The differential equation
satisfied by a plane curve of degree n
Alain Lascoux
Abstract
Eliminating the arbitrary coefficients in the equation of a generic
plane curve of order n by computing sufficiently many derivatives,
one obtains a differential equation. This is a projective invariant.
The first one, corresponding to conics, has been obtained by Monge.
Sylvester, Halphen, Cartan used invariants of higher order. The expression of these invariants is rather complicated, but becomes much
simpler when interpreted in terms of symmetric functions.
Résumé
L’expression différentielle des courbes planes de degré donné fournit un invariant projectif. Monge a obtenu celle des coniques planes,
Sylvester et Halphen ont généralisé l’équation de Monge aux courbes
planes de tout degré. Nous montrons que la théorie des fonctions
symétriques permet de retrouver ces invariants, et d’en donner des
expressions plus compactes.
Mots-clefs: Équation de Monge, invariant d’Halphen, invariants différentiels.
Key words: Monge equation, Halphen invariant, differential invariants.
A line in the plane can be written
y = ax +b ,
with arbitrary coefficients a, b, but it is more satisfactory to write it
y 00 = 0 .
More generally
y n + (?)y n−1 x + · · · + (?)xn + · · · + (?)y 0 x0 = 0
is the equation of a general planar curve of order n. Writing sufficiently many
derivatives of this equation, one can eliminate in their system the arbitrary
coefficients (?).
However, already in the case of a conic (solved by Monge), we have to
use the derivatives of order 3, 4, 5, and the outcome is not straightforward to
interpret. We need some method to perform the elimination.
1
di
,
dxi
It is convenient, instead of taking
Di =
to rather use the normalization
di
i! dxi
With these conventions, the Leibnitz formula loses its coefficients:
X
D n (f g) =
Di f Dj g .
i+j=n
We need the collection {D 0 y, D 1 y, D 2 y, . . .}, which we can write with
the help of a generating series
∞
X
z i Diy .
i=0
Symmetric function theory tells that we ought to formally factorize this
series, as we factorize the total Chern class of a vector bundle.
Thus we introduce a formal alphabet A and write
∞
∞
∞
X
X
Y
X
di y
i
i
=
z Dy=
(1 + za) =
z i Λi A ,
i
i!
dx
i=0
i=0
i=0
a∈A
(1)
denoting by Λi A the elementary symmetric functions in A, and thus identifying D i y to Λi A. We refer to [4] for what concerns the theory of symmetric
functions, and to [3] for its λ-ring approach.
Remember that taking k copies of an alphabet (we write k A) translates
into taking the k-th power of the generating function :
!k
∞
∞
X
X
i i
z ΛA =
z i Λi (k A) .
i=0
i=0
Adding r copies of a letter x to these k copies of A is written, at the level of
generating series, as
!k
∞
∞
X
X
(1 + zx)r
=
z i Λi (k A + r x) .
(2)
z i Λi A
i=0
i=0
P
Thus, instead of having a sum D n (y 2 ) = i+j=n D i y D j y to express the
derivatives of the square of y, one can now use the more compact notation
D n (y 2 ) = Λn (2A).
More generally, D n (y 3 ) = Λn (3A), D n (y 4 ) = Λn (4A), . . . and one has the
following easy lemma resulting from Leibnitz’ rule:
2
Lemma 1 Given n, k, r ∈ N, then
D n (xr y k ) = xr Λn (kA + r/x) .
(3)
We can now easily write the derivatives of any orders of the components
of the equation of a planar curve.
Let us look first at the case treated by Monge.
We start with
u = y 2 + c1 xy + c2 y + (?)x2 + (?)x + (?)
and take successive derivatives, starting from the third one (so that the part
depending on x only (where the coefficients (?) appear) has already been
eliminated).
