 0 = -a 1 = Y 2 1 + Y 2 2 + Y 1 Y 2 3
If Y 1 Y 2 = 0, then the MLE corresponds to a degenerate model (a

0 a 1 = 0). Otherwise let W = Y 2 1 +Y 2 2
2Y 1 Y 2 and the MLE is given as:

• the point in (3) if -2 < W < 0 • the point in (2) if 0 < W < 2
• the points in (1) otherwise.

Proof. Since we have a M A(1) process and Y = (Y 1 , Y 2 ), we have Σ = γ(0) γ (1) γ( 1) γ(0) , and the log-likelihood takes the form (a 0 , a 1 ) = -

1 2 log((a 2 0 + a 2 1 ) 2 -a 2 0 a 2 1 ) - 1 2 (Y 1 , Y 2 ) a 2 0 + a 2 1 a 0 a 1 a 0 a 1 a 2 0 + a 2 1 -1 (Y 1 , Y 2 ) ⊺ . (5.3)
There are generically four solutions to the system ∂ ∂a 0 = ∂ ∂a 1 = 0. This means the ML degree is 4. The critical points can be divided into the three groups (1), ( 2) and (3) of the statement. In order to find the MLE depending on the values of Y 1 , Y 2 , we evaluate the likelihood function at these 3 groups of points. In fact, substituting a 0 and a 1 from (1), ( 2) and (3) into the log-likelihood function, we obtain

(i) -1 2 log (Y 2 1 -Y 2 2 ) 2 -1 + log(2), (ii) -1 2 log 1 3 (Y 2 1 -Y 1 Y 2 + Y 2 2 ) 2 -1, (iii) -1 2 log 1 3 (Y 2 1 + Y 1 Y 2 + Y 2 2 ) 2 -1.
Computing (i) -(iii) gives the expression

1 2 log 4 Y 2 1 -Y 1 Y 2 + Y 2 2 2 -log 3 Y 2 1 -Y 2 2 2 , which is always nonnegative since 4 Y 2 1 -Y 1 Y 2 + Y 2 2 2 -3 Y 2 1 -Y 2 2 2 = Y 2 1 -4Y 1 Y 2 + Y 2 2 2 ≥ 0.
Analogously, (i) is greater than or equal to (ii) from

(Y 2 1 + 4Y 1 Y 2 + Y 2 2 )
2 ≥ 0. Hence, the first value (i) is always larger than or equal to the values (ii) and (iii), independently of Y 1 and Y 2 . We would conclude that the maximizers are always given by ( 1), but the points may not be real. Indeed, under (1), if

a 0 a 1 > 0 then a 2 0 + a 2 1 a 0 a 1 > 2 and thus W = Y 2 1 + Y 2 2 2Y 1 Y 2 > 2 while a 0 a 1 < 0 implies a 2 0 + a 2 1 a 0 a 1 < -2 and hence W < -2.
Direct inspection reveals that the likelihood for ( 2) is larger than the one for (3) if and only if W > 0. Note that when W = -2 the points (1) and (3) coincide, while W = 2 means that (1) and ( 2) coincide.

Compare our conditions for W with the similar ones found by [START_REF] Cryer | Small-sample properties of the maximum likelihood estimator in the first-order moving average model[END_REF] in their effort of computing the distribution of the MLE in this q = 1, n = 2 case (note the different parametrization in terms of σ, θ). Furthermore, it is gratifying to see that our computations provide a simple explanation for the 'curious' phenomenon that the MLE can belong to a non-invertible model. Algebraically, the points in (1) always maximize the likelihood, but for the specified region of Y 1 , Y 2 these points are strictly complex (even though evaluating at the likelihood yields real values!), which means then that (2) or (3) becomes the MLE.

In [START_REF] Zhang | Fitting MA(q) models in the closed invertible region[END_REF], standard numerical optimization routines were used to find the MLE in samples of M A(q) models with q = 1, 2, 3, 4. The simulations show the MLE can again lie on the non-invertible boundary.

Example 5.6. Consider a M A(1) process with n = 3 sample points Y = (Y 1 , Y 2 , Y 3 ). The ML degree is now 8. The expressions for the two non-invertible models a 0 = a 1 are:

a 0 = a 1 = 3x 2 1 + 4x 2 2 + 3x 2 3 -4x 1 x 2 + 2x 1 x 3 -4x 2 x 3 12
a 0 = -a 1 = 3x 2 1 + 4x 2 2 + 3x 2 3 + 4x 1 x 2 + 2x 1 x 3 + 4x 2 x 3 12
Obtaining closed form expressions for the other 6 critical points is also possible.

For n > 2, the matrix Σ is tridiagonal: it has γ(0) in the diagonal and γ(1) in the upper and lower diagonal. Our ML degree computations of M A(1) for n = 2, 3, . . . reveal the following pattern: Conjecture 5.7. The ML degree of M A(1) for n > 1 sample points is equal to 4(n -1).

