Joint Invariants of Linear Symplectic Actions

We discuss computations of joint invariants for symplectic and conformal symplectic group actions on subsets of symplectic and contact linear spaces and relate this to other equivalence problems and approaches.


Introduction
The classical invariant theory [6,14,11] investigates polynomial invariants of linear actions of a Lie group G on a vector space V , i.e. describes the ring (S V * ) G . For instance, the case of binary forms corresponds to G = SL(2, C) and V = C 2 ; equivalently for G = GL(2, C) one studies instead the algebra of relative invariants. The covariants correspond to invariants in the tensor product V ⊗ W for another representation W . Changing to the Cartesian product V × W leads to joint invariants of G.
In this paper we discuss joint invariants corresponding to the (diagonal) action of G on the iterated Cartesian product V ×m for increasing number of copies m ∈ N. We will focus on the case G = Sp(2n, R), V = R 2n and discuss the conformal G = CSp(2n, R) = Sp(2n, R) × R + and affine G = ASp(2n, R) = Sp(2n, R) ⋉ R 2n versions later.
This corresponds to invariants of m-tuples of points in V , i.e. finite ordered subsets, with respect to G. We describe the algebra of polynomial invariants given through generators and relations, also in the context of free resolutions, and in addition discuss the field of rational invariants. By the Hilbert-Mumford [6] and Rosenlicht [15] theorems, these can be interpreted as the algebra and field of functions on the quotient space V ×m /G respectively in the case of linear symplectic or conformal symplectic actions.
We also discuss invariants with respect to the group G = Sp(2n, R)×S m , i.e. invariants of non-ordered finite subsets of V . In this case description through generators and relations turns out to be much harder, but we provide a count of invariants via the Poincaré function and an explicit example. Another generalization that we discuss is the action of the same Lie group G on the contact space.
When approaching invariants of infinite sets, like curves or domains with smooth boundary, the theory of joint invariants is not directly applicable and the equivalence problem is solved via differential invariants [8]. In the case of group G and space V as above this problem was solved in [7]. We claim that the differential invariants from that reference can be obtained in a proper limits of joint invariants, i.e. via a certain discretization and quasiclassical limit, and demonstrate this explicitly in a simple case.
In this paper we focus on discussion of various interrelations of joint ivariants. In particular, at the conclusion we note that joint invariants can be applied to the equivalence problem of binary forms. Since these have been studied also via differential invariants [3,11] a further relation via the above symplectic discretization is possible. Moreover this can be generalized to n-ary forms, but we do not touch upon this here.
The relation to binary forms mentioned above is based on the Sylvester theorem [17], which in turn can be extended to more general Waring decompositions, important in algebraic geometry [1]. Our computations should carry over to the general case. This note is partially based on the results of [2], generalized and elaborated in several respects.

Recollection: Invariants
We briefly recall the basics of invariant theory, referring to [10,14] for more details. Let G be a Lie group acting on a manifold V . A point x ∈ V is regular if a neighborhood of the orbit G · x is fibered by G-orbits. In this case all these orbits are isomorphic homogeneous spaces G/H, where H is a stabilizer of a point in the orbit. A point x ∈ V is weakly regular, if this happens in a (not necessary G-invariant) neighborhood of x with respect to the Lie algebra g = Lie(G) action.
In general the action can lack regular points, but a generic point is weakly regular. For algebraic actions (in particular V is algebraic) a Zariski open set of points is regular.

Smooth invariants.
If G is just smooth and non-compact (and V is a smooth manifold), there is little one can do to guarantee regularity a priori. An alternative is to look for local invariants, i.e. functions I = I(x) in a neighborhood U ⊂ V such that I(x) = I(g · x) as long as x ∈ U and g ∈ G satisfy g · x ∈ U.
The standard method to search for such I is by elimination of group parameters, namely by computing quasi-transversals [14] or using normalization and moving frame [11]. Another way is to solve the linear PDE system L ξ (I) = 0 for ξ ∈ g = Lie(G).
Given the space of invariants {I} one can extend U ⊂ V and address regularity. In our case the invariants are easy to compute and we do not rely on any of these methods, however instead we describe the algebra and the field of invariants depending on specification of the type of functions I.