D3 u =
=
4
D u =
D5 u =
Λ3 (2A) + xc1 Λ3 (A + 1/x) + c2 Λ3 A
Λ3 (2A) + (xc1 + c2 )Λ3 (A) + c1 Λ2 A
Λ4 (2A) + (xc1 + c2 )Λ4 (A) + c1 Λ3 A
Λ5 (2A) + (xc1 + c2 )Λ5 (A) + c1 Λ4 A
(4)
(5)
(6)
(7)
Elimination of the coefficients among these three equations gives the vanishing :
2
30
21
Λ (A) Λ3 (A) Λ3 (2A) Λ2 Λ3
2Λ
+
2Λ
3
3
4
40
31
22
4
4
Λ (A) Λ (A) Λ (2A) = Λ Λ 2Λ + 2Λ + Λ ,
(8)
4
Λ (A) Λ5 (A) Λ5 (2A) Λ4 Λ5 2Λ50 + 2Λ41 + 2Λ32 writing Λi for Λi A, Λij for Λi AΛj A, . . . .
The last determinant can be simplified and becomes
00
2
000
y /2
Λ Λ3
y
/6
0
0
000
3
iv
00
2
4
22
Λ Λ Λ = y /6 y /24 (y /2) ,
4
Λ Λ5 2Λ32 y iv /24 y v /120 y 00 y 000 /6
(9)
which is Monge’s equation, after suppressing the extra factor y 00 /2 :
D 2 yD 2 yD 5 y − 3D 2 yD 3 yD 4 y + 2D 3 yD 3 yD 3 y = 0 .
(10)
The general case takes only a few lines more to be written down.
From Eq.2, one has
r n 0
r 2 n−2
n
n
n−1
x ΛA.
x Λ A+···+
Λ (A + rx) = Λ A + rx Λ A +
n
2
3
Let u be a polynomial in x, y of total degree n with leading term y n . The
equations
D n+1 u = 0 = · · · = D n(n+3)/2 u
are
0 = Λn+1 (nA) + c1,n−1 Λn ((n−1)A) + c2,n−1 Λn+1 ((n−1)A) + · · ·
+ c1,1 Λ2 A + c2,1 Λ3 A + · · · + Λn+1 A
0 = Λn+2 (nA) + c1,n−1 Λn+1 ((n−1)A) + c2,n−1 Λn+2 ((n−1)A) + · · ·
+ c1,1 Λ3 A + c2,1 Λ4 A + · · · + Λn+2 A
···
···
···
···
···
···
n(n+3)/2
0 = Λ
(nA)
+ c1,n−1 Λn(n+3)/2−1 ((n−1)A) + c2,n−1 Λn(n+3)/2 ((n−1)A) + · · ·
+ c1,1 Λn(n+1)/2+1 A + c2,1 Λn(n+1)/2+2 A + · · · + Λn(n+3)/2 A ,
where the coefficients ci,j are polynomials in x only.
Eliminating these coefficients, one obtains the vanishing of the following
determinant (we have written the columns in a different order):
n
n+1
Λ2 A Λ3 A · · · Λn+1 A · · ·
−1)A)
−1)A) Λn+1 (nA)
Λ
((n
Λ
((n
Λ3 A Λ4 A · · · Λn+2 A · · · Λn+1 ((n−1)A) Λn+2 ((n−1)A) Λn+2 (nA)
..
..
..
..
..
..
.
.
.
.
.
.
N
N −1
N
N
··· Λ A ··· Λ
((n−1)A) Λ ((n−1)A)
Λ (nA) (11)
with N = n(n + 3)/2, which is the differential equation satisfied by a planar
curve of order n.
This determinant has a simple structure, with blocks of n, n−1, . . . , 1
columns involving the elementary symmetric functions in A, 2A, . . . , nA.
One can simplify it a little. Because the image of a curve of degree n
under the transformation y → αy + β is still a curve of the same degree, the
value of the determinant is independent of y = Λ0 A and y 0 = Λ1 A, that one
can put both equal to 0.
Therefore, instead of the generating series (2), one can now take a second
alphabet D such that
Λi D = Λi+2 A , i = 0, 1, . . .
