Invariants of First Order Partial Diﬀerential Equations ∗

: In this paper we introduce the concepts of multiplicity and index of ﬁrst order partial diﬀerential equations. In particular, the concept of multiplicity coincides with the multiplicity of implicit diﬀerential equations given by Bruce and Tari in [2]. We also show that these concepts are invariants by smooth equivalences. Following the work [10] on implicit diﬀerential equations with ﬁrst integrals, we introduce a deﬁnition of multiplicity for this class of equations.


Introduction
Let F (x 1 , ..., x n , y, p 1 , ..., p n ) = 0 be a first order partial differential equation (first order PDE), where F is a smooth function on R 2n+1 (here smooth means C ∞ ). A classical solution of this equation is a smooth function f : U ⊂ R n −→ R such that y = f (x 1 , ..., x n ) and p i = ∂f ∂xi . If ∂F ∂pi (q 0 ) = 0 at q 0 ∈ R 2n+1 for some i ∈ {1, ..., n}, the first order PDE defines a family of classical solutions near q 0 (see [15]). The locus points where F = ∂F ∂p1 = ... = ∂F ∂pn = 0, denoted by Σ(F ), are called π-singular points. At such points, the notion of first order PDE with a singular solution was introduced in [12]. It was shown in [13] that the local normal form of such equation is y = 0, up to contactomorphism. In [11] Izumiya studied singularities of the first order PDEs, describing the singularities appearing in an open dense set in the space of all functions F with the Whitney C ∞ -topology. At a π-singular point the first order PDE generically have no singular solution and the set Σ(F ) consists of isolated singular points.

Lizandro Sanchez Challapa
When n = 1, the first order PDE is called implicit differential equation (IDE). A natural way to study IDEs is to lift the multi-valued direction field determined by the IDE to a single field on the surface F −1 (0). In [6], Davydov classified (following the work of Dara [5]) generic IDE's when the discriminant is a regular curve and showed that the topological normal form of the IDE acquires moduli when the discriminant is a cusp.
Bruce and Tari introduced in [2] the multiplicity of an IDE, at a singular point, as the maximum number of singular points of the IDE which emerge when perturbing the equation F . In [3] the author defined the index of an IDE and showed that this index is invariant by smooth equivalences.
In this work we introduce the concepts of multiplicity and index of a first order PDE at an isolated singular point. We shall use results in [7], where the authors defined the multiplicity and index of a 1-form on an isolated complete intersection singularity (ICIS). This concept of multiplicity extends the definition of multiplicity of an IDE given in [2]. The invariance of the multiplicity and index by smooth equivalences is proven in Section 3. We also define an invariant of first order PDEs by contactomorphism. Following the work [10] on implicit differential equations with first integrals, we introduce a definition of multiplicity for this class of equations (see Section 4). In the last section we give examples to distinguish normal forms given in [10] and [17].