In contrast, the pattern for M A(2) is not as clear. The first values for n = 3, 4, . . . are recorded in Table 3 Not unusually, Gröbner basis computations quickly become prohibitive. However, this does not mean that our algebraic approach is not useful. In applied algebraic geometry, this often means one needs to go into numerical techniques. Indeed, as far as we know, the algebraic nature of the ML problem has not been exploited yet, and a numerical algebraic geometry approach brings both a fresh perspective and efficient computational tools. Knowing the ML degree beforehand helps homotopy continuation and monodromy methods find all solutions to the critical equations and thus guarantee that the MLE will be found. In contrast, classical local search methods may only find a local maximum of the likelihood function. One way to compare these methods is to conduct simulation studies such as the one in the next example.

Example 5.8. We simulate 500 independent paths of a M A(1) process with n = 8 observations for each path. A sample path for this process is illustrated in Figure 2.

As moving average parameters we take a 0 = 1 and a 1 = 0.5.

The M A(1) process is driven by i.i.d. standard Gaussian noise. Having simulated the M A(1) process, we proceed by estimating the model parameters with the MLE in (5.2) in two different ways.

For our first approach we use the standard R command optim for minimization of the objective function. As it is standard in time series analysis, we take the output of the innovations algorithm as the initial value for the optimization routine (cf. [START_REF] Brockwell | Simple consistent estimation of the coefficients of a linear filter[END_REF]Section 2]).

For our second approach we differentiate the likelihood function with respect to the moving average parameters and set the derivatives to zero. In order to compute the critical points of the likelihood function we solve the resulting polynomial system using homotopy continuation. This is implemented in the julia package HomotopyContinuation [START_REF] Breiding | Homotopycontinuation. jl: A package for homotopy continuation in julia[END_REF].

Finally, we evaluate the likelihood at every critical point and choose the maximal one. The summary of the estimation results are given in Tables 4 and5 below. We observe that using homotopy continuation reduces the bias for both a 0 and a 1 , whereas it increases the standard deviation for a 1 and decreases the standard deviation for a 0 .

True Value Mean

Bias Std a 0 1.0000 0.8642 -0.1358 0.2459 a 1 0.5000 0.4503 -0.0497 0.4071 Further study of the autocovariance varieties of these models with their ED and ML degrees, as well as robust numerical (algebraic) estimation of their parameters is a natural direction for future work.

Remark 3 . 5 . 7 )

 357 Substituting the parametrization of Example 1.1 into (3.2) to (3.4), we find that the three irreducible components of the singular locus correspond to the three conditions a 10 = a 01 and a 00 = a 11 , (3.5) a 10 = -a 01 and a 00 = -a 11 , (3.6) and a 00 a 11 = a 01 a 10 .(3.These conditions represent submodels and we will analyze Equation (3.7) in more detail in Example 4.5.

  x 1 x 2 = (b 10 + b 11 x 1 )(b 20 + b 21 x 2 ) for some real-valued coefficients b 10 , b 11 , b 20 , b 21 , we can deduce (3.7

Proof.

  The first part is a consequence of Proposition 3.2. In fact, for d = 1 the unique critical point for (5.1) is γ * n = γn . For the second we use the following M2 code: R = QQ[g00,g01,gm11,g10,g11] I = ideal(g01^2*gm11^2-g00*g01*gm11*g10+g01^2*g10^2+gm11^2*g10^2+ g00^2*gm11*g11-2*g01^2*gm11*g11-4*gm11^3*g11-g00*g01*g10*g11 -2*gm11*g10^2*g11+g01^2*g11^2+8*gm11^2*g11^2+g10^2*g11^2-4*gm11*g11^3) sing = ideal singularLocus I u = {5,7,13,11,3}; M = (matrix{apply(# gens R,i->(gens R)_i-u_i)})||(transpose(jacobian I)); time J = saturate(I + minors(2,M), sing); dim J, degree J

Figure 1 :

 1 Figure 1: M A(1, 1) random field in Example 5.2

Figure 2 :

 2 Figure 2: Sample path of a M A(1) process.

Finally, we closeProposition 5 . 9 .γ 00 γ 01 γ 10 γ 11 γ 01 γ 00 γ 1 -1 γ 10 γ 10 γ 1 - 1 γ.

 59111 this section by reporting the ML degree of M A(1, 1): Assume that n = 4 sample points Y = (Y 11 , Y 12 , Y 21 , Y 22 ) over the lattice L = {1, 2} 2 of a M A(1, 1) random field are given. The autocovariance matrix Σ of Y is 00 γ 01 γ 11 γ 10 γ 01 γ 00The ML degree of the model is 192 over P 3 .

  2 12 g 1-2 g 11 g 02 g 01 -16g 12 g 2 1-2 g 11 g 02 g 01 + 4g 3 1-2 g 11 g 02 g 01 + 4g 1-2 g 3 11 g 02 g 01 -2g 12 g 11 g 3 02 g 01 + 2g 1-2 g 11 g 3 02 g 01 + 4g 3 12 g 02 g 1-1 g 01 -16g 2 12 g 1-2 g 02 g 1-1 g 01 + 12g 12 g 2 1-2 g 02 g 1-1 g 01 + 4g 12 g 2 11 g 02 g 1-1 g 01 -8g 1-2 g 2 11 g 02 g 1-1 g 01 + 2g 12 g 3 02 g 00 -8g 12 g 1-2 g 11 g 02 g 1-1 g 00 -4g 2 12 g 02 g 2 1-1 g 00 -4g 2 12 g 1-2 g 11 g 01 g 00 + 12g 12 g 2 1-2 g 11 g 01 g 00 + g 12 g 11 g 2 02 g 01 g 00 + g 1-2 g 11 g 2 02 g 01 g 00 + 12g 2 12 g 1-2 g 1-1 g 01 g 00 -4g 12 g 2