(as usual Λi = 0 for P
i < 0).
In other words, i z i Λi D = Λ2 A + zΛ3 A + · · · , and for what concerns
its powers, one has that
Λi (kD) = Λi+2k (kA) ,
4
i, k ∈ N .
The equation of a planar curve can now be rewritten
1−n
Λ0 D · · · Λn−1 D Λ−1 (2D) · · · Λn−3 (2D) · · ·
Λ
(nD)
0
n−2
2−n
Λ1 D · · · Λ n D
Λ (2D) · · · Λ (2D) · · ·
Λ (nD) ..
..
..
..
..
.
.
.
.
.
N
N −2
N −2n+2
··· Λ D
··· Λ
(2D)
Λ
(nD)
(12)
with N = (n − 1)(n + 4)/2, and, apart from notations, this is the equation
given by Sylvester.
For the conic, this equation is the determinant (9) that we have seen
above :
0
Λ D Λ1 D Λ−1 (2D) Λ0 D Λ1 D
0
1
Λ D Λ2 D Λ0 (2D) = Λ1 D Λ2 D Λ0 D Λ0 D (13)
2
Λ D Λ3 D Λ1 (2D) Λ2 D Λ3 D 2Λ1 D Λ0 D
The equation
Λ 0 D
Λ 1 D
..
.
5
Λ D
In terms
1 Λ1
Λ1 Λ2
Λ2 Λ3
Λ3 Λ4
Λ4 Λ5
Λ5 Λ6
of a planar cubic is :
Λ1 D Λ2 D Λ−1 (2D) Λ0 (2D) Λ−2 (3D)
Λ2 D Λ3 D Λ0 (2D) Λ1 (2D) Λ−1 (3D)
=0.
..
..
..
..
..
.
.
.
.
.
6
7
4
5
3
Λ D Λ D Λ (2D) Λ (2D) Λ (3D) (14)
of Λi D only, written Λi , and putting Λ0 = 1, the determinant reads
Λ2
0
1
0
Λ3
1
2Λ1
0
1
Λ4
2Λ1
Λ21 + 2Λ2
2
Λ5
Λ1 + 2Λ2
2Λ1 Λ2 + 2Λ3
3Λ1
2
2
3Λ1 + 3Λ2
Λ6
2Λ1 Λ2 + 2Λ3
Λ2 + 2Λ1 Λ3 + 2Λ4
2
3
Λ7 Λ + 2Λ1 Λ3 + 2Λ4 2Λ2 Λ3 + 2Λ1 Λ4 + 2Λ5 Λ + 6Λ1 Λ2 + 3Λ3 2
1
which expands into the rather less attractive expression : 6Λ71 Λ2 Λ23 −30Λ42 Λ21 Λ5 −
10Λ33 Λ1 Λ5 +5Λ2 Λ7 Λ23 −2Λ5 Λ7 Λ31 +5Λ7 Λ21 Λ32 +30Λ4 Λ52 Λ1 +6Λ5 Λ22 Λ23 +6Λ51 Λ6 Λ4 −
6Λ61 Λ6 Λ3 +3Λ6 Λ1 Λ24 −3Λ26 Λ1 Λ2 −20Λ41 Λ24 Λ3 −60Λ4 Λ31 Λ42 +5Λ2 Λ5 Λ24 +10Λ5 Λ4 Λ23 −
3Λ81 Λ5 Λ2 + 20Λ22 Λ24 Λ3 − 3Λ4 Λ1 Λ25 + 10Λ22 Λ24 Λ31 − 15Λ7 Λ41 Λ22 + 2Λ26 Λ31 + 4Λ7 Λ42 +
