Abbas Ali
email: ali.abbas@math.univ-angers.fr

Assi Abdallah

Semigroup associated with a free polynomial

Keywords: Semigroup, Formal power series, Newton-Puiseux expansions AMS Classification: 05E40, 20M14

come

Introduction

Let K be an algebraically closed field of characteristic zero and let K [[x]] be the ring of formal power series in x = (x 1 , . . . , x e ) over K. Let f = y n + a 1 (x)y n-1 + • • • + a n (x) be a nonzero polynomial of degree n in K[[x]][y]. Suppose that f is a quasi-ordinary polynomial, i.e its discriminant ∆ y (f )(the yresultant of f and its y-derivative), is of the form ∆ y (f ) = x α .ε(x), where ε(x) is a unit in K [[x]] (Note that this is always the case if e = 1). If f is irreducible then, by the Abhyankar-Jung theorem, there exists y = p c p x p ∈ K[[x

n ]] such that f (x, y) = 0. Define the support of y to be the set Supp(y) = {p ∈ N e |c p = 0}. In [START_REF] Lipman | Quasi-ordinary singularities of embedded surfaces[END_REF], Lipman proved that there exists a sequence of elements m 1 n , ..., m h n ∈ Supp(y) such that:

(i) m 1 < m 2 < • • • < m h coordinate wise. (ii) If m n ∈ Supp(y), then m ∈ (nZ) e + h i=1 m i Z.
Moreover, m i / ∈ (nZ) e + j<i m j Z for all i = 1, ..., h.

The semigroup of f is defined to be the set Γ(f ) = {O(f, g), g ∈ K[[x]][y]\(f )}, where O(f, g) is the order of the initial form of the y-resultant of f and g with respect to a fixed order on N e (we also have O(f, g) = ng(x, y(x). Now we can associate with f the following sequences: the D-sequence of f is defined to be D 1 = n e , and for all 1 ≤ i ≤ h, D i is the gcd of the (e, e) minors of the matrix [nI e , m T 1 , ..., m T i ], where T denotes the transpose of a vector. We have D 1 > ... > D h+1 = n e-1 . Then 1 we define the e-sequence to be e i = D i D i+1 for all 1 ≤ i ≤ h, and the r-sequence r 1 0 , ..., r e 0 , r 1 , ..., r e to be r 1 = m 1 , r i = e i-1 r i-1 + m i -m i-1 for all 2 ≤ i ≤ h, and r 1 0 , ..., r e 0 is the canonical basis of Z e . The sequence {r 1 0 , ..., r e 0 , r 1 , ..., r h } is a system of generators of Γ(f ). Moreover, there exists a special set of polynomials g 1 , ..., g h (the approximate roots of f ), such that O(f, g i ) = r i for all i ∈ {1, ..., h}( see [START_REF] Assi | Irreducibility criterion for quasi-ordinary polynomials[END_REF]). The aim of this article is to generalize these results to a wider class of polynomials. Namely let C be a line free rational convex cone in R e and let K C [[x]] be the ring of power series whose exponents are in C. Let f = y n + a 1 (x)y n-1 + • • • + a n (x) be a nonzero polynomial of K C [[x]][y]. We say that f is free if it is irreducible in K C [[x]][y] and if it has a root (then all its roots) y(x)

∈ K C [[x 1 n ]].
Note that irreducible quasi-ordinary polynomials are free with respect to the cone R e + . In general, we associate with a free polynomial f its set of characteristic exponents and characteristic sequences. We also associate with f its set of pseudo-approximate roots and we prove that the set of orders (with respect to a fixed order on Z e ∩ C) of these polynomials generate the semigroup of f , which is defined to be the set of orders of polynomials of K C [[x]][y]. Finally we prove that the semigroup is also generated by the set of orders of approximate roots of f (see Section 3). Note that the semigroup is free in the sense of [START_REF] Assi | The Frobenius vector of a free affine semigroup[END_REF]. This explain the notion of free polynomial. In Section 4 we apply our results to polynomials of K

[[x]][y] = K R e + [[x]][y]. An irreducible polynomial f ∈ K[[x]][y] is not free in general. Our main result is that f becomes free in K C [[x]][
y] for a specific cone, after a preparation result. More precisely let ∆ y (f ) be the y-discriminant of f . If f is a prepared polynomial (in the sense of Remark 4) then f is equivalent, modulo a birational transformation, to a quasi-ordinary polynomial F . This transformation is used in order to go from roots of F to roots of f , and these roots are in

K C [[x 1
n ]] for the cone introduced in Proposition 17.

G-adic expansion and Approximate roots

In this section we recall the notion of G-adic expansion and the notion of approximate roots (see [START_REF] Abhyankar | Expansion Techniques in Algebraic Geometry[END_REF]). Let R[Y ] be the polynomial ring in one variable over an integral domain R.

Proposition 1 Let f be a polynomial of degree n in R[Y ] and let d be a divisor of n. Let g be a monic polynomial of degree n d , then there exist unique polynomials a 1 , ...,

a d ∈ R[Y ] with deg Y (a i ) < n d
for all i ∈ {1, ..., d} such that a i = 0, and

f = g d + a 1 g d-1 + . . . + a d .
This expression is called the g-adic expansion of f . The Tschirnhausen transform of g with respect to f is defined to be τ f (g) = g + d -1 a 1 . Note that the τ f (g) is a monic polynomial of degree n d and so we can define recursively the i th Tschirnhausen transform of g to be τ i f (g) = τ f (τ

(i-1) f (g)) with τ 1 f (g) = τ f (g). By [2], τ f (g) = g if and only if a 1 = 0 if and only if deg(f -g d ) < n -n d .
In this case g is said to be the d th approximate root of f . For every divisor d of n there exists a unique d th approximate root of f . We denote it by App(f, d).

More generally let n = d 1 > d 2 > ... > d h be a sequence of integers such that d i+1 divides d i for all i ∈ {1, ..., h -1}, and set

e i = d i d i+1 , 1 ≤ i ≤ h -1
, and e h = +∞. For all i ∈ {1, ..., h} let G i be a polynomial of degree

n d i (in particular deg Y G 1 = 1) and let G = (G 1 , . . . , G h ). Let B = {b = (b 1 , ..., b h ) ∈ N h , 0 ≤ b i < e i ∀1 ≤ i ≤ h}. Then f can be written in a unique way as f = b∈B c b G b 1 1 . . . G b h h .
We call this expression the G-adic expansion of f .

Line Free Cones

In this section we recall the notion of line free cones, which will be used later in the paper. Let C ⊆ R e . We say that C is a cone if for all s ∈ C and for all λ ≥ 0, λs ∈ C. A cone C is said to be finitely generated if there exists a finite subset {s 1 , ..., s k } of C such that for all s ∈ C, s = λ 1 s 1 + . . . + λ k s k for some λ 1 , ..., λ k ∈ R. If s 1 , . . . , s k can be chosen to be in Q e , then C is said to be rational. From now on we suppose that all considered cones are finitely generated and rational.

Definition 1 Let C be a cone, then C is said to be a line free cone if ∀v ∈ C -{0}, -v / ∈ C.

Given a line free cone, we can define the set of formal power series in several variables with exponents in

C, denoted K C [[x]]. More precisely an element y ∈ K C [[x]
] is of the form y = p=(p 1 ,...,pe)∈C α p x p 1 1 . . . x pe e . It follows from [START_REF] Monforte | Formal Laurent series in several variables[END_REF] that this set is a ring.

Definition 2 Let ≤ be a total order on Z e , then ≤ is said to be additive if for all m, n, k ∈ Z e we have : m ≤ n =⇒ m + k ≤ n + k. An additive order on Z e is said to be compatible with a cone C if m ≥ 0 = (0, ..., 0) for all m ∈ C ∩ Z e .

With these notations we have the following: Proposition 2 (see [START_REF] Monforte | Formal Laurent series in several variables[END_REF]) Let C is a line free cone. There exists an additive total order ≤ which is compatible with C. Moreover, if ≤ is such an total orde, then ≤ is a well-founded order on C ∩ Z e , i.e, every subset of C ∩ Z e contains a minimal element with respect to the chosen order, and this minimal element is unique.

Let y = p c p x p be an element in K C [[x]]. The support of y, denoted Supp(y), is defined to be the set of elements p ∈ C such that c p = 0. It results from Proposition 2 that elements in Supp(y) can be written as an increasing sequence with respect to the chosen additive order on C.

We shall now introduce the notion of free polynomials.

Definition 3 Let C be a line free cone and let

f = y n + a 1 (x)y n-1 + . . . + a n (x) ∈ K C [[x]][y]. Then f is said to be a free polynomial if f is irreducible in K C [[x]][y] and if it has a root y(x) in K C [[x 1 n ]].

Characteristic sequences of a free polynomial

In this section we will introduce the set of characteristic sequences associated with a free polynomial as well as its semigroup. Let C be a line free cone and let ≤ be an additive order on Z e compatible with C.