Polynomial invariants.
If G is semi-simple and V is linear, then by the Hilbert-Mumford theorem generic orbits can be separated by polynomial invariants I ∈ (S V * ) G , where S V * = ⊕ ∞ k=0 S k V * is the algebra of polynomials on V . Moreover, by the Hilbert basis theorem, the algebra of polynomial invariants A G = (S V * ) G is Noetherian, i.e. it has a finite number of generators a j ∈ A G , 1 ≤ j ≤ s.
Let V = R n (x 1 , ..., x n ), and let x = (x 1 , ..., x n ) represent the linear coordinates. Then we can identify S V * = R[x] as the algebra of homogeneous polynomials in x i . Then a j = a j (x), and we set a = (a 1 , . . . , a s ).
Denote R = R[a] the free commutative R-algebra generated by a = (a 1 , . . . , a s ). It forms a free module F 0 over itself. We can also consider A G as a module over R, with surjective R-homomorphism φ 0 : F 0 → A G , φ 0 (a j ) = a j (x). The first syzygy module is S 1 = Ker(φ 0 ), so we get the exact sequence: A syzygy is an element of S 1 , i.e. a relation r = r(a) between the generators of A G of the form k p=1 r ip a jp = 0, r ip ∈ R. The module S 1 is Noetherian, i.e. finitely generated by some b = (b 1 , . . . , b t ). Denote the free R-module generated by b by , defines the second syzygy module S 2 = Ker(φ 1 ), and we can continue obtaining , etc. This yields the exact sequence of R-modules: . . .
If for some q ∈ N the module S q is free, we can stop and get the free resolution of A G . The Hilbert syzygy theorem states that q-th module of syzygies S q is free for q ≥ s = #a. In particular, the minimal free resolution exists and has length ≤ s, see [5].
To emphasize the generating sets we adopt the following convention for depicting free resolutions:

Rational invariants.
If G is algebraic, in particular reductive, then by the Rosenlicht theorem [15] generic orbits can be separated by rational invariants I ∈ F G . Here R(x) is the field of rational functions on V and F G = R(x) G . Let d be the transcendence degree of F G . This means that there exist (a 1 , . . . , a d ) =ā, a j ∈ F G , such that F G is an algebraic extension of R(ā). Then either F G = R(a) for a =ā or F G is generated by a set a ⊃ā, which by the theorem on primitive element can be assumed of carinality s = #a = d + 1, i.e. a = (a 1 , . . . , a d , a d+1 ). In the latter case there is one algebraic relation on a. Note that d ≤ n because R(ā) ⊂ R(x).
We adopt the following convention for depicting this: 2.4. Our setup. If the Lie group G acts effectively on V , then for some q it acts freely on V ×q , and hence on all V ×m for m ≥ q. The number of rational invariants separating a generic orbit in V ×m is equal to the codimension of the orbit. It turns out that knowing all those invariants I on V ×q is enough to generate the invariants in V ×m for m > q. Indeed, let π i 1 ,...,iq : V ×m → V ×q be the projection to the factors (i 1 , . . . , i q ). Then the union of π * i 1 ,...,iq I for I from the field F G (V ×q ) gives the generating set of the field F G (V ×m ), and similarly for the algebra of invariants.
Below we denote A m G = A G (V ×m ) and F m G = F G (V ×m ).
2.5. The equivalence problem. Since polynomial invariants are rational and thus F G is obtained from A G by localization (field of fractions), it is enough to describe a solution to the equivalence problem for the rational case. Let I 1 , . . . , I s be the generating set of the action of G on V ×q . If s = d + 1, this set of generators is subject to the algebraic condition, which constrains the generators to an algebraic set Σ ⊂ R s . This is the signature space, cf. [12]. Now the q-tuple of points X = (x 1 , . . . , x q ) is mapped to I 1 (X), . . . , I s (X) ∈ Σ. Denote this map by Ψ. Two generic configuration of points X ′ , X ′′ ∈ V ×q are Gequivalent iff their signatures coincide Ψ(X ′ ) = Ψ(X ′′ ).