5Λ62 Λ3 − 8Λ2 Λ4 Λ3 Λ1 Λ5 − 5Λ2 Λ6 Λ21 Λ5 − 10Λ23 Λ22 Λ1 Λ4 + 5Λ31 Λ34 + 12Λ71 Λ5 Λ3 −
6Λ2 Λ51 Λ24 +5Λ21 Λ4 Λ33 +5Λ21 Λ3 Λ25 −30Λ41 Λ2 Λ33 −50Λ42 Λ23 Λ1 −12Λ5 Λ61 Λ22 +6Λ22 Λ4 Λ71 −
10Λ4 Λ42 Λ3 − 10Λ4 Λ32 Λ5 − 10Λ61 Λ33 + 10Λ2 Λ4 Λ41 Λ5 − 7Λ21 Λ6 Λ4 Λ3 + 12Λ2 Λ7 Λ61 +
10Λ32 Λ33 + Λ21 Λ5 Λ24 + 3Λ3 Λ6 Λ1 Λ5 − 3Λ3 Λ7 Λ1 Λ4 − 2Λ4 Λ6 Λ5 − 15Λ2 Λ6 Λ4 Λ31 +
5Λ2 Λ7 Λ21 Λ4 − 2Λ2 Λ6 Λ4 Λ3 + 10Λ23 Λ1 Λ24 + 6Λ71 Λ24 − 10Λ2 Λ1 Λ34 − 10Λ1 Λ2 Λ43 +
3Λ5 Λ7 Λ1 Λ2 − 5Λ31 Λ6 Λ23 + 10Λ41 Λ5 Λ23 − 3Λ81 Λ4 Λ3 + 30Λ51 Λ4 Λ23 − 6Λ61 Λ4 Λ5 −
4Λ51 Λ7 Λ3 −2Λ51 Λ25 −10Λ6 Λ42 Λ1 −6Λ6 Λ32 Λ3 −4Λ7 Λ22 Λ4 +4Λ6 Λ22 Λ5 +30Λ6 Λ32 Λ31 −
5
3Λ81 Λ7 +54Λ22 Λ23 Λ51 +50Λ22 Λ33 Λ21 +24Λ4 Λ51 Λ32 −5Λ6 Λ33 −30Λ32 Λ31 Λ23 +45Λ32 Λ41 Λ5 +
60Λ52 Λ21 Λ3 −45Λ42 Λ41 Λ3 −10Λ32 Λ1 Λ24 −10Λ32 Λ61 Λ3 −5Λ3 Λ34 +3Λ52 Λ51 +10Λ2 Λ21 Λ24 Λ3 −
14Λ7 Λ1 Λ22 Λ3 + 10Λ62 Λ31 + 5Λ2 Λ31 Λ25 − 27Λ6 Λ51 Λ22 − 15Λ72 Λ1 + 40Λ32 Λ3 Λ1 Λ5 +
6Λ2 Λ6 Λ71 − 8Λ2 Λ3 Λ25 + 6Λ53 + Λ35 + Λ91 Λ6 + Λ26 Λ3 − 20Λ2 Λ4 Λ33 + 10Λ4 Λ21 Λ32 Λ3 +
45Λ4 Λ41 Λ3 Λ22 − 40Λ5 Λ31 Λ3 Λ22 − 15Λ6 Λ21 Λ3 Λ22 + 14Λ6 Λ4 Λ1 Λ22 + 10Λ2 Λ7 Λ3 Λ31 +
18Λ2 Λ6 Λ1 Λ23 − 5Λ2 Λ5 Λ21 Λ23 − 42Λ2 Λ4 Λ61 Λ3 − 20Λ2 Λ4 Λ31 Λ23 − 12Λ2 Λ5 Λ51 Λ3 +
20Λ2 Λ6 Λ41 Λ3 + Λ7 Λ24 + Λ21 Λ7 Λ23 − Λ5 Λ7 Λ3 Recall that Λ0 = y 00 /2!, Λ1 =
y 000 /6!, . . . , Λ7 = y vii /7!, so that the equation of a cubic is a differential polynomial of order 15, and degree 10.
Invariance under the projective group
We did not yet use that the image of a curve under a projective transformation of the plane is still a curve of the same degree, and therefore that
the differential equation is a projective invariant.