Let f ∈ K C [[x]][y] be a free polynomial and let y ∈ K C [[x 1 n ]] be a root of f . Let L be the field of fractions of K C [[x]] and set L 1 = L(x 1 n 1 ), L 2 = L 1 (x 1 n 2 ), ..., L n = L n-1 (x 1 n e ) = L(x 1 n 1 , ..., x
1 n e ). Then L n is a galois extension of L of degree n e . Let finally U n be the set of n th roots of unity in K.

Let θ ∈ Aut(L n /L). For all i = 1, ..., e we have θ(x

1 n i ) = ω i x 1 n i for some ω i ∈ U n . Then θ(x p n ) = kx p n ,
where k is a non zero element of K. Let Roots(f ) = {y 1 , . . . , y n } be the conjugates of y over L, with the assumption that y 1 = y = c p x p n . Then for all 2 ≤ i ≤ n there exists an automorphism θ ∈ Aut(L n /L) such that y i = θ(y), hence

y i = θ(y) = c p k p x p n , k p ∈ K * , and consequently Supp(y) = Supp(y i ). Let z(x) ∈ K C [[x 1 n ]].
Then Supp(z(x)) can be arranged into an increasing sequence with respect to ≤. We define the the order of z, denoted O(z), to be O(z) = inf ≤ (Supp(z)) if z = 0, and O(0) = -∞. We set LM (z) = x p n where p = O(z), and we call it the leading monomial of z. We set LC(z) = c O(z) and we call it the leading coefficient of z. We finally set Inf o(z) = LC(z)LM (z) and we call it the initial form of y.

Definition 4 Let the notations be as above with {y 1 , ..., y n } = Roots(f ) and y 1 = y. The set of characteristic exponents of f is defined to be {O(y i -y j ), y i = y j }. Similarly we define the set of characteristic monomials of f to be {LM (y i -y j ), y i = y j }.

Next we will give some properties of the set of characteristic exponents.

Proposition 3 Let the notations be as above. Then the set of characteristic exponents of f is equal to the set {O(y k -y), y k = y}. In particular the set of characteristic monomials of f is given by

{LM (y k -y), k = 2, ..., n} = {LM (θ(y) -y), θ(y) = y, θ ∈ Aut(L n /L)}.
Proof. We only need to prove that any characteristic exponent is of the form O(y k -y) for some k. Let 1 ≤ i = j ≤ n and let c ij = LC(y i -y j ) and M ij = LM (y i -y j ), then y i -

y j = c ij M ij + ij where ij ∈ L n and O( ij ) > O(M ij ). Let θ ∈ Aut(L n /L), such that θ(y j ) = y, then θ(y i ) = y k for some 1 ≤ k ≤ n, and θ(y i -y j ) = θ(y i ) -θ(y j ) = y k -y = c k1 M k1 + k1 = θ(c ij M ij + ij ) = c ij αM ij + θ( ij ) with α = 0, O( k1 ) > O(M k1 ), and O(θ( ij )) > O(M ij ). Hence M k1 = M ij = LM (y i -y j )
. This proves our assertion.

Let {M 1 , . . . , M h } be the set of characteristic monomials of f and write M i = x m i n . Then {m 1 , . . . , m h } is the set of characteristic exponents of f . We shall suppose that m 1 < m 2 < ... < m h . Proposition 4 Let the notations be as above. We have

L(y) = L(M 1 , ..., M h ). Proof. Let θ ∈ Aut(L n /L(y)), then θ is an L-automorphism of L n with θ(y) = y. We have θ(y) = θ( c p x p n ) = c p θ(x p n ) = c p k p x p n = y = c p x p n
, with k p = 0 for all p ∈ Supp(y), and so θ(x p n ) = x p n . Hence x p n ∈ L(y) for all p ∈ Supp(y). In particular, since M 1 , ..., M h are monomials of y, then M 1 , ..., M h ∈ L(y), and so

L(M 1 , ..., M h ) ⊂ L(y). Conversely, if θ ∈ Aut(L n /L(M 1 , .., M h )), i.e if θ is an L automorphism of L n such that θ(M i ) = M i ∀ i = 1, ..., h, then θ(y) = y. In fact if θ(y) = y then θ(y) -y = cM i + i for some characteristic monomial M i , hence θ(M i ) = M i
which contradicts the hypothesis. This proves our assertion.

Note that for all i ∈ {1, ..., h}, L(M 1 , ...

, M i ) = L[M 1 , ..., M i ] since M i is algebraic over L.
Proposition 5 Let the notations be as above. If m ∈ Z e ∩ Supp(y) then m ∈ (nZ) e + h i=1 m i Z.

4 Proof. Write M = x m n . Since M is a monomial of y, then M ∈ L(y) = L[M 1 , ..., M h ], hence M = f 1 g 1 M α 1 1 1 . . . M α 1 h h + . . . + f l g l M α l 1 1 . . . M α l h h for some f 1 , ..., f l , g 1 , ..., g l ∈ K C [[x]
] and l ∈ N * , and so g 1 . . .

g l M = f 1 g 2 . . . g l M α 1 1 1 . . . M α 1 h h + . . . + f l g 1 . . . g l-1 M α l 1 1 . . . M α l h h . Comparing both sides we get that x b = LM (g 1 . . . g l M ) = x a M α i 1 1 . . . M α i h
h for some i ∈ {1, ..., l} and a ∈ Z e . In particular

nb + m = na + α i 1 m 1 + ... + α i h m h , and so m = n(a -b) + α i 1 m 1 + ... + α i h m h ∈ (nZ) e + h i=1 m i Z.
Remark 1 Write F 0 = L and for all i ∈ {1, . . . , h},

F i = L[M 1 , ..., M i ] = F i-1 [M i ]. Also let G 0 = (nZ)
e and for all i ∈ {1, . . . , h}, G i = (nZ) e + i j=1 m j Z. As in Proposition 5, we can prove that for any monomial

M = x m n with m ∈ C, we have M ∈ F i ⇔ m ∈ G i .
Next we will define the set of characteristic sequences associated with f . Definition 5 Let the notations be as above and let {m 1 , ..., m h } be the set of characteristic exponents of f . Let I e be the e × e identity matrix. We shall introduce the following sequences:

• The GCD-sequence {D i } 1≤i≤h+1 , where D 1 = n e and for all i ∈ {1, ..., h}, D i+1 = gcd(nI e , m T 1 , . .., m T i ), the gcd of the (e, e) minors of the e × (e + i) matrix (nI e , m T 1 , ..., m T i ).

• The d-sequence {d i } 1≤i≤h+1 , where d i = D i D h+1 . • The e-sequence {e i } 1≤i≤h , where e i = D i D i+1 = d i d i+1 .
• The r-sequence {r 1 0 , ..., r e 0 , r 1 , ..., r h }, where (r 1 0 , ...r e 0 ) is the canonical basis of (nZ) e , r 1 = m 1 , and for all i ∈ {2, ..., h} r i = e i-1 r i-1 +m i -m i-1 . Note that for all i ∈ {2, ..., h}, r

i d i = r 1 d 1 + i k=2 (m k - m k-1 )d k = i-1 k=1 (d k -d k+1 )m k + m i d i .
Remark 2 Let the notations be as in Definition 5 and let v be a non zero vector in Z e . Let D be the gcd of the (e, e) minors of the matrix (nI e , m T 1 , ..., m T i , v T ), then v ∈ (nZ) e + i j=1 m j Z if and only if D i+1 = D. More generally, D i+1 D .v ∈ (nZ) e + i j=1 m j Z and if D i+1 > D then for all

1 ≤ k < D i+1 D , kv / ∈ (nZ) e + i j=1 m j Z.
Proposition 6 For all i = 1, ..., h -1 let

H i = L(M = x m n , m ∈ Supp(y), m < m i+1 ). Then we have (i) F i = H i and m i does not belong to F i-1 (ii) [F i : F i-1 ], the degree of extension of F i over F i-1 , is equal to e i .
Proof. (i) Since m j < m i+1 for all j = 1, ..., i, then m 1 , ..., m i ∈ H i , and so F i ⊆ H i . In order to prove that H i ⊆ F i , consider a monomial M of y such that M < M i+1 . For each θ ∈ Aut(L n /F i ), θ is an L automorphism of L n and θ(M j ) = M j for all j < i + 1. Hence LM (θ(y) -y) ≥ M i+1 , and so θ(M ) = M for all M < M i+1 , hence M ∈ F i . Finally we get that H i = F i . Now to prove that m i / ∈ F i-1 , let θ ∈ Aut(L n \L) such that θ(y) -y = cM i + ε with O(ε) > m i and c a non zero constant (such a θ obviously exists since M i is a characteristic monomial of f ), then θ(M j ) = M j for all j = 1, ..., i -1 and θ(M i ) = M i , and so θ ∈ Aut(L n \F i-1 ) with θ(

M i ) = M i , hence M i does not belong to F i-1 . (ii) Since M i / ∈ F i-1 , then m i / ∈ G i-1
, and so D i > D i+1 . Moreover e i m i ∈ G i-1 and for all 0 < α < e i we have αm i / ∈ G i-1 . Now let g = y l + a 1 y l-1 + ... + a l be the minimal polynomial of M i over F i-1 and suppose that l < e i . Since g(M i ) = 0, then there exists some k ∈ {0, ..., l-1} such that x l m i n = x α n x km i n for some α ∈ G i-1 , and so (l -k)m i = α ∈ G i-1 with 0 < l -k < e i which is a contradiction. Hence l ≥ e i . But g divides Y e i -x e i • m i n . Hence g = Y e i -x e i • m i n , and consequently [F i :

F i-1 ] = e i .
Proposition 7 Let the notations be as above. For all i ∈ {1, ..., h} we have e i r i ∈ (nZ) e + i-1 j=1 r j Z. Moreover, αr i / ∈ (nZ) e + i-1 j=1 r j Z for all 1 ≤ α < e i .