	+ 2g 12 g 2 02 g 10 g 2 01 + 2g 1-2 g 2 02 g 10 g 2 01 -g 2 02 g 2 10 g 2 01 -4g 2 1-2 g 2 11 g
	2 12 g 2 02 g 2 01 + 2g 12 g 1-2 g 2 02 g 2 01 -g 2 1-2 g 2 02 g 2 01 -g 2 11 g 2 02 g 2 01 + 2g 11 g 2 02 g 1-1 g 2 01 -g 2 02 g 2 1-1 g 2 01

02 g 1-1 g 01 -2g 1-2 g 3 02 g 1-1 g 01 -8g 12 g 11 g 02 g

2

1-1 g 01 + 4g 1-2 g 11 g 02 g 2 1-1 g 01 + 4g 12 g 02 g 3 1-1 g 01 -8g

2

1-2 g 11 g 02 g 10 g 01 -2g 11 g 3 02 g 10 g 01 -8g 2 12 g 02 g 1-1 g 10 g 01 -2g 3 02 g 1-1 g 10 g 01 + 4g 1-2 g 11 g 02 g 2 10 g 01 + 4g 12 g 02 g 1-1 g 2 10 g 01 -g

Table 1 :

 1 Summary of first autocovariance varieties for d = 2

	1 quartic

  ). One of the four preimage points in the fiber of γ is given by the equations

	a 01 =	1 2	γ 00 -γ 2 00 -4 (γ 2 01 + γ 2 10 -4γ 2 11 ) --2γ 00 γ 2 00 -4 (γ 2 01 + γ 2 10 -4γ 2 11 ) + 2γ 2 00 -4 (γ 2 01 + γ 2 10 ),
	a 10 =	1 2	γ 00 -γ 2 00 -4 (γ 2 01 + γ 2 10 -4γ 2 11 ) + -2γ 00 γ 2 00 -4 (γ 2 01 + γ 2 10 -4γ 2 11 ) + 2γ 2 00 -4 (γ 2 01 + γ 2 10 ),
	a 11 =	1 2	γ 00 + γ 2 00 -4 (γ 2 01 + γ 2 10 -4γ 2 11 ) + 2γ 00 γ 2 00 -4 (γ 2 01 + γ 2 10 -4γ 2 11 ) + 2γ 2 00 -4 (γ 2 01 + γ 2 10
	a 00 =	1 2	γ 00 + γ 2 00 -4 (γ 2 01 + γ 2 10 -4γ 2 11 ) -2γ 00 γ 2 00 -4 (γ 2 01 + γ 2 10 -4γ 2 11 ) + 2γ 2 00 -4 (γ 2 01 + γ 2 10 ),

Table 2 :

 2 3, we expect 16 complex critical points, and we compute them numerically. ED degree of the variety MA q

	Six of them are real
	(87.1147, 18.6511, -33.4739, 5.78808, -17.312),
	(80.8137, 30.7661, -23.1126, -3.96875, -28.7833),
	(61.9284, -24.7157, -16.0001, 1.76548, 19.994),
	(55.2165, 8.80716, 26.5528, 0.977029, 8.45708),
	(71.9207, -7.85594, -8.51067, 35.9693, 0.649541),
	(63.1632, -18.9463, -12.5151, 0.0189543, 24.6219).
	The first line has the lowest Euclidean distance to the estimated point
	γn = (86.6439, 19.1877, -34.2433, 6.6726, -17.3195),
	and therefore
	γ * n = (87.1147, 18.6511, -33.4739, 5.78808, -17.312).
	Moreover, we have that
	γ -γn = 5.2604 and γ -γ * n = 5.0711,
	so that projecting onto the autocovariance variety improves the empirical estimate.
	q	ED degree
	(1,1)	16
	(1,2)	169
	(1,3)	1600
	(1,4)	14641

Table 3 :

 3 . ML degrees of the M A(2) model by number of sample points

	n ML degree
	3	29
	4	69
	5	129
	6	205

Table 4 :

 4 Parameter estimation results for M A(1) with n = 8 and R command optim.

		True Value Mean	Bias	Std
	a 0	1.0000 0.8818 -0.1182 0.2129
	a 1	0.5000 0.4678 -0.0322 0.5094

Table 5 :

 5 Parameter estimation results for M A(1) with n = 8 and homotopy continuation.

Acknowledgments. We are grateful to Claudia Klüppelberg for her support during this project. We thank Daniele Agostini for helpful conversations. Carlos Améndola was partially supported by the Deutsche Forschungsgemeinschaft (DFG) in the context of the Emmy Noether junior research group KR 4512/1-1. Viet Son Pham acknowledges support from the graduate program TopMath at the Technical University of Munich and the Studienstiftung des deutschen Volkes.