Invariants on Symplectic Vector Spaces
The group G = Sp(2n, R) acts almost transitively on V = R 2n (one open orbit that complements the origin), therefore there are no (continuous) invariants of the action. Already for two copies of V there appears one invariant -the symplectic area.
We will first consider in detail two partial cases n = 1, 2 and then pass to a description of the general case.
3.1. The case of dimension 2n = 2. Here V = R 2 (x, p) is endowed with the standard symplectic form ω = dx∧dp. Since the action is linear, the origin O(0, 0) is preserved and a pair of points A 1 (x 1 , p 1 ), A 2 (x 2 , p 2 ) forms the triangle OA 1 A 2 for which the symplectic area is an invariant of G. To avoid the factor 1 2 it is convenient to compute with the area of the parallelogram. Denote this invariant by This generates pairwise invariants on V ×m for m ≥ 2 induced through the pull-back of the projection π i,j : V ×m → V × V to the corresponding factors. In other words, we have Here the algebra is generated by one element, and we get the following minimal free resolution: Here the action is free on the level of m = 3 copies of V and we get 3 = dim V ×3 − dim G independent invariants a 12 , a 13 , a 23 . They generate the entire algebra, and we get the following minimal free resolution: 12 , a 13 , a 23 ]. Also F 3 G = R(a 12 , a 13 , a 23 ).
To obtain a relation, we try eliminating the variables x 1 , x 2 , x 3 , x 4 , p 1 , p 2 , p 3 , p 4 , but this fails with the standard Maple command. Yet, using the transitivity of the G-action we fix A 1 at (1, 0) and A 2 at (0, a 12 ), and then obtain the only relation b 1234 := a 12 a 34 − a 13 a 24 + a 14 a 23 = 0 that we identify as the Plücker relation. Thus the first syzygy is a module over R := R[a] with one generator, hence the minimal free resolution is: From the rational invariants viewpoint our 6 generators are not independent, and we can solve for any one of them from the relation b 1234 = 0, for instance: Hence, we can drop a 34 from the set of generators, getting This time the number of generators is 10, while codimension of the orbit is 10 − 3 = 7. Using the same method we obtain that the first syzygy module is generated by the Plücker relations b ijkl := a ij a kl − a ik a jl + a il a jk = 0. We have 5 of those: b = {b ijkl : 1 ≤ i < j < k < l ≤ 5}. Thus there should be relations among relations, or equivalently the second syzygies.
then this module is S 2 = Ker(φ 1 : F 1 → S 1 ⊂ F 0 ). Using elimination of parameters we find that S 2 is generated by c = {c i : 1 ≤ i ≤ 5} with Thus the minimal free resolution of A 5 G is (note that here, as well as in our other examples, the length of the resolution is smaller than what the Hilbert theorem predicts): As before, to generate the field of rational invariants, we express superfluous generators in terms of the others using the first syzygies. Namely, we express a 34 , a 35 , a 45 from the relations b 1234 , b 1235 , b 1245 ; the other 2 syzygies follow from the higher syzygies. Removing these generators we obtain a set of 7 independent generatorsā = a\{a 34 , a 35 , a 45 } whence The previous arguments generalize straightforwardly to conclude that A m G is generated by a = {a ij : 1 ≤ i < j ≤ m}. The first syzygy module is generated by the Plücker relations b = {b ijkl : 1 ≤ i < j < k < l ≤ m}. In other words we have: Similarly, the field of rational invariants is generated by a, yet all of them except for a 1j , a 2j can be expressed (rationally) through the rest via the Plücker relations b 12kl . Denoteā := {a 12 , a 13 , . . . , a 1m , a 23 , . . . , a 2m }, #ā = 2m − 3. Then we get for m ≥ 2: Taking instead the area of the parallelogram we get the invariant a 12 .
More generally for V ×m and 1 ≤ i < j ≤ m there is a G-invariant For m = 2 this gives the minimal free resolution: For m = 3 we have dim V ×3 = 12, dim G = 10 and the codimension of generic orbits is 3. In fact, the algebra A 3 G is generated by a = {a 12 , a 13 , a 23 } and the minimal free resolution is: For m = 4 the action of G becomes free, the codimension of generic orbits is 6 and the algebra A 4 G is generated by a = {a 12 , a 13 , a 14 , a 23 , a 24 , a 34 }. The minimal free resolution is: The same happens for m = 5: with a = {a ij : 1 ≤ i < j ≤ 5} the minimal free resolution is Note that this is a cubic relation in contrast with the quadratic relations for n = 1. Since the first syzygy is generated by one relation, we get the following minimal free resolution with R = R[a]: Thus we remove a 56 from the generating set,ā = a \ {a 56 }, and obtain 3.2.6. V ×7 . Here for 21 generators a = {a ij : 1 ≤ i < j ≤ 7}. we get 6 relations b ijklmn :=a ij a kl a mn − a ij a km a ln + a ij a kn a lm − a ik a jl a mn + a ik a jm a ln − a ik a jn a lm + a il a jk a mn − a il a jm a kn + a il a jn a km − a im a jk a ln + a im a jl a kn − a im a jn a kl + a in a jk a lm − a in a jl a km + a in a jm a kl = 0.
The minimal free resolution becomes longer and we do not provide it. Again, from the syzygies we can rationally express a 56 , a 57 , a 67 and lettingā = a\{a 56 , a 57 , a 67 } we obtain the following description of the field of rational invariants: 3.2.7. General V ×m . The previous arguments generalize straightforwardly to conclude that A m G is generated by a = {a ij : 1 ≤ i < j ≤ m}. The first syzygy module is generated by the cubic relations b = {b ijklst : 1 ≤ i < j < k < l < s < t ≤ m}. In other words we have: Similarly, the field of rational invariants is generated by a, yet all of them except for a sj , 1 ≤ s ≤ 4, can be rationally expressed through the rest via the cubic relations b 1234kl . Denoteā := {a 1j , a 2j , a 3j , a 4j }, #ā = 4m − 10. Then we get for m ≥ 4: 3.3. The general case. Let V = R 2n (x 1 , . . . , x n , p 1 , . . . , p n ) be equipped with the standard symplectic form ω = dx 1 ∧ dp 1 + · · · + dx n ∧ dp n . We consider the action of G on m-tuples of points V ×m . As before for a pair of points A i , A j ∈ V we have the invariant Let us count the number of local smooth invariants. The action of G on V is almost transitive, so the stabilizer of a nonzero point A 1 has dim G A 1 = 2n+1 2 − 2n = 2n 2 . For generic A 2 there is only one invariant a 12 (the orbit has codimension 1) and the stabilizer of . For generic A 3 there is only two more new invariants a 13 , a 23 (the orbit has codimension 2 + 1 = 3) and the stabilizer of . By the same reason for k ≤ 2n the stabilizer of generic k-tuple of points A 1 , . . . , A k has dim G A 1 ,...,A k = 2n−k+1 2 .
Finally for k = 2n the stabilizer of generic A 1 , . . . , A 2n is trivial.
Thus we get the expected number of invariants a ij . For m ≤ 2n + 1 there are no relations between them, and the first comes at m = 2n + 2. To see why, let us examine the first syzygies for increasing n as follows.
For n = 1 they have generators Inductively we obtain relations for dim V = 2n: (Again we identify S 2n+1 with the subgroup {σ ∈ S 2n+2 : σ(1) = 1}.) This resembles the decomposition of the determinant by a row (or column). And it is indeed the case, but for the Pfaffian: The algebra of G-invariants is generated by a with syzygies b: Let us fist prove that the invariants a ij generate the field F m G of rational invariants. We use the symplectic analog of Gram-Schmidt normalization. Namely, given m = 2n points A 1 , . . . , A 2n in general position, we normalize them using G = Sp(2n, R) as follows.
Let e 1 , . . . , e 2n be the symplectic basis of V with only nontrivial relations ω(e 2k−1 , e 2k ) = 1. At first A 1 can be mapped to the vector e 1 . The point A 2 can be mapped to the line Re 2 , and with the invariant condition ω(OA 1 , OA 2 ) = a 12 it is mapped to the vector a 12 e 2 . Next in mapping A 3 we have two constraints ω(OA 1 , OA 3 ) = a 13 , ω(OA 2 , OA 3 ) = a 23 , and the point can be mapped to the space spanned by e 1 , e 2 , e 3 satisfying those constraints. Continuing like this we arrive to the following matrix with columns OA i : where b 1234 = a 12 a 34 − a 13 a 24 + a 14 a 23 (but this does not vanish in general if n > 1) and by * we denote some rational expressions in a ij that do not fit the table.
If m < 2n then only the first m columns of this matrix have to be kept. If m > 2n then the remaining point A 2n+1 , . . . , A m have all their coordinates invariant as the stabilizer of the first 2n points is trivial. In any case we see that the invariants of A 1 , . . . , A m are expressed rationally in a ij .
To obtain polynomial invariants one clears the denominators in these rational expressions, so A m G is generated by a as well. Now the Pfaffian of the skew-symmetric matrix (a ij ) 2k×2k is the square root of the determinant of the Gram matrix of the vectors OA i , 1 ≤ i ≤ k, with respect to ω. If we take k = n + 1 then the vectors are linearly dependent and therefore the Pfaffian vanishes. Thus b are syzygies among the generators a. That they form a complete set follows from the same normalization procedure as above.
To obtain the generators of the field of rational invariants we use the syzygy to express all invariants throughā = {a ij : 1 ≤ i ≤ 2n; i < j} (for m ≥ 2n, otherwiseā = a). This is possible to do rationally, and we have #ā = 2nm − n(2n + 1). In summary: for m ≤ 2n.