Proof. We can easily prove that r i = m i + i-1 j=1 (e j -1)r j for all i ∈ {2, ..., h}, hence each of the sequences (m k ) 1≤k≤h and (r k ) 1≤k≤h can be obtained from the other and (nZ) e + i j=1 r j Z = (nZ) e + i j=1 m j Z for all i ∈ {1, ..., h}. In particular, for all α ∈ N, αr i ∈ (nZ) e + i-1 j=1 r j Z if and only if αm i ∈ (nZ) e + i-1 j=1 m j Z. Let i ∈ {1, ..., h}. By Remark 2, e i m i = D i D i+1 m i ∈ (nZ) e + i-1 j=1 m j Z and αm i / ∈ (nZ) e + i-1 j=1 m j Z for all 1 ≤ α < e i . Hence e i r i ∈ (nZ) e + i-1 j=1 r j Z and αr i / ∈ (nZ) e + i-1 j=1 r j Z for all 1 ≤ α < e i . For all i ∈ {1, . . . , h}, define the following sets:

Q(i) = {θ ∈ Aut(L n /L)|O(y -θ(y)) < m i }, R(i) = {θ ∈ Aut(L n /L)|O(y -θ(y)) m i } and S(i) = {θ ∈ Aut(L n /L)|O(y -θ(y)) = m i }.
With these notations we have the following:

Proposition 8 #R(i) = D i and #S(i) = D i -D i+1
, where # stand for the cardinality.

Proof. We have θ ∈ R(i) ⇔ θ(M j ) = M j for all j < i ⇔ θ ∈ Aut(L n /L(M 1 , ..., M i-1 )), hence #R(i) = #Aut(L n /L(M 1 , ..., M i-1 )) = [L n : L(M 1 , ..., M i-1 )] = [L n : F i-1 ]. By proposition 6 we have [F i-1 : L] = [F i-1 : F i-2 ] • • • [F 1 : L] = e i-1 • • • e 1 = D 1 D i = n e D i . But [L n : L] = [L n : F i-1 ][F i-1 : L] = n e , then [L n : F i-1 ] = D i , and so #R(i) = D i . Now R(i + 1) ⊂ R(i) and θ ∈ S(i) if and only if O(y -θ(y)) = m i if and only if θ ∈ R(i) and θ / ∈ R(i + 1), hence #S(i) = D i -D i+1 .
The statement and the proof of Proposition 8 imply the following: for all i ∈ {1, . . . , h}, let R(i) = {y k |O(y -y k ) ≥ m i } and S(i) = {y k |O(y -y k ) = m i }. We have:

Proposition 9 # R(i) = d i and # S(i) = d i -d i+1 .

Pseudo roots, semigroup, and approximate roots of a free polynomial

Let the notations be as above. For all i ∈ {1, ..., h} we will define a specific free polynomial G i , called the i th pseudo root of f such that O(G i (x, y(x))) = r i . Also we will define the semigroup Γ(f ) of f and we will construct a system of generators of Γ(f ). Finally we will prove that O(f, App(f, d i )) = r i for all i ∈ {1, . . . , h}. Let y(x) = c p x p n be a root of f and let m ∈ Supp(y). We set y <m = p<m c p x p n and we call y <m the m-truncation of y.

Definition 6 For all i ∈ {1, . . . , h}, we define the i th pseudo root of f to be the minimal polynomial of y <m i over L. We denote it by G i .

In the following we shall study the properties of G i . In particular we shall prove that O(f, G i ) = r i .

Proposition 10 Let the notations be as above. For all i = 1, ..., h, deg y

(G i ) = n e D i = n d i .
Proof. By proposition 6 we have L(

y <m i ) = L(M 1 , .., M i-1 ). In particular deg y (G i ) = [L(y <m i ) : L] = [L(M 1 , ..., M i-1 ) : L] = n e D i = n d i .
Proposition 11 The polynomial G i is free, and its characteristic exponents are m 1 d i , ..., m i-1 d i .

Proof. The polynomial G i is free from the definition. We shall prove that

y <m i ∈ K C [[x 1 n d i ]]. Let x
λ n be a monomial of y <m i , then λ ∈ (nZ) e + i-1 j=1 m j Z. Let D be the gcd of the minors of the matrix (m 1 0 , ..., m e 0 , m 1 , ..., m i-1 , λ), then D = D i . For all l ∈ {1, ..., e} the matrix A l = (m 1 0 , ..., m l-1 0 , λ, m l+1 0 , ..., m e 0 ) is one of the minors of the matrix (m 1 0 , ..., m e 0 , m 1 , ..., m i-1 ), then D i divides Det(A l ). Write λ = (λ 1 , ..., λ e ), then obviously Det(A l ) = n e-1 λ l , and so D i divides n e-1 λ l for all l ∈ {1, ..., e}. It follows that n e-1 λ D i = λ d i ∈ Z e . Moreover, since λ ∈ C, and

1 d i ≥ 0, then λ d i ∈ C. Hence x λ n = x λ n d i where λ = λ d i , and so x λ n ∈ K C [[x 1 n d i ]].
Let θ(y <m i ) be a conjugate of y <m i , then obviously LM (θ(y <m i ) -

y <m i ) = x m j n
for some j ∈ {1, ..., i -1}. But

m j n = m j d i n d i , hence the set of characteristic monomials of G i is { m 1 d i , ..., m i-1 d i }.
Proposition 12 Let the notations be as above. For all i ∈ {1, . . . , h}, we have O(f (x,

y <m i (x)) = r i d i n .
Proof. We have f (x,

y <m i ) = n k=1 (y <m i -y k ) with the assumption that y = y 1 . Clearly O(y <m i - y k ) = O(y 1 -y k ) if O(y 1 -y k ) < m i
n and m i n otherwise. It follows from Proposition 9 that O( n k=1 (y <m i -

y k ) = 1 n ( i-1 k=1 (d k -d k+1 )m k + d i m i ), which is equal to r i d i n by Definition 5. Let g = y m + b 1 (x)y m-1 + . . . + b m (x) be free polynomial of K C [[x]]
[y] and let z 1 , . . . , z m be the set of roots of

g in K[[x 1 m ]]. We set O(f, g) = n i=1 O(g(x, y i (x)). Clearly O(f, g) = m j=1 O(f (x, z j (x)) = O(g, f ) = O(Res y (f, g)),
where Res stand for the y-resultant of f, g. As a corollary of Proposition 12 we get the following:

Corollary 1 With the notations above, we have O(f, G i ) = r i Proof. In fact, O(f, G i ) = O(G i , f ) = n d i O(f (x, y <m i )) = r i .
As a corollary we get the following:

Proposition 13 Let {G 1 , ..., G h } be the set of pseudo roots of f . Let i ∈ {1, ..., h}, then we have O(G i , G j ) = r j d i for all j ∈ {1, ..., i -1}.
Proof. This is an immediate consequence of Corollary 1 because G 1 , . . . , G i-1 is the set of pseudoapproximate roots of G i and the r sequence of G i is given by

r 1 0 d i , . . . , r e 0 d i , r 1 d i , . . . , r i-1 d i . Definition 7 Given g ∈ K C [[x]][y], f |g, we set O(f, g) = n i=1 O(g(x, y i )) = nO(g(x, y(x)). Clearly O(f, g 1 g 2 ) = O(f, g 1 )+O(f, g 2 ). It follows that Γ(f ) = {O(f, g)|g ∈ K C [[x]][y]\(f )} is a subsemigroup of Z e .
We call it the semigroup associated with f .

In the following we will be prove that (r 1 0 , . . . , r e 0 , r 1 , . . . , r h ) is a system of generators of Γ(f ). We shall need the following result: Lemma 1 Let the notations be as above and let α = (α 1 0 , . . . , α e 0 , α 1 , . . . , r h ), β = (β 1 0 , . . . , β e 0 , β 1 , . . . , β h ) be two elements of Z e × N h such that 0 ≤ α i , β i < e i for all i ∈ {1, . . . , h}. If a = e i=1 α i 0 r i 0 + h j=1 α j r j = e i=1 β i 0 r i 0 + h j=1 β j r j then α = β.

Proof. Suppose that α = β and let k be the smallest integer ≥ 1 such that

α i = β i for all i ≥ k + 1. Suppose that α k > β k . We have (α k -β k )r k = e i=1 (β i 0 -α i 0 )r i 0 + k-1 j=1 (β j -α j )r j . This contradicts Proposition 7. Lemma 2 Let g ∈ K C [[x]]
[y] and suppose that f |g. There exists a unique θ = (θ 1 0 , . . . , θ e 0 , θ 1 , . . . , θ h ) ∈ Z e ×N h such that 0 ≤ θ j < e j for all j ∈ {1, . . . , h} and O(f, g) = e i=1 θ i 0 r i 0 + h j=1 θ j r j . In particular Γ(f ) is generated by r 1 0 , . . . , r e 0 , r 1 , . . . , r h .