Conjecture 5.3. The ED degree of MA (1,k) equals (3 k+1 -1) 2

Authors' addresses:

Zentrum Mathematik, Technische Universität München Boltzmannstrasse 3, 85748 Garching (b. München), Germany carlos.amendola@tum.de , vietson.pham@tum.de


Guillaume Tronchet

on different levels (local, national and global). At the end of the war, the Ministry of Public Instruction clearly set the tone. At the beginning of the 1920s, it concluded a dozen academic and scientific agreements between France and other countries, mainly for the exchange of professors: with

Introduction

From the late nineteenth century to the 1930s, an increasing number of international students chose to study in France. They were only 500 in 1868. From 6,000 regularly enrolled students before World War I, their number reached to more than 17,000 in the interwar period, making the French university market one of the most attractive in the world. At the same time, the French Third Republic created a high-level academic network with around twenty institutes abroad, several hundred teachers of French language, and university chairs all over the world.

It is generally accepted that this development was a key part of French cultural diplomacy, in a sense a compensatory mechanism for restoring national prestige after the Franco-Prussian War of 1870. But international competition between states does not explain everything, and the concept of "cultural diplomacy" may be a trap for historians in understanding the richness and complexity of international higher education history. The international life of universities involved multiple transnational connections (mainly between institutions, academics, etc.) regardless of governments. Moreover, we must never forget that the concept of "cultural diplomacy," before being a useful category of analysis, was a category of practice, which foreign ministries used to incorporate under their administrative control all international activities distinct from military, economic or political considerations. [START_REF] Bourdieu | Esquisse d'une théorie de la pratique[END_REF] Under the French Third Republic, international higher education was a social space for struggles between different actors in order to dominate the field. Using the concept of "cultural diplomacy" therefore abides by the views of the winners, at the expense of studying specific social dynamics and actors forgotten by history. [START_REF] Tronchet | Savoirs en diplomatie. Une histoire sociale et transnationale de la politique universitaire internationale de la France, années 1870-années 1930[END_REF] For these reasons, it is better to use the concept of "academic diplomacy," meaning the international and transnational actions developed in higher education, whether by state or nonstate actors, provided that such actions bear witness to the autonomy of academia from diplomacy. This chapter attempts to sketch out the rise of French academic diplomacy and the changing landscape of international academic policies throughout the French Third Republic, at the end of which the monopoly of the academic field declined in favor of the diplomatic field and its cultural diplomacy. This chapter will especially focus on mobility scholarships and exchange programs launched during this period, not just as useful examples to reveal what happened, but also to consider them as factors in the progressively weakening of university autonomy in international relations.

A Multilevel Approach

The beginning of the French Third Republic was a key moment for the emergence of international higher education in France and thinking on academic questions in a global context. This process resulted from interactions occurring on different scales (local, national and international), within which new scholarships and exchange programs were founded for the benefit of a small French elite.

National Trends

This chapter appears in a larger collection published by Berghahn Books (https://www.berghahnbooks.com/title/tournesglobal). Tronchet, Guillaume. 2017. "The Defeat of University Autonomy. French Academic Diplomacy, Mobility Scholarships and Exchange Programs (1880s-1930s)." In Global Exchanges. Scholarship and Transnational Circulations in the Modern World, eds. Giles Scott-Smith and Ludovic Tournès. New York: Berghahn Books.

60

On a national scale, two trends of internationalization can be identified. The first one resulted from the fact that republicans who were involved in the reform of the French higher education system (members of the French government, senior members of the Ministry of Public Instruction and academics close to the government) looked beyond the French borders to learn from foreign models-particularly Germany-in order to upgrade the training of French elites [START_REF] Weisz | The Emergence of Modern Universities in France, 1863-1914[END_REF] and strengthen the spread of French language and culture, in accordance with the imperialism of the Third Republic. In this context, in 1873 the Ministry of Public Instruction created the first national scholarships for mobility abroad, mainly to go to German universities. Six French students were selected annually for this award. 4 The same process was related to the foundation, in 1882, of a government office (Service des échanges universitaires) dedicated to the exchange of academic dissertations and scientific publications between French and thirty-nine foreign universities, mostly in Germany (about 45,000 copies were send abroad and 100,000 copies were received in France before 1914). [START_REF] Tronchet | Savoirs en diplomatie[END_REF] The second national trend of internationalization was supported by academics (mainly in modern languages, history and archeology) who were interested in the creation of international mobility scholarships (between 1883 and 1914, about twenty young teachers in modern languages received a government-funded scholarship to study abroad each year) [START_REF] Tronchet | Savoirs en diplomatie[END_REF] or in the foundation of new French schools abroad, in order to develop their own fields of teaching and research (like the Ecole française de Rome, which was founded in 1873 to train a new generation of historians and archeologists).

By the 1880s, these proponents of internationalization met together to share ideas and projects within national and extraparliamentary fora (Alliance française, Société de l'enseignement supérieur, Société pour la propagation des langues étrangères, etc.). These formed a "reformist nebula" 7 in which certain norms of internationalization emerged or

61

stabilized in the 1880-1890s. The "intellectual expansion" of France abroad was not a new theme, 8 but higher education now occupied a central role.