Expressed in terms of D, invariance under some subgroup of the projective
group amounts to belonging to the kernel of
∇D :=
d
d
d
d
+ 2Λ1 D
+ 3Λ2 D
+ 4Λ3 D
+···
d Λ1
d Λ2
d Λ3
d Λ4
This operator appears in the theory of binary forms in x, y, and simply
expresses the invariance under the translation x → x + 1. Elements of this
kernel are called semi-invariants in the theory of binary forms.
To simplify the operator ∇D , let us introduce another alphabet E :
Λi E = Λi D/i! =
di+2 y
.
i! (i +2)! dxi+2
1
(15)
Under the change of alphabet, our projective invariants belong to the
kernel of
d
d
d
d
∇E :=
+ Λ1 E
+ Λ2 E
+ Λ3 E
+···
d Λ1
d Λ2
d Λ3
d Λ4
But this kernel is very easy to determine. Indeed, ∇E sends ψ1 E onto 1,
and it sends all the other power sums ψi E, i = 2, . . . , n onto 0, n being the
cardinality of E. Therefore a semi-invariant is a polynomial in ψ2 E, ψ3 E, . . ..
Monge’s element is of degree 3 in E. Therefore it must be proportional
to ψ3 E. Indeed, the equation of a conic is
ψ3 E = 0 ,
(16)
or, in terms of elementary symmetric functions,
ψ3 E = 3Λ3 E − 3Λ12 E + Λ111 E = 0
6
(17)
4
3
5
d y
d y
d y
, Λ2 E = 2! 4!!
, Λ3 E = 1! 3!!
, into the
which rewrites, with Λ1 E = 1! 3!!
dx3
dx4
dx5
equation that we have already seen:
00 2 v
000 3
y
y 00 y 000 y iv
y
y
3
−3
+
=0.
2!
3!5!
2! 1!3! 2!4!
1!3!
The equations of curves of higher degree can be similarly treated, using
the usual theory of binary forms, and involve generalizations of the Hessian.
The next projective invariant, after the invariant of Monge, has been
found by Halphen[2]. Cartan[1] takes it as the projective analogue of the
curvature. The equation of the cubic is a polynomial in it and Monge invariant. Halphen invariant, in terms of power sums in E, is
48ψ5 ψ3 − 20ψ32 ψ2 − ψ24 + 12ψ22 ψ4 − 36ψ42
(18)
and is easier to handle than determinants of the type displayed in Eq. (14).
References
[1] E. Cartan. Leçons sur la théorie des espaces à connexion projective,
chapitre II, Paris, Gauthier Villars (1937).
[2] G.H. Halphen. Thèse sur les invariants différentiels, Paris (1878).
[3] A. Lascoux. Symmetric functions & Combinatorial operators on polynomials, CBMS/AMS Lectures Notes 99, (2003).
[4] I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford University Press, Oxford, 2nd edition (1995).
[5] P.A. MacMahon. Collected Papers, 2 vol., MIT Press (1978).
Vol. II: Operators in the theory of seminvariants, p. 562–565.
Memoir on seminvariants, p.566–583.
On perpetuant reciprocants, p. 584–596.
The invariants of the Halphenian homographic substitution, p. 798–828.
On a modified form of pure reciprocants, p. 829–832.
[6] J.J. Sylvester. Collected Work, Chelsea reprints.
Vol IV: Note on Captain Mac Mahon’s transformation of the theory of
invariants, p. 236–237.
On the method of reciprocants, p.278–302.
7
Lectures on the theory of reciprocants, 303–513.
Note sur les invariants différentiels, p.520–523.
Sur l’équation différentielle d’une courbe, p.524–526.
Alain Lascoux
CNRS, Institut Gaspard Monge, Université de Marne-la-Vallée
77454 Marne-la-Vallée Cedex, France
Alain.Lascoux@univ-mlv.fr
8