Proof. Let g = θ c θ (x)G θ 1 1 . . . G θ h h f θ h+1 be the expansion of g with respect to (G 1 , . . . , G h , f ) and recall that for all θ, if c θ = 0 then θ = (θ 1 , ..., θ h+1 ) ∈ {(β 1 , ..., β h+1 ) ∈ N h+1 , 0 ≤ β j < e j ∀j = 1, ..., h}. The hypothesis implies that there exists at least one θ such that c θ = 0 and θ h+1 = 0. Let In fact, in this case, any monomial M of the expansion of g with respect to (G 1 , . . . , G h , f ) is a monomial in G 1 , . . . , G i-1 . Hence this expansion coincides with that of g with respect to (G 1 , . . .

M = c θ (x)G θ 1 1 . . . G θ h h , N = c θ (x)G θ 1 1 . . . G
, G i-1 , G i ). If M is the unique monomial such that O(f, g) = O(f, M ) then M is the unique monomial such that O(G i , g) = O(G i , M ). But O(f, M ) = d i O(G i , M
). This proves our assertion.

The next Proposition shows that we can calculate a system of generators of Γ(f ) only with the set of approximate roots of f . Proposition 14 For all i ∈ {1, . . . , h}, let

g i = App(f, d i ). We have O(f, g i ) = r i . Proof. Let i = h and consider the G h -adic expansion of f , f = G d h h + C 1 (x, y)G d h -1 h + . . . + C d h (x, y) = d h k=0 C k (x, y)G d h -k h where C 0 = 1 and C k (x, y) ∈ K C [[x]][y] with deg y (C k (x, y)) < n d h for all k = 1, ..., d h . Consider the Tschirnhausen transform of G h with respect to f given by τ f (G h ) = G h + d -1 h C 1 (x, y). We have O(f, G h ) = r h , hence we need to prove that O(f, C 1 ) > r h . Let k ∈ {0, ..., d h -1}. For all α = k, we have O(f, C α G d h -α h ) = O(f, C k G d h -k h ). In fact, suppose that O(f, C α G d h -α h ) = O(f, C k G d h -k h ), that is O(f, C α ) + (d h -α)r h = O(f, C k ) + (d h -k)r h . Suppose that α > k, then (α -k)r h = O(f, C α ) -O(f, C k ). But deg y (C α ), deg y (C k ) < n d h , then by Remark 4, O(f, C α ), O(f, C k ) ∈ (nZ) e + r 1 Z + . . . + r h-1 Z, and so (α -k)r h ∈ (nZ) e + r 1 N + . . . + r h-1 N, with 0 < α -k < d h = e h . This contradicts Proposition 7. Now a similar argument shows that O(f, C k G d h -k h ) = O(f, C k ) + (d h -k)r h = O(f, C d h ). As f (x, y(x)) = 0, we get that O(f, C d h ) = O(f, G d h h ) = r h d h < O(C k G d h -k k , hence O(f, C k ) > kr h . This is true for k = 1, consequently O(f, C 1 ) > r h , and O(f, τ f (G h )) = r h . Repeating this process, we get that O(f, τ l f (G h )) = r h for all l ≥ 1. But g h = App(f, d h ) = τ l 0 f (G h )
for some l 0 . Hence O(f, g h ) = r h . Now suppose that O(f, g k ) = r k for all k > i, and let us prove that O(f, g i ) = r i . Note that g i = App(g i+1 , e i ). Let

g i+1 = G e i i + β 1 (x, y)G e i -1 i + . . . + β e i (x, y) (1)
be the G i -adic expansion of g i+1 and consider O(f, g i+1 ). For all k ∈ {1, . . . , e i },

O(f, β k G e i -k k ) = O(f, β k ) + (e i -k)r i . But O(f, β k ) ∈ (nZ) e + i-1 j=1 r j N because deg y β k ) < n d i ,
e i = O(f, G e i i ) = O(f, β e i ) < O(β 1 G e i -1 i ). Hence O(f, β 1 ) > r i . In particular O(f, τ g i+1 (G i )) = O(f, G i + 1 e i β 1 ) = r i
Applying the same process to f and τ g i+1 (G i ) instead of f and G i . We get that O(f, τ 2 g i+1 (G i )) = r i . But g i = τ l i g i+1 (G i )) for some l i , hence O(f, g i ) = O(f, τ e i g i+1 (G i )) = r i . This proves our assertion.

Remark 5 The semigroup Γ(f ) is a free affine semigroup with respect to the arrangement (r 1 0 , . . . , r e 0 , r 1 , . . . , r h ) (see [START_REF] Assi | The Frobenius vector of a free affine semigroup[END_REF] for the definition and properties of free affine semigroup). This explains the notion of free polynomials introduced in the paper. In the next remark we will show how to prepare our polynomial so that the smallest homogeneous component u a of ∆ contains a monomial in x 1 .

Solutions of formal power series

Remark 6 (Preparation) Consider the mapping ξ : K[[x]] → K[[x]], defined by ξ(x 1 ) = X 1 and ξ(x i ) = X i + tX 1 for all i ∈ {2, ..., e}, where t is a parameter to be determined. Let ψ : K

[[x]][y] → K[[X]][
y] be the map defined as follows: if H = h 0 (x)y m + . . . + h m-1 (x)y + h m (x) ∈ K[[x]][y] then ψ(H) = ξ(h 0 (x))y m +. . .+ξ(h m-1 (x))y +ξ(h m (x)). Then we easily prove that ψ is an isomorphism. If ∆ is the discriminant of ψ(f ) and if v d (X) = u d (X 1 , X 2 + tX 1 , ..., X e + tX 1 ) then ∆ = d≥a v d . But v d (X) = ε d (t)X d 1 + v d , where v d is a homogeneous polynomial of degree d, and ε d (t) is a polynomial in t. Since K is an infinite filed, then we can choose t ∈ K such that ε a (t) = 0.

In the following we shall say that a polynomial f is prepared if it satisfies the condition of Remark 6, i.e. its discriminant is of the form ∆ = d≥0 u d such that the smallest homogeneous component is of the form u a = c a x a 1 + u a with c a = 0 and u a ∈ K[x]. The next proposition shows that a prepared polynomial is birationally equivalent to a quasi-ordinary polynomial.

Remark 3

 3 Since [L(y) : L] = n, then it follows from proposition 6 that [L(y) : L] = e 1 . . . e h = D 1 D h+1 . But [L(y); L] = n and D 1 = n e , hence D h+1 = n e-1 . It follows that d 1 = n and d h+1 = 1.

θ hhRemark 4

 4 be two distinct monomials of g. It follows from Lemma 1 that O(f, M ) = O(f, N ). Hence there exists a unique monomial M of g such that O(f, g) = O(f, M ). This proves our assertion. In the Lemma above, if deg y g < n d i for some i ∈ {1, . . . , h}, then O(f, g) ∈ (nZ) e + i-1 k=1 r k N. Moreover, O(f, g) = d i O(G i , g).

  Let f (x, y) = y n + a 1 (x)y n-1 + • • • + a n-1 (x)y + a n (x) be a polynomial of degree n in K[[x]][y]. In this Section we shall prove that, modulo an automorphism of K[[x]][y], and under an irreducibility condition, f is free in K C [[x]][y] for some specific line free cone C. Let ∆(x) be the discriminant of f in y, and write ∆(x) = p∈N e c p x p = d≥0 u d (x) where for all d ≥ 0, u d is the homogeneous component of degree d of ∆. Let a = inf {d, u d = 0}. If a = 0, then f is a quasi-ordinary polynomial. Suppose that a > 0, and, without loss of generality, that u a / ∈ K[[x 2 , . . . , x e ]].

  and r i+1 / ∈ (nZ) e + i j=1 r j N. Now a similar argument as above shows that r i

Proposition 15 With the notations above, if f is a prepared polynomial then F (X 1 , ..., X e , y) = f (X 1 , X 2 X 1 , . . . , X e X 1 , y) is a quasi-ordinary polynomial.

Proof. Let ∆ be the discriminant of f . The discriminant ∆ N of F is ∆ N = ∆(X 1 , X 2 X 1 , . . . , X e X 1 ). Write ∆ = d≥a u d , where u d is the homogeneous component of degree d of ∆ and u a = 0, then ∆ N = d≥a w d (X) with w d (X) = u d (X 1 , X 2 X 1 , . . . , X e X 1 ). For all d ≥ a, we have

where

We will now introduce the following line free cone. x Along this Section, C will denote the cone defined in proposition 16.

x a 2 2 . . . x ae e . In particular Supp(y

The following proposition characterizes the irreducibility of elements of K

] for all i = 1, ..., m, then by Lemma 3 we get that a i (x 1 , x 2 x -1 1 , . . . , x e x -1

], then a 1 ≥ -(a 2 + . . . + a e ) for all (a 1 , ..., a e ) ∈ Supp(a(x)). It follows that a 1 + a 2 + . . . + a e ≥ 0 for all (a 1 , ..., a e ) ∈ Supp(a(x)). Hence, a(X 1 , X 2 X 1 , ..., X e X 1 )

y).