International Trends

At the same time, this internationalization process was also supported by social dynamics on a global scale. First, as Victor Karady has argued, rising levels of international migration in the 1890s led to the development of French courses being offered to foreigners, in order to meet the growing demand coming from national elites who lacked a higher education system, or people forced into academic exile because they had lost the right of access to higher education in their home country. [START_REF] Karady | La République des lettres des temps modernes. L'internationalisation des marchés universitaires occidentaux avant la Grande Guerre[END_REF] Among the forty nationalities represented in French universities before World War I, 51 percent of the international students came from the Russian Empire (including many Jewish refugees), while a majority of others came from Eastern Europe and the Ottoman Empire. At the time, there were no French scholarships for these international students.

In addition, the process was strengthened by the fact that French reformers and academics established new international standards that circulated via the 'global academic web' woven by the back-and-forth movement of actors and ideas in the worldwide higher education community. 10 Through organizing and attending international congresses and events in higher education, French participants benefited from discussions with their peers and ministry officials on topics ranging from the equivalence of degrees between different countries, the reception of international students and the general organization of higher education systems. [START_REF] Tronchet | Savoirs en diplomatie[END_REF] This chapter appears in a larger collection published by Berghahn Books (https://www.berghahnbooks.com/title/tournesglobal). Tronchet, Guillaume. 2017. "The Defeat of University Autonomy. French Academic Diplomacy, Mobility Scholarships and Exchange Programs (1880s-1930s)." In Global Exchanges. Scholarship and Transnational Circulations in the Modern World, eds. Giles Scott-Smith and Ludovic Tournès. New York: Berghahn Books.

62

Local trends

Local initiatives were also crucial in the rise of French academic diplomacy and mobility scholarships. With new standards for international development circulating within the global academic web and the French government keen to reform the university system, French universities launched initiatives to promote themselves on a global scale and thereby increase their number of students. They were assisted by local actors (mayors, lawyers, doctors and merchants) who wanted to develop their social capital, international networks, tourism and other economic opportunities for their cities. The national competition among French universities, especially between Paris and the rest and also between provincial universities themselves, was here a key factor in this international scramble. From the 1890s, academics and philanthropic citizens founded "committees of patronage for foreign students" in a dozen French university towns, such as Paris (1890), Lyon (1891), Nancy (1893) and Grenoble (1896), in order to raise funds, welcome international students and finance international activities. These committees provided information for foreign students on living facilities and finding accommodation in hotels or private homes in university towns. French universities did not have a campus [START_REF] Tronchet | Savoirs en diplomatie[END_REF] and student housing was therefore a key area for helping international students when they arrived in France. "The goal of these committees is to make the student's stay in France agreeable as well as profitable," said one observer. [START_REF]The Reception of Foreign Students in French Universities and Schools[END_REF] Above all, holiday courses (or summer schools) for international students were created, particularly under the influence of the Oxford Summer Meetings, launched in 1888, and the German Ferienkurse, created in 1889. Prior to World War I, French universities took the leading position in organizing such events. [START_REF] Tronchet | Savoirs en diplomatie[END_REF] These initiatives were significant for university budgets, contributed toward developing university course catalogs and stimulated the creation of new local academic degrees highly

63

valued by international students. [START_REF]French universities created about 190[END_REF] To promote these developments, universities also drew up marketing campaigns with the financial support of the committees of patronage, which customized marketing materials (handbooks, posters and advertisements in the international press in many languages). In order to study in France, international students still had to rely on their own funding because the field of scholarships was still limited to French mobility abroad only. Some of the French universities accepted private donations for funding two or three mobility scholarships a year (like the Albert Kahn Around-the-World Scholarships funded by the University of Paris in 1898), [START_REF] Tronchet | Les bourses de voyage[END_REF] while other universities invested in building branch campuses abroad at the beginning of the 1900s in order to mobilize transnational networks of municipal communities. In this way, the University of Lyon was very active in the Middle East with the foundation of a Law School in Beirut in 1913 (where some local students could learn French law), while Grenoble turned to Italy, Bordeaux and Toulouse to Spain, and Lille to Great Britain. The first French institutes of Florence, Madrid and London were also created in the early twentieth century. 17

Toward a State Public Policy

The 1910s represented a turning point in the history of international academic relations for France. For the first time, and with different names-"foreign policy of universities" and "university expansion"-French academic diplomacy no longer depended only on uncoordinated local initiatives, intellectual debates and public authorities, but became a state public policy, conceptualized as a whole, and openly debated in Parliament and the national press. The beginning of the first exchange programs of students and professors launched or controlled by the government introduced a new framework for the landscape of scholarships.

The Nationalization of French Academic Diplomacy

This nationalization process saw a new generation of actors, often born in the 1870s, who thought that international higher education policies could serve their own careers, such as Government administration therefore became more present in the process, but the lack of centralized and dedicated offices for international academic affairs caused the organization of French academic diplomacy to develop as a federal apparatus. Provincial university initiatives, which sometimes tried to group themselves together for international activities without governmental agreement, were frozen. This nationalization process was not peculiar to France: national regulation of international academic affairs also took place in Germany, Spain (with the creation in 1907 of the Junta para Ampliacion de estudios e investigaciones scientificas) and Britain (with the creation in 1912 of the Universities Bureau of the British Empire). Far from narrowing the field of possibilities, these new national strata elevated existing initiatives, but did not replace them, instead shaping the opportunities for new repertoires of action. Ultimately, interstate cooperation would be improved by World War I.