Multiplicity and index of first order PDE
As mentioned in the introduction, a first order partial differential equation is an equation of the form F (x 1 , ..., x n , y, p 1 , ..., p n ) = 0, (2.1) where F is a smooth function on R 2n+1 . Consider the projection π : R 2n+1 −→ R n+1 given by π(x 1 , ..., x n , y, p 1 , ..., p n ) = (x 1 , ..., x n , y). The set of critical points of the restriction of π to F −1 (0) is called the criminant of the first order PDE and is given by the equations F = F p1 = ... = F pn = 0, where F pi = ∂F ∂pi . These points are called π-singular points and their locus is denoted by Σ(F ). The image of Σ(F ) by the projection π is called discriminant of the first order PDE.
Let ω = dy − n i=1 p i dx i be the canonical contact 1-form on R 2n+1 . Since we will only study local properties, a solution of the first order PDE (2.1) is a submanifold germ (L, q 0 ) ⊂ R 2n+1 such that L ⊂ F −1 (0), dim(L) = n and ω |L = 0, where ω |L is the restriction of the 1-form ω to L. Let (L, q 0 ) be a solution of the first order PDE, then (L, q 0 ) is said to be classical solution if there exist a function germ f : is a classical solution if and only if q 0 is a regular point of the map π |L .
The notion of a singular solution is defined as follows. If the set Σ(F ) is a solution of the first order PDE, then we call it a singular solution of the first order 135 PDE. The zeros of the 1-form ω | F −1 (0) correspond to zeros of the vector field . These zeros are called contact singular points. We shall denote by Σ 2 (F ) the set of critical points of π | Σ(F ) . This set is given by the equations F = F p1 = ... = F pn = det(F pipj ) = 0.
Definition 2.1. We say that q 0 ∈ R 2n+1 is a singular point or a zero of the first order PDE (2.1) if q 0 is a contact singular point or a zero of the 1-form ω on Σ 2 (F ).
This definition coincides with Definition 2.3 given in [2] when the first order PDE is an implicit differential equation. Note that, by definition, the singular points of the first order PDE lie on the criminat of F . We denote by (F, q 0 ) the germ of the first order PDE (2.1) at an isolated singular point q 0 . If F is a real analytic function we say that (F, q 0 ) is an analytic germ. The concept of multiplicity of an IDE given in [2] motivates the following definition.
Definition 2.2. Let (F, q 0 ) be an analytic germ of first order PDE. The multiplicity M (F, q 0 ) of (F, q 0 ) is the maximum number of zeros that can appear in a deformation of the equation F = 0 (including complex zeros).
Note that the multiplicity is not defined if the 1-form ω vanishes identically on both Σ 2 (F ) and F −1 (0). The general problem of computing the multiplicity and index of the zero of a 1-form on an isolated complete intersection singularity (ICIS) has been considered by Ebeling and Gusein-Zade in [8]. They also give an algebraic formula for computing the multiplicity of a 1-form. We use this algebraic formula to define the following ideal. Definition 2.3. Let θ = n i=1 a i dx i be a smooth 1-form on R n and let f = (f 1 , ..., f k ) : (R n , 0) −→ R k be a smooth map germ, where n ≥ k + 1. We define I(f −1 (0), θ) as the ideal generated by f 1 , ..., f k and the (k + 1) × (k + 1)-minors of the matrix Remark 2.4. The following holds for a function germ λ : (R n , 0) −→ R with λ(0) = 0: The next lemma characterizes the zeros of the 1-form θ on f −1 (0) that are regular points of f −1 (0).
coincides with the multiplicity of the ideal I(f −1 (0), θ) at 0, the result follows from Proposition 2.1 in [4]. ✷ Lemma 2.7. Let (F, 0) be a germ of first order PDE. Suppose that I(Σ 2 (F ), ω) is generated by 2n + 1 elements. Then there exists a smooth family of functions for t = 0 sufficiently close to zero, where p t and q t are isolated zeros of the 1-form ω on Σ 2 (F t ) and F −1 t (0), respectively, and E pt n (resp. E qt n ) the ring of function germs on R n at p t (resp. q t ).
Proof: Note that I(F −1 (0), ω) = F, F p1 , ..., F pn , F x1 + p 1 F y , ..., F xn + p n F y is generated by 2n + 1 elements. Using Thom's transversality Theorem, we obtain a smooth family of mapping F t with F 0 = F such that 0 is a regular value of (F t , F tp 1 , ..., F tp n , F tx 1 + p 1 F ty , ..., F tx n + p n F ty ), for t = 0 sufficiently close to zero.
Analogously one proves the other equality. ✷ We can state the following consequences from previous results. (b) If 0 is not a contact singular point, then the multiplicity of (F, 0) is given by (c) If 0 is a contact singular point and det(F pipj )(0) = 0, then the multiplicity of (F, 0) is the sum of the numbers holding in (a) and (b).
Proof: Lemmas 2.6 and 2.7 remain valid in the complex analytic case. The result follows by complexifying the algebras E 2n+1 /I(F −1 (0), ω) and E 2n+1 /I(Σ 2 (F ), ω). ✷ From Proposition 2.1, the multiplicity of the germ of first order PDE is invariant by deformations of the complexification of F . It is not true that M 1 (F, 0) and M 2 (F, 0) are invariants by real deformations of F . We denote by deg 0 (f ) the degree of f : (R n , 0) −→ (R n , 0) at 0 (see [9] for more details).
Definition 2.8. Let (F, 0) be a germ of first order PDE. We define the index of (F, 0) as the integer for t = 0 sufficiently close to zero, where q i are the contact singular points of the first order partial differential equation F t .
Definition 2.10. We say that (F, q 0 ) and (G, q 1 ) are equivalent if there exist a germ of diffeomorphism h : (R n+1 , π(q 0 )) −→ (R n+1 , π(q 1 )) and a function germ A diffeomorphism H : R 2n+1 −→ R 2n+1 is said to be a contactomorphism (or contact diffeomorphism ) if H * (ω) = λω for some nowhere zero function λ. Another equivalence relation of germs of first order PDEs is introduced in [15]. This relation is defined as follows.
Definition 2.11. We say that (F, q 0 ) and (G, q 1 ) are contact equivalent if there exists a germ of contactomorphism H : (R 2n+1 , q 0 ) −→ (R 2n+1 , q 1 ) and a function germ γ : It is clear that equivalence of germs of first order PDEs implies contact equivalent. The converse is not true in general.    by cancelling the s-th column. Using (2.6), we deduce I(f −1 (0), θ) ⊂ I(g −1 (0), θ). The opposite inclusion follows by applying the same argument to C −1 f =g. Therefore, Since the determinant is linear in each row, we deduce from the above equality that I(g −1 (0), h * (θ)) ⊂ h * (I(g −1 (0), θ)). The opposite inclusion follows fromg = g •h −1 and the result follows from Equation (2.7). ✷ Let f : (R n , 0) −→ R k be a map germ. We denote by ind 0 (θ | f ) the index of the 1-form θ on f −1 (0) at 0, introduced by Ebeling and Gusein-Zade in [8]. We also denote by B(r) ⊂ R n the open ball of radius r centered at 0.
Lemma 2.14. Let f : (R n , 0) −→ R k be a smooth map germ and let λ : (R n , 0) −→ R be a function germ with λ(0) = 0. If θ is a 1-form on R n , then Proof: The proof follows from Definition 1 in [8]. ✷
Proof: The proof follows by using the formula of Eisenbud and Levine [9]. ✷