This contradicts the hypothesis.

In the following we give a characterization for the polynomial f to be free.

then it is free.

Proof. By Proposition 15, F (X 1 , ..., X e , y) = f (X 1 , X 2 X 1 , . . . , X e X 1 , y) is a quasi-ordinary polynomial of K[[X]][y], and by Proposition 17 we get that F is an irreducible quasi-ordinary polynomial in K[[X]][y] of degree n, then by the Abhyankar-Jung theorem there exists a formal power series Z in

. This proves our assertion.

Remark 7 In Propositions 17 and 18, if

is not irreducible, then it decomposes into quasi-ordinary polynomials, hence f itself decomposes into free polynomials in

As for reducible quasi-ordinary polynomials, we can associate with f the set of characteristic sequences of its irreducible components as well as a semigroup defined from the set of semigroups of these components.

Next we prove that the approximate roots of a prepared free polynomial are free polynomials.

Proposition 19 Suppose that f is prepared and let d be a divisor of n.

Proof. By Propositions 15, 18 and Lemma 17, the polynomial F (X, y


Elisabeth Polard

Adeline Degremont
email: adeline.degremont@chu-rennes.fr

Caroline Rault

Béranger Le Nautout

Emmanuel Oger

Arnaud Biraben

Emmanuel Nowak

Trends in sodium valproate prescriptions among children aged 0 to 14 years between 2010 and 2016 A study based on the French National Health Insurance Database

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

1) Introduction

Epilepsy is one of the commonest serious chronic neurological disorders. Worldwide there are at least 65 million people living with epilepsy. The incidence of epilepsy in the infantile period is high, with an estimation of about 70 per 100,000 inhabitants in European countries. [1] The management of epilepsy among children is different from that among adults because of a broader spectrum of syndromes in children and because evidence on the effectiveness of antiepileptic drugs (AEDs) is not always established. [2][3][4] A therapeutic choice must still be made even if a precise diagnosis is not straightforward at the onset of epilepsy.

In the 2000s, new AEDs arrived on the drug market and changed practices in the treatment of epilepsy for both children and adults. [5,6] The changes could also be related to an awareness of the risks associated with in utero exposure to earlier antiepileptic drugs, such as sodium valproate (VPA), which has been integrated into clinicians' practices, especially neurologists, as demonstrated in Europe and across the Atlantic. [7][8][9] In 2014, as recommended by the European Medicines Agency (EMA), the French Agency for the safety of medicines and health products (ANSM) initiated a communication strategy with a "Dear Doctor" letter [10] to remind healthcare professionals of the teratogenic effects of VPA (major congenital malformations [11], neuro-developmental delays [12], autism spectrum disorders [13,14]). A few months later, in May 2015, French clinicians were also informed that the conditions for prescription and issue of VPA and its derivatives would be reinforced. [15] Since the 1st of January 2016, VPA and its derivatives should, whenever it is possible, not be prescribed to women of childbearing age, except in case of intolerance or ineffectiveness of alternatives.

Nonetheless, the unchallenged efficacy of VPA, with its broad spectrum of activity, makes it a drug of choice for the first-line treatment of epilepsy. [16] More recently in Europe, likewise, VPA has been contraindicated for women of reproductive potential, except in case of intolerance or ineffectiveness of alternatives.

We aimed to evaluate trends in AED prescriptions, especially in the period 2015-2016, to tailor a communication strategy, focusing on young epileptic girls, for whom VPA should not be chosen as the first-line treatment.

1) Methods

We conducted a nationwide study using the French National Health Insurance Database (SNIIRAM), over 7 years (2010)(2011)(2012)(2013)(2014)(2015)(2016), to describe the trends in VPA prescription among epileptic girls 0 to 14 years of age, particularly as first-line treatment, and we compared this with prescribing trends for boys aged 0 to 14 years.

Source

SNIIRAM anonymously and comprehensively links a health reimbursement database (DCIR) to the French hospital discharge database (PMSI). The DCIR contains i) demographic data such as date of birth, gender and information on complementary insurance systems or the presence of CMUc (Couverture médicale universelle complémentaire) (Complementary Universal Health Insurance), an insurance cover for low-income status individuals ii) medication, recorded as dispensed packs, including a single national registration code (CIP), and Anatomical Therapeutic Chemical (ATC) code is also provided with the date of prescription, the profession of the prescriber and date of issue iii) the presence of long-term illness (LTI) status and date of first registration of chronic illness status, which gives registered patients full coverage for all medical expenses related to the illness. PMSI covers all overnight or day hospitalizations in the public and private sectors, and includes short-term stays in medical, surgical or obstetric facilities. It collects information on the patients and their diagnoses (primary diagnoses (DP-diagnostic principal /DR-diagnostic relié) (main and related diagnoses)) and comorbidities or complications (DAS-Diagnostic associé significatif) (significant related diagnoses) using ICD10 codes (International Classification of Diseases, 10th revision), surgical/interventional procedures (CCAM-Classification Commune des Actes Médicaux) and prescription of particularly expensive drugs.

The SNIIRAM database was developed to ensure the reimbursement of individual medical claims but was not intended to serve medical research. It does not comprise any clinical information concerning results related to consultations, prescriptions or examinations. Using it to follow individual patients over time and across different data sets can therefore be challenging.

It has previously been used to describe prescription trends with respect to recommendations [17] in the context of epilepsy management [18] and VPA exposure [19].

Patient selection

All patients aged 0 to 14 years with at least one issue of antiepileptic drug (AED) recorded between 2010 and 2016 were selected. An AED was defined as a drug licensed for the treatment of epilepsy in France, i.e. drugs listed under the ATC code N03A, plus two benzodiazepines, diazepam (N05BA01) and clobazam (N05BA09). As diazepam is mainly authorized for febrile convulsions, patients who were prescribed diazepam only were excluded. Patients, who were prescribed sodium divalproex (N03AG01) or valpromide (N03AG02) were also excluded because these medications are mostly indicated in psychiatric setting.

Data collection

For descriptive purposes different patient characteristics were collected: date of birth to determine the age at the beginning of the study (January 1 st , 2010), gender, LTI status, CMUc status, which provides information on patient socio-economic level. Prescription characteristics were obviously studied: date of prescription, date of issue, drug issued, number of boxes and also information on the prescriber, as working in a healthcare facility or in the community.

Statistical analysis

For each patient matching the selection criteria and for each prescription we determined whether it was a prevalent or an incident use (first delivery after birth or after one year without AED issue). A one-year history was necessary to obtain this information and this is the reason why we retrospectively collected data up to 2009. We also determined which AEDs were prescribed (grouping them by ATC7 Code). The number of different ATC7 codes prescribed defined the polytherapy criterion.

For each year, we first calculated the percentage of VPA and that of each ATC7 code other than VPA among incident uses, and plotted these proportions by gender and age category. We use the following age categories: 0-2 years, 3-6 years, 7-10 years and 11-14 years. For purposes of comparison, we calculated the relative reduction in prescriptions between the percentage of VPA prescribed in 2010-2014 and the one in 2016. We intentionally excluded the year 2015, during which time the EMA and ANSM promoted a huge campaign to avoid the use of VPA among women of childbearing age. The impact of the 2015 risk minimization measure was likely to be more obvious in 2016, the year we chose for comparison.

We then calculated the incidence rate of VPA prescription among prevalent uses of any other AEDs and plotted them by gender and age category: the number of VPA initiations was divided by the number of person-years (PY) with a prevalent use of another AED. The types of AED before VPA initiation were described.

Finally we calculated the incidence rate for VPA discontinuations among prevalent users of VPA and the proportion for each ATC7 code among AED prescriptions after VPA discontinuation. VPA discontinuation was defined by the last VPA delivery date followed by at least one year without VPA issue, assuming one year of follow-up was available.

Statistical analyses were performed using SAS 9.4 (SAS INSTITUTE, North Carolina, USA) via SAS Enterprise Guide®.

Regulatory approval

The study was granted regulatory approval as part of a larger program on VPA exposure (CNIL 03/11/2016). The study was conducted with direct access to the SNIIRAM database under ANSM agreement.

2) Results

General characteristics of the study population

In the SNIIRAM database, 113,362 children fulfilled the selection criteria, 61,259 boys and 52,103 girls. Almost all patient characteristics were similar between boys and girls (Table 1).

The number of incident uses over the study period (seven years) was 93,557: 42,493 among girls and 50,064 among boys. The median age at the beginning of the study was five years.

The most frequently prescribed antiepileptic drugs were, in decreasing order: sodium valproate, largely predominant, followed by diazepam, lamotrigine (LTG), levetiracetam (LEV), clobazam and clonazepam. Most of the drugs were prescribed in monotherapy.