World War I and Interstate Cooperation

The outbreak of war obviously impacted the international academic arena. [START_REF] Irish | The University at War, 1914-25: Britain, France, and the United States[END_REF]

66

(allied, neutral or enemy). The war severed connections with German higher education, while it strengthened those with Allied nations. This situation disrupted the transnational dynamics of cooperation between academics and institutions, and led French academic diplomacy into interstate cooperation. The first step of this process was the arrival of thousands of Serb refugees, including students, and the conclusion of an academic exchange agreement between the two countries in November 1916, which provided the first French government scholarships for foreign students. The second step concerned the presence of US demobilized soldier-students in France, which led in 1919-20 to the organization by the ONUEF of an academic program for 5,000-6,000 American students in French universities. [START_REF] Tronchet | Savoirs en diplomatie[END_REF] These two episodes pushed all actors involved to take into account the increasing need for reciprocity in international higher education standards, building a bridge between the French imperialism of the nineteenth century and the intellectual cooperation that would shape international academic affairs in the 1920s. The decision to allow universities the right to confer honorary degrees on international figures-the Sorbonne conferred the first French degree of Doctor Honoris causa on President Woodrow Wilson in December 1918-was one of the symbols of this new state of affairs.

The French-Serbian Academic Exchange Agreement of 1916

From the beginning of World War I, France took several academic initiatives to support her Serbian ally: a Source: Annuaire statistique de la France, 1916-18, 1919-20 At the same time, French decision-making in international academic affairs increasingly followed a top-down process. With the general increase of the government's power during the war [START_REF] Roussellier | La force de gouverner. Le pouvoir exécutif en France, XIXe-XXe siècles[END_REF] and the ONUEF gradually becoming a pro-internationalization official higher education agency, universities were no longer able to stay at the forefront of international innovation. Gone were the days when back and forth was the norm between actors operating

69

of France after World War I, this leading position was due to the increasing investment of the government in the field of international higher education, which pursued a very liberal policy until the mid 1930s. [START_REF] Weil | How to Be French: Nationality in the Making since 1789[END_REF] But academia no longer enjoyed the autonomy it had possessed in the 1900s for its international activities. The universities and the Ministry of Public Instruction were of course still involved in academic diplomacy, especially through the ONUEF, but the main change in the field of international academic affairs concerned the French Ministry of Foreign Affairs developing its own cultural diplomacy to compete with other nations, especially the fascist countries. In 1920, the Service des oeuvres françaises à l'étranger was created to reorganize academic mobility abroad, to stimulate the foundation of French cultural institutes in strategic countries and to manage grants for international students in

France. It was a turning point for academic exchanges, which not only became part of an imperial policy of academic grandeur, but were also progressively integrated into French foreign policy, under the control of diplomats. At the beginning of the twentieth century, academic exchange programs had not been based on diplomatic efforts to isolate Germany, like the Franco-Russian Alliance or the Entente Cordiale. Following World War I, academic activities at the international level were openly in line with the new goals of French foreign policy, mainly toward Central and Eastern Europe.

For example, new French Institutes were founded in Prague (1920), Zagreb (1922), Sofia (1922), Damascus (1922), Bucharest (1924), Warsaw (1925), Berlin (1930) Europe. Each year, about 800 international students (approximately 5 percent of the total number of students who came to France) benefited from fee reductions and government scholarships created by the Ministry of Public Instruction. The Russians were the most numerous (about 300 students), followed by the Yugoslavs (about 200) and Romanians (about 100). The Ministry of Foreign Affairs also created its own scholarships for about 100 international students selected each year by French embassies and consulates, with a focus on Czechs (fifteen to twenty scholarships per year), Yugoslavs (about fifteen scholarships),

Poles (about fifteen scholarships), Hungarians (between eight and ten scholarships) and Bulgarians (between six and nine scholarships). [START_REF] Tronchet | Savoirs en diplomatie[END_REF] To ensure that the accommodation conformed to new hygiene and healthcare standards, As there were frequent conflicts within the French government between actors related either to the Ministry of Public Instruction or to the Ministry of Foreign Affairs, [START_REF] Tronchet | Un bluff perpétuel'. Les dessous de la présence française dans l'Université turque (années 1930)[END_REF] the This instrumental political reaction was part of a growing global trend. At the end of the interwar period, international higher education was certainly not just an arena reserved for academics or pacifists. Specialists in comparative education were under no illusions. In 1933, in a critical review of a book on intellectual relations, an observer pointed out this ambiguity: "many will be inclined to wonder what meaning is to be attached to the phrase 'international understanding' when the funds expended by the French Government in bringing students from Yugoslavia to Paris are cited as evidence of its pursuit in the section dealing with International Student Exchange." [START_REF][END_REF] In the middle of the 1930s, international understanding or peace was clearly not a French priority in relation to organizing academic exchanges.