First order PDEs with first integral
When n = 1, the first order PDE is called implicit differential equation (IDE). At points where the partial derivative F y = 0, the IDE is locally the image of a germ of an immersion (R 2 , 0) −→ (R 3 , 0). Conversely, the image of every germ of an immersion f : (R 2 , 0) −→ (R 3 , 0) define an germ of IDE and is denoted by (R f , 0).  It is not hard to see that if the critical set of g is nowhere dense, then f is uniquely determined by (g, µ).
commutes for some germs of diffeomorphisms k, ψ and φ.
The following proposition reduces the equivalence problem for IDEs, which admit independent first integral, to that for the corresponding induced integral diagrams. Let (g, µ) be an integral diagram. We denote by m(g, µ) = dim R E 2 /I(J(g) −1 (0), dµ), where J(g) is the determinant of the jacobian matrix of g.

The cases n=1 and n=2
In this section we give examples to distinguish normal forms given in [10] and [17]. When n=1, we have M 1 (F, 0) = dim R E 3 / F, F p , F x + pF y and M 2 (F, 0) = dim R E 3 / F, F p , F pp . This shows that the multiplicity of (F, 0) coincides with the multiplicity introduced by Bruce and Tari in [2]. A particular class of implicit differential equations that have been most intensively studied are the IDEs that define at most two directions in the plane. This class of equations are called binary differential equations and are of the form F (x, y, p) = p 2 − δ(x, y) = 0. (5.1) In this case, M 1 (F, 0) = dim R E 2 /(δ, δ x ), M 2 (F, 0) = 0, ind(F, 0) = deg 0 (δδ y , δ x ).
In [17], Tari studied the singularities of codimension 2 of binary differential equations. He also obtained the topological normal forms of these singularities. We calculate in Table 1 the index and multiplicity of this class of equations. When n = 2, we have M 2 (F, 0) = dim R E 5 / F, F p1 , F p2 , R, det(B) and M 1 (F, 0) = dim R E 5 / F, F p1 , F p2 , F x1 + p 1 F y , F x2 + p 2 F y , where R = det(F pipj ) and