Prescribing trends for sodium valproate

Figure 1 shows the trends in VPA initiations over the study period by gender and age category. Focusing on children aged 0-10 years and incident uses only in 2010, the proportion of VPA prescriptions ranged between 40 and 50%, without a clear gap between boys (nearly 30%) and girls (20%). Globally, the proportion remained quite stable until 2014 and then decreased. In 2016, the proportion of VPA still accounted for 11% of AED initiations among girls aged 11-14 years. Between 2010 and 2016, the proportions of prescriptions for LEV, LTG, ethosuximide, gabapentin in incident use increased among girls aged 0-14 years (Figure 2). Table 2 reports the proportion of VPA among AED initiations according to prescriber category. In 2010, initiation of VPA among girls by specialists (including hospital practitioner and neurologist or pediatrician working in community) was relatively frequent (over 40%). This frequency was lower in 2016 for neurologists (-24.5% between 2010 and 2016), hospital practitioners (-20.6%) and pediatricians (-15.6%). In contrast, the frequency of VPA use among AED initiations by General Practitioners (GPs) was relatively stable over the period, at around 20-25%. It can be noted that the incidence rate for VPA prescription among teenage girls (11-14 years of age) and young girls (7-10 years of age) was relatively low over the study period (less than 10 for 100 PY). The most frequently prescribed AEDs just before VPA initiation were benzodiazepines, lamotrigine and vigabatrine, mostly in monotherapy (91% in 2010; 89% in 2015). Discontinuation of VPA was defined as an issue of VPA that was not followed later in the year by another issue of VPA, assuming one year of follow-up was available. Discontinuation of VPA was quite high (around 35 per 100 child-years) but quite stable over the study period and similar in girls and in boys. It was higher among young children (0-6 years old) than among older children (7-14 years old) (40 to 60 per 100 PY versus 30 per 100 PY).

In around 70% of cases, VPA discontinuation was not followed by another AED. Before withdrawal of sodium valproate treatment, the antiepileptic treatment duration was for the majority 1 year or more (60%), while in 18% of cases it was less than 1 month (only 1 issue).

It was different among children aged 0-2 years, for whom only 1 month of antiepileptic treatment before VPA withdrawal was observed in 30% of cases.

The remaining 30% "switched" from VPA to another AED, mostly in monotherapy.

Lamotrigine and levetiracetam were the most frequently prescribed AEDs.

Discussion

Our study evaluating the trends in sodium valproate (VPA) prescription among children highlighted a number of points.

Firstly, the proportion of VPA prescriptions among incident uses of antiepileptic drugs decreased in both genders over the study period, but more so among girls, especially in teenage girls. Over the 1993 to 2006 period, the prevalence of VPA per 1000 prescriptions among females (12-18 years old) significantly decreased from 0.94 (95% CI 0.80 to 1.09) to 0.63 (95% CI 0.55 to 0.72). [7]. Over the 2000 to 2010 period, girls were prescribed sodium valproate significantly less often as a first-line AED (74% versus 46%). [20] Our study enables these observations to be updated and it points to a substantial decrease in VPA use among teenage girls, possibly as a result of new information released in the media on its teratogenic effects. [10] Secondly, the alternatives to VPA, for which frequencies increased during the study period, were mainly levetiracetam, then lamotrigine, followed by ethosuximide and gabapentin.

Increases were always more pronounced for girls than for boys. Changes in the choice of the first AED for a new onset of epilepsy are similar to those observed in previous years in other European countries. [7,9,20] During the 2000s, it was established that the use of firstgeneration AEDs (VPA, carbamazepine, phenytoin, phenobarbital) decreased over time, at least in developed countries, concomitantly with the arrival of new AEDs on the market. [5,9,21,22] More recently, a trend towards more LEV and LTG prescriptions could also be explained by prescribers' knowledge of the significantly lower rates of major congenital malformations for these drugs. [11] Thirdly, the proportion of VPA initiations among girls already under treatment greatly decreased for specialists but remained stable over the study period for GPs. The proportions of AED prescriptions by three different prescriber categories (GPs, specialists, psychiatrists) were studied by Hollingworth & al. in Australia over a 5-year period (2002)(2003)(2004)(2005)(2006)(2007). They observed an increase in prescribing of new AEDs, such as levetiracetam, topiramate and lamotrigine, by GPs. Also, the proportion of prescriptions by GPs was the highest for phenytoin, followed by sodium valproate. [23] Prescriber data in this study indicated a shift from specialist to GP prescribing. Neurologists commonly initiate AED prescription for epilepsy and the subsequent prescribing is usually taken on by GPs. This pattern should be applied in France according to the latest recommendations. Nevertheless, VPA accounted for 20% of AEDs initiated by a GP. In the clinical context of epilepsy, primary care should lead to an initiation of antiepileptic treatment to limit possible seizures and should not have to be delayed until the final therapeutic choice made by a specialist. Continuing prescription of VPA as first-line treatment by GPs could reflect difficulties in healthcare trajectories for the French epileptic population, with difficult access to specialized care (EEG, neurologist, tertiary care hospitals), compelling GPs to ensure the care of first seizures with VPA, on the grounds that this AED has a broad spectrum of activity.

Finally, discontinuation of VPA was stable over the study period, among girls but also among boys, and did not seem to be influenced by the recommendations. In 70% of cases, the withdrawal of VPA was not followed by another AED, and this was all the more true among children aged 0-2 years. The exact reasons why treatments were discontinued is not known, but some situations can justify it. For instance, treatment with VPA can be discontinued after a diagnosis of recovery. VPA is the drug of choice for many epileptic syndromes, and widely prescribed by GPs. After specialist consultation or access to electroencephalogram (EEG), a diagnosis other than epilepsy can be made. An antiepileptic treatment can also be prescribed on a temporary basis, for example in case of accidental seizure after meningitis, head trauma, or febrile seizures. In these cases, the treatment duration is short (a few months) and pursuit of antiepileptic treatment is not expected. Among older children, discontinuation of the antiepileptic treatment can occur in certain types of syndrome (childhood absence epilepsy, benign rolandic epilepsy) after a seizure-free period, or to limit the neurocognitive impact of AEDs. Prolonged discontinuations do not systematically lead to seizure recurrence and are possible for some patients. [30,31] Compared to previous studies, our investigation, by using a large population database over a recent period of time, is the first one that evaluates latest trends in prescribing valproate focusing on children.

A first limitation is the definition of incident use, which could include a re-instatement of treatment, as it was based on a one-year AED-free period only. However, recommendations on VPA use encompass AED initiation as well as resumption of treatment. A second limitation is the fact that the SNIIRAM database does not enable the pathological context of prescriptions to be identified because no clinical data is available except for the hospital discharge diagnoses.. Nevertheless, we believe that AEDs are primarily prescribed in children for anticonvulsant purposes and, more importantly, our objective was to observe a decrease in valproate prescribing trends regardless of indication. Finally, patients in long-term facilities were not included in this study because AED issues for these patients were not available in the database. These patients often have pharmaco-resistant epilepsy, or at least epilepsy that is hard to stabilize with different AED prescription strategies. [23,24]

Conclusion

To the best of our knowledge, this is the first study describing VPA and alternative prescription patterns in a large pediatric population over a long period of time taking into account the new French recommendations concerning VPA use. This large-scale study describes changes in paediatric prescription patterns following the implementation of a national information campaign on the appropriate use of VPA.

The observed trend was as expected, with a decrease in VPA initiations, especially after 2015, among children whether or not already treated with an AED. This decrease was observed in both genders, more pronounced for girls, and sharper in 2016. The proportion of VPA prescriptions among the different AED prescriptions decreased between 2010 and 2016 in favor of LTG and LEV, often used as monotherapy. Whatever the nature of the prescription (initiation or not), they came mainly from hospital prescribers (specialists) and from neurologists or pediatricians.

All prescribers reduced VPA prescriptions between 2010 and 2016, especially the neurologists. VPA initiation remained stable among GPs. In 2016, the Prescription and Drug Supply Conditions changed for VPA in France: it is now mandatory for it to be initiated by a specialist. Furthermore, the European Medicines Agency recently changed the recommendations for VPA use: it should not be prescribed to women of childbearing potential except if there is no other treatment option, and provided the woman is on adequate contraception. Further evaluations in 2016, 2017 and 2018 could be valuable.

Highlights:

 Since 2010, initiations of VPA have decreased among girls aged 0 to 14 years.

 Initiations of LTG and LEV increased among girls over the study period.

 In 2016, GPs were still initiating VPA among girls aged 11-14 years.

 GPs remain the primary healthcare professionals for epileptic patients.

 VPA is still the drug of choice for first-line treatment.

Highlights:

 Since 2010, initiations of VPA have decreased among girls aged 0 to 14 years.

 Initiations of LTG and LEV increased among girls over the study period.

 In 2016, GPs were still initiating VPA among girls aged 11-14 years.

 GPs remain the primary healthcare professionals for epileptic patients. Epilepsy is one of the commonest serious chronic neurological disorders. Worldwide there are at least 65 million people living with epilepsy. The incidence of epilepsy in the infantile period is high, with an estimation of about 70 per 100,000 inhabitants in European countries.

Abbreviations:

AED

[1]

The management of epilepsy among children is different from that among adults because of a broader spectrum of syndromes in children and because evidence on the effectiveness of antiepileptic drugs (AEDs) is not always established. [2][3][4] A therapeutic choice must still be made even if a precise diagnosis is not straightforward at the onset of epilepsy.