Conclusion

The economic crisis and the rise of nationalisms in the 1930s, in shrinking the access to national academic markets, generated a contraction in the internationalization of higher education and an increasing role for governments in academic affairs, both of which induced a decline in university autonomy. In this context, reciprocity was less an academic ideal than

  newly elected members of Parliament seeking a national cause to distinguish themselves in the political sphere or academics interested in going into senior public service. In 1910, all these actors teamed together in the Office national des universités et écoles françaises (ONUEF), an interest group that obtained investment credits from the Parliament for French academic diplomacy. A new budget line was introduced in 1912 via the Ministry of Public Instruction to support international academic activities. The ONUEF itself received subsidies to advise on the organization of French academic diplomacy and to propose reforms. Up to 1914, its director, Professor Jules Coulet, was very active. Social rights for French teachers who wanted to teach abroad were obtained in 1913 (their years abroad being recognized as part of their career progression); job offers for teachers of French were methodically listed by the ONUEF; the creation of French studies departments in foreign universities was encouraged (the University of Reykjavik in 1911, the universities of London and Amsterdam in 1912, the universities of Copenhagen, Helsinki and Oslo in 1913, and the universities of Kansas and Minnesota in 1914); a French handbook of higher education was published in 1914; and more equal terms for international students were introduced (equivalence of degrees and equal tuition fees for French and foreign students). The first exchange programs between French and U.S. universities were negotiated and managed at a national level by the ONUEF. In 1911 an agreement was made between Harvard and the Sorbonne for an annual exchange of two professors (the first exchange professors were William M. Davis, Professor of Geology at Harvard, and Charles Diehl, Professor of Byzantine History at the Sorbonne),This chapter appears in a larger collection published by Berghahn Books (https://www.berghahnbooks.com/title/tournesglobal). Tronchet, Guillaume. 2017. "The Defeat of University Autonomy. French Academic Diplomacy, Mobility Scholarships and Exchange Programs (1880s-1930s)." In Global Exchanges. Scholarship and Transnational Circulations in the Modern World, eds. Giles Scott-Smith and Ludovic Tournès. New York: Berghahn Books. 65 and in 1912 a similar arrangement was made for a young French professor to reside at Columbia in New York.

  Some projects were brought to a standstill, like those for the development of student exchange. By 1915, French universities had lost 70 percent of their international students. Conversely, other sectors were boosted, like the sending of books and academic speakers abroad for French propaganda. In 1915 the ONUEF was charged by the government to send hundreds of thousands of copies of the Manifesto by French Universities to the Universities of Neutral Countries, published in November 1914, all over Europe. But the main change was the reorganization of French academic relationships with foreign universities according to the diplomatic position of their country vis-à-vis France

"

  Serbian Day" was launched in March 1915 by the Ministry of Public Instruction to raise funds for Serbian school children; conferences on Serbia were organized in universities; and propaganda books were published. In the autumn of 1915, the retreat of the Serbian army caused a wave of refugees. French academic diplomacy was mobilized and the Ministry of Public Instruction founded a committee in December 1915 to coordinate the This chapter appears in a larger collection published by Berghahn Books (https://www.berghahnbooks.com/title/tournesglobal). Tronchet, Guillaume. 2017. "The Defeat of University Autonomy. French Academic Diplomacy, Mobility Scholarships and Exchange Programs (1880s-1930s)." In Global Exchanges. Scholarship and Transnational Circulations in the Modern World, eds. Giles Scott-Smith and Ludovic Tournès. New York: Berghahn Books. 67 reception of Serbian refugees in the French education system. The ONUEF led the operations for the reception of hundreds of Serbian students in universities. In order to strengthen cooperation, France and Serbia signed a convention in Corfu, on 9 November 9, 1916, and another one in Paris, on 27 November 27, 1917. These agreements provided scholarships for Serbian students who wanted to study in France (400 were created by the Serbian Government and 100 by the French government each year); courses in Serbian for students in French universities (a chair of Serbian language and literature was created at the Ecole nationale des langues orientales); and conferences on Serbia for French students. This was the first academic agreement concluded between two states on such a large scale (in 1918, 44 percent% of international students in French universities were Serbs) and based on reciprocity. It became a model for all of the academic agreements concluded in the 1920s by France with other countries.

the

  figures of transnational philanthropy (such as Murry Guggenheim, John D. Rockefeller, Jr. and David David-Weill).[START_REF]La Babel étudiante. La Cité internationale universitaire de Paris[END_REF] Before World War II, up to 2,500 French and international

  , and scholarships and exchange programs founded after World War I were under the control of diplomatic even when programs were still managed by universities or the Ministry of Public Instruction. This "diplomatization" process was not temporary either. World War II and the postwar context would not fundamentally change the new dynamics of international educational exchange. Guillaume Tronchet is professeur agrégé d'histoire, research associate at the Institut d'histoire moderne et contemporaine (ENS, University of Paris 1 Panthéon-Sorbonne, CNRS) and special adviser to the director of the Ecole normale supérieure. He holds a Ph.D. from the University of Paris 1 Pantheon-Sorbonne and specializes in international history of higher education. He has coedited a book on the history of the Cité internationale universitaire de Paris with Dzovinar Kévonian: La Babel étudiante. La Cité internationale universitaire de Paris (Presses universitaires de Rennes, 2013). He is preparing a single-authored monograph on the internationalization of French Higher Education in the nineteenth and twentieth centuries.