Symmetric joint invariants
Now we discuss invariants of the extended groupĜ = Sp(2n, R) × S m on V ×m or, equivalently, G-invariants on configurations of unordered sets of points V ×m /S m (which is an orbifold but for rational invariants we can localize to the regular part).
Denote the algebra of polynomialĜ-invariants on V ×m by S m G ⊂ A m G . We have the projection π : A m G → S m G , given by As a Noetherian algebra S m G is finitely generated, yet it is not easy to establish its generating set explicitly. All linear terms average to zero, π(a ij ) = 0, but there are several invariant quadratic terms in terms of the homogeneous decomposition A m G = ⊕ ∞ k=0 A m k . For example, for n = 1, m = 4 we have A m 0 = R, A m 1 = R 6 = a 12 , a 13 , a 14 , a 23 , a 24 , a 34 , A m 2 = R 20 (21 monomials a ij a kl modulo 1 Plücker relation), etc. Then we have π(A m 0 ) = R, π(A m 1 ) = 0, π(A m 2 ) = R 2 with the generators 6π(a 2 12 ) = a 2 12 + a 2 13 + a 2 14 + a 2 23 + a 2 24 + a 2 34 , 12π(a 12 a 13 ) = a 12 a 13 + a 12 a 14 + a 13 a 14 − a 12 a 23 − a 12 a 24 + a 23 a 24 + a 13 a 23 − a 13 a 34 − a 23 a 34 + a 14 a 24 + a 14 a 34 + a 24 a 34 .
Theorem 4.1. The algebra S m G is generated by π(f ) for homogeneous polynomials f in a = (a ij ) of some degree d = d(m) independent of n.
Proof. From the previous section we know that A m G is generated by a ij with the Pfaffian relations. However these relations b 2n i 1 i 2 ...i 2n+2 are skew-symmetric in indices (i j ). Indeed, if σ ∈ S 2n+2 is a permutation and S → S σ is its action on skew-symmetric matrices, then Pf(S σ ) = sgn(σ) Pf(S). Thus all relations disappear in symmetrization and S m G is the image of the symmetrization operator π acting on the free algebra R[a]. From this moment on, the number n does not show in our symbolic computations.
Let us recall the idea of the proof of Hilbert's theorem on invariants [6]. (The above π is the Reynolds operator used there.) Consider the ideal I ⊂ R[a] generated by the homogeneous invariants of positive degrees. It is finitely generated (as an ideal) and its generators also generate the algebra of invariants (as an algebra).
To trace those generators, introduce the lexicographic order on monomials a 12 > a 13 > . . . > a 23 > . . . of cardinality N = m 2 . Select the leading monomials among S m -orbits of those that average to 0. Their multi-index powers generate a subset Σ ⊂ Z N ≥0 . The complement is a monoid. Let S be its set of generators. Then the highest degree d(m) of elements in S is the required number.