In the 2000s, new AEDs arrived on the drug market and changed practices in the treatment of epilepsy for both children and adults. [5,6] The changes could also be related to an awareness of the risks associated with in utero exposure to earlier antiepileptic drugs, such as sodium valproate (VPA), which has been integrated into clinicians' practices, especially neurologists, as demonstrated in Europe and across the Atlantic. [7][8][9] In 2014, as recommended by the European Medicines Agency (EMA), the French Agency for the safety of medicines and health products (ANSM) initiated a communication strategy with a "Dear Doctor" letter [10] to remind healthcare professionals of the teratogenic effects of VPA (major congenital malformations [11], neuro-developmental delays [12], autism spectrum disorders [13,14]). A few months later, in May 2015, French clinicians were also informed that the conditions for prescription and issue of VPA and its derivatives would be reinforced. [15] Since the 1st of January 2016, VPA and its derivatives should, whenever it is possible, not be prescribed to women of childbearing age, except in case of intolerance or ineffectiveness of alternatives.

Nonetheless, the unchallenged efficacy of VPA, with its broad spectrum of activity, makes it a drug of choice for the first-line treatment of epilepsy. [16] More recently in Europe, likewise, VPA has been contraindicated for women of reproductive potential, except in case of intolerance or ineffectiveness of alternatives.

We aimed to evaluate trends in AED prescriptions, especially in the period 2015-2016, to tailor a communication strategy, focusing on young epileptic girls, for whom VPA should not be chosen as the first-line treatment.

1) Methods

We conducted a nationwide study using the French National Health Insurance Database It has previously been used to describe prescription trends with respect to recommendations [17] in the context of epilepsy management [18] and VPA exposure [19].

Patient selection

All patients aged 0 to 14 years with at least one issue of antiepileptic drug (AED) recorded between 2010 and 2016 were selected. An AED was defined as a drug licensed for the treatment of epilepsy in France, i.e. drugs listed under the ATC code N03A, plus two benzodiazepines, diazepam (N05BA01) and clobazam (N05BA09). As diazepam is mainly authorized for febrile convulsions, patients who were prescribed diazepam only were excluded. Patients, who were prescribed sodium divalproex (N03AG01) or valpromide (N03AG02) were also excluded because these medications are mostly indicated in psychiatric setting.

Data collection

For descriptive purposes different patient characteristics were collected: date of birth to determine the age at the beginning of the study (January 1 st , 2010), gender, LTI status, CMUc status, which provides information on patient socio-economic level. Prescription characteristics were obviously studied: date of prescription, date of issue, drug issued, number of boxes and also information on the prescriber, as working in a healthcare facility or in the community.

Statistical analysis

For each patient matching the selection criteria and for each prescription we determined whether it was a prevalent or an incident use (first delivery after birth or after one year without AED issue). A one-year history was necessary to obtain this information and this is the reason why we retrospectively collected data up to 2009. We also determined which AEDs were prescribed (grouping them by ATC7 Code). The number of different ATC7 codes prescribed defined the polytherapy criterion.

For each year, we first calculated the percentage of VPA and that of each ATC7 code other than VPA among incident uses, and plotted these proportions by gender and age category. We use the following age categories: 0-2 years, 3-6 years, 7-10 years and 11-14 years. For purposes of comparison, we calculated the relative reduction in prescriptions between the percentage of VPA prescribed in 2010-2014 and the one in 2016. We intentionally excluded the year 2015, during which time the EMA and ANSM promoted a huge campaign to avoid the use of VPA among women of childbearing age. The impact of the 2015 risk minimization measure was likely to be more obvious in 2016, the year we chose for comparison.

We then calculated the incidence rate of VPA prescription among prevalent uses of any other AEDs and plotted them by gender and age category: the number of VPA initiations was divided by the number of person-years (PY) with a prevalent use of another AED. The types of AED before VPA initiation were described.

Finally we calculated the incidence rate for VPA discontinuations among prevalent users of VPA and the proportion for each ATC7 code among AED prescriptions after VPA discontinuation. VPA discontinuation was defined by the last VPA delivery date followed by at least one year without VPA issue, assuming one year of follow-up was available.

Statistical analyses were performed using SAS 9.4 (SAS INSTITUTE, North Carolina, USA) via SAS Enterprise Guide®.

Regulatory approval

The study was granted regulatory approval as part of a larger program on VPA exposure (CNIL 03/11/2016). The study was conducted with direct access to the SNIIRAM database under ANSM agreement.

2) Results

General characteristics of the study population

In the SNIIRAM database, 113,362 children fulfilled the selection criteria, 61,259 boys and 52,103 girls. Almost all patient characteristics were similar between boys and girls (Table 1).

The number of incident uses over the study period (seven years) was 93,557: 42,493 among girls and 50,064 among boys. The median age at the beginning of the study was five years.

The most frequently prescribed antiepileptic drugs were, in decreasing order: sodium valproate, largely predominant, followed by diazepam, lamotrigine (LTG), levetiracetam (LEV), clobazam and clonazepam. Most of the drugs were prescribed in monotherapy. Discontinuation of VPA was defined as an issue of VPA that was not followed later in the year by another issue of VPA, assuming one year of follow-up was available. Discontinuation of VPA was quite high (around 35 per 100 child-years) but quite stable over the study period and similar in girls and in boys. It was higher among young children (0-6 years old) than among older children (7-14 years old) (40 to 60 per 100 PY versus 30 per 100 PY).

Prescribing trends for sodium valproate

In around 70% of cases, VPA discontinuation was not followed by another AED. Before withdrawal of sodium valproate treatment, the antiepileptic treatment duration was for the majority 1 year or more (60%), while in 18% of cases it was less than 1 month (only 1 issue).

It was different among children aged 0-2 years, for whom only 1 month of antiepileptic treatment before VPA withdrawal was observed in 30% of cases.

The remaining 30% "switched" from VPA to another AED, mostly in monotherapy.

Lamotrigine and levetiracetam were the most frequently prescribed AEDs.

Discussion

Our study evaluating the trends in sodium valproate (VPA) prescription among children highlighted a number of points.

Firstly, the proportion of VPA prescriptions among incident uses of antiepileptic drugs decreased in both genders over the study period, but more so among girls, especially in teenage girls. Over the 1993 to 2006 period, the prevalence of VPA per 1000 prescriptions among females (12-18 years old) significantly decreased from 0.94 (95% CI 0.80 to 1.09) to 0.63 (95% CI 0.55 to 0.72). [7]. Over the 2000 to 2010 period, girls were prescribed sodium valproate significantly less often as a first-line AED (74% versus 46%). [20] Our study enables these observations to be updated and it points to a substantial decrease in VPA use among teenage girls, possibly as a result of new information released in the media on its teratogenic effects. [10] Secondly, the alternatives to VPA, for which frequencies increased during the study period, were mainly levetiracetam, then lamotrigine, followed by ethosuximide and gabapentin.

Increases were always more pronounced for girls than for boys. Changes in the choice of the first AED for a new onset of epilepsy are similar to those observed in previous years in other European countries. [7,9,20] During the 2000s, it was established that the use of firstgeneration AEDs (VPA, carbamazepine, phenytoin, phenobarbital) decreased over time, at least in developed countries, concomitantly with the arrival of new AEDs on the market. [5,9,21,22] More recently, a trend towards more LEV and LTG prescriptions could also be explained by prescribers' knowledge of the significantly lower rates of major congenital malformations for these drugs. [11] Thirdly, the proportion of VPA initiations among girls already under treatment greatly decreased for specialists but remained stable over the study period for GPs. The proportions of AED prescriptions by three different prescriber categories (GPs, specialists, psychiatrists)

were studied by Hollingworth & al. in Australia over a 5-year period (2002)(2003)(2004)(2005)(2006)(2007). They observed an increase in prescribing of new AEDs, such as levetiracetam, topiramate and lamotrigine, by GPs. Also, the proportion of prescriptions by GPs was the highest for phenytoin, followed by sodium valproate. [23] Prescriber data in this study indicated a shift from specialist to GP prescribing. Neurologists commonly initiate AED prescription for epilepsy and the subsequent prescribing is usually taken on by GPs. This pattern should be applied in France according to the latest recommendations. Nevertheless, VPA accounted for 20% of AEDs initiated by a GP. In the clinical context of epilepsy, primary care should lead to an initiation of antiepileptic treatment to limit possible seizures and should not have to be delayed until the final therapeutic choice made by a specialist. Continuing prescription of VPA as first-line treatment by GPs could reflect difficulties in healthcare trajectories for the French epileptic population, with difficult access to specialized care (EEG, neurologist, tertiary care hospitals), compelling GPs to ensure the care of first seizures with VPA, on the grounds that this AED has a broad spectrum of activity.