  This chapter appears in a larger collection published by Berghahn Books (https://www.berghahnbooks.com/title/tournesglobal).Tronchet, Guillaume. 2017. "The Defeat of University Autonomy. French Academic Diplomacy, Mobility Scholarships and ExchangePrograms (1880s-1930s)." In Global Exchanges. Scholarship and Transnational Circulations in the Modern World, eds. Giles Scott-Smith and Ludovic Tournès. New York: Berghahn Books.

  This chapter appears in a larger collection published by Berghahn Books (https://www.berghahnbooks.com/title/tournesglobal).Tronchet, Guillaume. 2017. "The Defeat of University Autonomy. French Academic Diplomacy, Mobility Scholarships and ExchangePrograms (1880s-1930s)." In Global Exchanges. Scholarship and Transnational Circulations in the Modern World, eds. Giles Scott-Smith and Ludovic Tournès. New York: Berghahn Books.

  This chapter appears in a larger collection published by Berghahn Books (https://www.berghahnbooks.com/title/tournesglobal).Tronchet, Guillaume. 2017. "The Defeat of University Autonomy. French Academic Diplomacy, Mobility Scholarships and ExchangePrograms (1880s-1930s)." In Global Exchanges. Scholarship and Transnational Circulations in the Modern World, eds. Giles Scott-Smith and Ludovic Tournès. New York: Berghahn Books.

	64

  This chapter appears in a larger collection published by Berghahn Books (https://www.berghahnbooks.com/title/tournesglobal).Tronchet, Guillaume. 2017. "The Defeat of University Autonomy. French Academic Diplomacy, Mobility Scholarships and ExchangePrograms (1880s-1930s)." In Global Exchanges. Scholarship and Transnational Circulations in the Modern World, eds. Giles Scott-Smith and Ludovic Tournès. New York: Berghahn Books.

Table 2 . 1

 21 Serbian students in French universities, 1914-21

1914 1915 1916 1917 1918 1919 1920 1921

	Serbian students	103	15	125	586 1,416 1,546 1,437 1,179
	Percentage of Serbians
		1.6	0.8	6.4 24.4 44.2 25.6 28.3 18.2
	among foreign students
	French scholarships for
		-	-	-	84	112	105	107	106
	Serbian students

  This chapter appears in a larger collection published by Berghahn Books (https://www.berghahnbooks.com/title/tournesglobal).Tronchet, Guillaume. 2017. "The Defeat of University Autonomy. French Academic Diplomacy, Mobility Scholarships and ExchangePrograms (1880s-1930s)." In Global Exchanges. Scholarship and Transnational Circulations in the Modern World, eds. Giles Scott-Smith and Ludovic Tournès. New York: Berghahn Books.

  and Vienna (1930). The directors, often academics, were not appointed by universities or the Ministry of Public Instruction, as was the case for French institutes in the 1910s, but by the Ministry of Foreign Affairs. At the same time, many government scholarships, based on the model established by French-Serbian cooperation during World War I, were created as part of a new state strategy to dominate the market of student elites coming from Eastern and CentralThis chapter appears in a larger collection published by Berghahn Books (https://www.berghahnbooks.com/title/tournesglobal). Tronchet, Guillaume. 2017. "The Defeat of University Autonomy. French Academic Diplomacy, Mobility Scholarships and Exchange Programs (1880s-1930s)." In Global Exchanges. Scholarship and Transnational Circulations in the Modern World, eds. Giles Scott-Smith and Ludovic Tournès. New York: Berghahn Books.

	70

  This chapter appears in a larger collection published by Berghahn Books (https://www.berghahnbooks.com/title/tournesglobal).Tronchet, Guillaume. 2017. "The Defeat of University Autonomy. French Academic Diplomacy, Mobility Scholarships and Exchange Programs (1880s-1930s)." In Global Exchanges. Scholarship and Transnational Circulations in the Modern World, eds. Giles Scott-Smith and Ludovic Tournès. New York: Berghahn Books.71academic diplomats tried to preserve their autonomy throughout the 1930s, but without success. The international academic policy of France gradually came under the control of governmental cultural diplomacy. In 1932, when Paul-Henri Siriex, a student in political sciences, benefited from the first scholarship provided by Oxford University to a French student (under a general agreement negotiated by the ONUEF with the universities of Oxford and London to exchange five French and five British students), he clearly received instructions to spy on the British student aristocracy, as well as the activities of German students on the campus, and his reports were duly sent by the ONUEF to the Ministry of Foreign Affairs.[START_REF] Siriex | Souvenirs en vérité[END_REF]

  This chapter appears in a larger collection published by Berghahn Books (https://www.berghahnbooks.com/title/tournesglobal).Tronchet, Guillaume. 2017. "The Defeat of University Autonomy. French Academic Diplomacy, Mobility Scholarships and ExchangePrograms (1880s-1930s)." In Global Exchanges. Scholarship and Transnational Circulations in the Modern World, eds. Giles Scott-Smith and Ludovic Tournès. New York: Berghahn Books.

French Academic Diplomacy, Mobility Scholarships and Exchange Programs (1880s-1930s)