4.1.
Example. Let us illustrate how this works in the first nontrivial case m = 3 (as noted in the proof the value of n is not essential, it enters the formulae for a ij = a ij (x) but if we work only with symbols a ij then it plays no role).
These functions have the following form for some polynomial q(z): Finally let us note that from the general results [16] it follows that the field of fractions of S m G is the field of rational invariants, which is an extension of F m G of order m! and, in particular, it has the same transcendence degree d(m, n) as F m G .

Variation on the group and space
It is also popular to include scaling and translations to the transformation group G.

5.1.
Conformal and affine symplectic groups. For the group G 1 = CSp(2n, R) = Sp(2n, R) × R + the scaling makes the invariants a ij relative, yet of the same weight, so their ratios [a 12 : a 13 : · · · : a m−1,m ] or simply the invariants I ij = a ij a 12 are absolute invariants. These generate the field of invariants of transcendence degree d(m, n) − 1.
For the group G 2 = ASp(2n, R) = Sp(2n, R)⋉R 2n the translations do not preserve the origin O and this makes a ij non-invariant. However due to the formula 2ω(A 1 A 2 A 3 ) = a 12 + a 23 − a 13 (or more symmetrically: a 12 + a 23 + a 31 ), with the proper orientation of the triangle A 1 A 2 A 3 , we easily recover the absolute invariants a ij + a jk + a ki .
Alternatively, using the translation freedom we can move the point A 1 to the origin O, then its stabilizer in G 2 is G = Sp(2n, R) and we compute the invariants of (m − 1) tuples of points A 2 , . . . , A m as before. In particular they generate the field of invariants of transcendence degree d(m − 1, n).