Finally, discontinuation of VPA was stable over the study period, among girls but also among boys, and did not seem to be influenced by the recommendations. In 70% of cases, the withdrawal of VPA was not followed by another AED, and this was all the more true among children aged 0-2 years. The exact reasons why treatments were discontinued is not known, but some situations can justify it. For instance, treatment with VPA can be discontinued after a diagnosis of recovery. VPA is the drug of choice for many epileptic syndromes, and widely prescribed by GPs. After specialist consultation or access to electroencephalogram (EEG), a diagnosis other than epilepsy can be made. An antiepileptic treatment can also be prescribed on a temporary basis, for example in case of accidental seizure after meningitis, head trauma, or febrile seizures. In these cases, the treatment duration is short (a few months) and pursuit of antiepileptic treatment is not expected. Among older children, discontinuation of the antiepileptic treatment can occur in certain types of syndrome (childhood absence epilepsy, benign rolandic epilepsy) after a seizure-free period, or to limit the neurocognitive impact of AEDs. Prolonged discontinuations do not systematically lead to seizure recurrence and are possible for some patients. [30,31] Compared to previous studies, our investigation, by using a large population database over a recent period of time, is the first one that evaluates latest trends in prescribing valproate focusing on children.

A first limitation is the definition of incident use, which could include a re-instatement of treatment, as it was based on a one-year AED-free period only. However, recommendations on VPA use encompass AED initiation as well as resumption of treatment. A second limitation is the fact that the SNIIRAM database does not enable the pathological context of prescriptions to be identified because no clinical data is available except for the hospital discharge diagnoses.. Nevertheless, we believe that AEDs are primarily prescribed in children for anticonvulsant purposes and, more importantly, our objective was to observe a decrease in valproate prescribing trends regardless of indication. Finally, patients in long-term facilities

were not included in this study because AED issues for these patients were not available in the database. These patients often have pharmaco-resistant epilepsy, or at least epilepsy that is hard to stabilize with different AED prescription strategies. [23,24]

Conclusion

This large-scale study describes changes in paediatric prescription patterns following the implementation of a national information campaign on the appropriate use of VPA.

The observed trend was as expected, with a decrease in VPA initiations, especially after 2015, among children whether or not already treated with an AED.

Figure 1 .

 1 Figure 1. Trends in sodium valproate initiations over the study period by gender and age category. (2-column fitting image)

Figure 2 .Figure 3

 23 Figure 2. Trends in antiepileptic treatment initiations over the study period among girls aged 0-14 years. (1-column fitting image)

Figure 3 .

 3 Figure 3. Trends in sodium valproate initiations (add-on or switch) among children already treated with another antiepileptic drug over the study period, by gender and age category. (2column fitting image)

(

  SNIIRAM), over 7 years(2010)(2011)(2012)(2013)(2014)(2015)(2016), to describe the trends in VPA prescription among epileptic girls 0 to 14 years of age, particularly as first-line treatment, and we compared this with prescribing trends for boys aged 0 to 14 years.SourceSNIIRAM anonymously and comprehensively links a health reimbursement database (DCIR)to the French hospital discharge database (PMSI). The DCIR contains i) demographic data such as date of birth, gender and information on complementary insurance systems or the presence of CMUc (Couverture médicale universelle complémentaire) (Complementary Universal Health Insurance), an insurance cover for low-income status individuals ii) medication, recorded as dispensed packs, including a single national registration code (CIP), and Anatomical Therapeutic Chemical (ATC) code is also provided with the date of prescription, the profession of the prescriber and date of issue iii) the presence of long-term illness (LTI) status and date of first registration of chronic illness status, which gives registered patients full coverage for all medical expenses related to the illness. PMSI covers all overnight or day hospitalizations in the public and private sectors, and includes short-term stays in medical, surgical or obstetric facilities. It collects information on the patients and their diagnoses (primary diagnoses (DP-diagnostic principal /DR-diagnostic relié) (main and related diagnoses)) and comorbidities or complications (DAS-Diagnostic associé significatif) (significant related diagnoses) using ICD10 codes (International Classification of Diseases, 10th revision), surgical/interventional procedures (CCAM-Classification Commune des Actes Médicaux) and prescription of particularly expensive drugs.The SNIIRAM database was developed to ensure the reimbursement of individual medical claims but was not intended to serve medical research. It does not comprise any clinical information concerning results related to consultations, prescriptions or examinations. Using it to follow individual patients over time and across different data sets can therefore be challenging.

Figure 1

 1 Figure1shows the trends in VPA initiations over the study period by gender and age category. Focusing on children aged 0-10 years and incident uses only in 2010, the proportion of VPA prescriptions ranged between 40 and 50%, without a clear gap between boys (nearly 30%) and girls (20%). Globally, the proportion remained quite stable until 2014 and then decreased. In 2016, the proportion of VPA still accounted for 11% of AED initiations among girls aged 11-14 years.

Figure 1 .

 1 Figure 1. Trends in sodium valproate initiations over the study period by gender and age category. (2-column fitting image)

Figure 2 .Figure 3

 23 Figure 2. Trends in antiepileptic treatment initiations over the study period among girls aged 0-14 years. (1-column fitting image)

Figure 3 .

 3 Figure 3. Trends in sodium valproate initiations (add-on or switch) among children already treated with another antiepileptic drug over the study period, by gender and age category. (2column fitting image)

Table 1 :

 1 Characteristics of the study population.

	Girls (N=52103)	Boys (N=61259)

Table 2 :

 2 Proportion of VPA among AED treatment initiations, by prescriber category.

	Sodium valproate		2010	2011	2012	2013	2014	2015	2016*
	Girls	Hospital	N 4243	4330	4325	4320	4195	4081	2255
	0-14 years	practitioner**	n % 42.59 1807	1740 40.18	1703 39.38	1721 39.84	1552 37.00	1343 32.91	495 21.95
		General	N 1901	1684	1378	1243	1223	1184	645
		practitioner	n	438	406	426	372	360	281	130
			% 23.04	24.11	30.91	29.93	29.44	23.73	20.16
		Pediatrician	N 288	275	275	237	246	233	130
			n	138	122	130	116	107	110	42
			% 47.92	44.36	47.27	48.95	43.50	47.21	32.31
		Neurologist	N 428	348	338	338	339	336	172
			N 177	145	131	132	116	80	29
			% 41.36	41.67	38.76	39.05	34.22	23.81	16.86
	Boys	Hospital	N 5250	5357	5199	5054	5071	4913	2853
	0-14 years	practitioner**	n % 44.72 2348	2323 43.36	2199 42.30	2145 42.44	2119 41.79	1946 39.61	1082 37.92
		General	N 2083	1979	1590	1437	1401	1273	653
		practitioner	n	523	546	602	469	421	351	153
			% 25.11	27.59	37.86	32.64	30.05	27.57	23.43
		Pediatrician	N 299	295	327	307	319	289	149
			n	142	129	130	152	164	131	66
			% 47.49	43.73	39.76	49.51	51.41	45.33	44.30
		Neurologist	N 401	357	311	357	310	308	167
			n	177	171	159	162	146	126	58
			% 44.14	47.90	51.13	45.38	47.10	40.91	34.73
	N = number of prescriptions initiated by the prescriber category
	n = number of prescriptions of VPA among the previous N
	*up to 31/07/2016
	**physician working in a hospital, considered as a specialist

Table 2

 2

	reports the proportion of VPA among AED initiations according to prescriber
	category. In 2010, initiation of VPA among girls by specialists (including hospital practitioner
	and neurologist or pediatrician working in community) was relatively frequent (over 40%).
	This frequency was lower in 2016 for neurologists (-24.5% between 2010 and 2016), hospital
	practitioners (-20.6%) and pediatricians (-15.6%). In contrast, the frequency of VPA use
	among AED initiations by General Practitioners (GPs) was relatively stable over the period, at
	around 20-25%.

  This decrease was observed in both genders, more pronounced for girls, and sharper in 2016. The proportion of VPA prescriptions among the different AED prescriptions decreased between 2010 and 2016 in favor of LTG and LEV, often used as monotherapy. Whatever the nature of the prescription (initiation or not), they came mainly from hospital prescribers (specialists) and from neurologists or pediatricians.All prescribers reduced VPA prescriptions between 2010 and 2016, especially the neurologists. VPA initiation remained stable among GPs. In 2016, the Prescription and Drug Supply Conditions changed for VPA in France: it is now mandatory for it to be initiated by a specialist. Furthermore, the European Medicines Agency recently changed the recommendations for VPA use: it should not be prescribed to women of childbearing potential except if there is no other treatment option, and provided the woman is on adequate contraception. Further evaluations in 2016, 2017 and 2018 could be valuable.

Funding Source:

This work was supported by an ANSM grant, as part of the Pharmaco-Epidemiology for health Product Safety (PEPS consortium) research program.

Abstract

Purpose

After a huge campaign of information on the teratogenic risk of sodium valproate (VPA) having taken place in France we aimed to evaluate the trend of its prescriptions in young epileptic girls.

Method

Using the French National Health Insurance Database we searched for patients aged 0 to 14 years being supplied an antiepileptic drug (AED) between 2010 and 2016.

Results

113,362 children received at least one AED, 61,259 boys and 52,103 girls. Compared to 2010-2014 years, VPA was less prescribed in 2016 as first AED (29% vs 37.3% respectively). The difference between the two periods was greater for girls (-41%) than for boys (-12%).

Conclusion

The changing trend of VPA as first AED prescribed, particularly in girls, reflects published evidence in terms of safety.