5.2.
Invariants in the contact space. Infinitesimal symmetries of the contact structure Π = Ker(α), α = du − p dx in the contact space M = R 2n+1 (x, p, u), where x = (x 1 , . . . , x n ), p = (p 1 , . . . , p n ), are given by the contact vector field X H with the generating function H = H(x, p, u). Taking quadratic functions H with weights w(x) = 1, w(p) = 1, w(u) = 2 results in the conformally symplectic Lie algebra, which integrates to the conformally symplectic group G 1 = CSp(2n, R) (taking H of degree ≤ 2 results in the affine extension of it).
Alternatively, one considers the natural lift of the linear action of G = Sp(2n, R) on V = R 2n to the contactization M and makes a central extension of it. We will discuss the invariants of this action. Note that this action is no longer linear, so the invariants cannot be taken to be polynomial, but can be assumed rational.
5.2.1. The case n = 1. In the 3-dimensional case the group G 1 = GL(2, R) acts on M = R 3 (x, p, u) as follows: This action is almost transitive (no invariants), however there are singular orbits and a relative invariant R = xp − 2u. Extending the action to multiple copies of M, i.e. considering the diagonal action of G 1 on M ×m , results in m copies of this relative invariant, but also in the lifted invariants from various V ×2 : These are all relative invariants of the same weight, therefore their ratios are absolute invariants: Since u k enter only R k there are no relations involving those, and the relations on I ij are the same as for a ij , namely they are Plücker relations (since those are homogeneous, they are satisfied by both R ij and I ij ). As previously, we can use them to eliminate all invariants except forĪ = {I k , I 1i , I 2i }: The field of rational invariants for m > 1 is then described as follows:

5.2.2.
The general case. In general we also have no invariants on M and the following relative invariants on M ×m resulting in absolute invariants I k , I ij given by the same formulae. Again using the Pffafian relations we can rationally eliminate superfluous generators, and denote the resulting set byĪ = {I k , I ij : 1 ≤ k < m, i < j ≤ m, 1 ≤ i ≤ 2n}. This set is independent and containsd(m, n) elements, wherē Thisd(m, n) is thus the transcendence degree of the field of rational invariants:

From joint to differential invariants
When we pass from finite to continuous objects the equivalence problem is solved through differential invariants. In [7] this was done for submanifolds and functions with respect to our group G. Let us briefly recall the results for the curves.
In the case of dimension 2n = 2, the jet-space is J k (V, 1) = R k+2 (x, y, y 1 , . . . , y k ). Here G = Sp(2, R) has an open orbit in J 1 (V, 1), and the first differential invariant comes in order 2: There is also an invariant derivation By differentiation we get new differential invariants I 3 = ∇I 2 , I 4 = ∇ 2 I 2 , etc. The entire algebra of differential invariants is free: In the case of dimension 2n = 4, the algebra of differential invariants is also free: where I 2 and ∇ are correspondingly modified, but the new generators I 3 , I 4 (the subscript indicates the differential order) are quite complicated, see formulae in [7,Appendix].
In the general case we denote the canonical coordinates on V = R 2n by (t, x, y, z), where x and z and (n − 1)-dimensional vectors. Then the invariant derivation is equal to and the first differential invariant of order 2 is There is one invariant I 3 of order 3 independent of I 2 , ∇(I 2 ), one invariant I 4 of order 4 independent of I 2 , ∇(I 2 ), I 3 , ∇ 2 (I 2 ), ∇(I 3 ), and so on up to order 2n. Then the algebra of differential invariants of the G-action on J ∞ (V, 1) is freely generated as follows [7]: 6.2. Symplectic discretization. Consider, for simplicity, the case n = 1 with coordinates (x, y) on V = R 2 . Let A i = (x i , y i ), i = 0, 1, 2, be three close points lying on the curve y = y(x).
Similarly we obtain the invariant derivation (it uses only two points and hence is of the first order) The other generators I 3 , I 4 , . . . (important for n > 1) can be obtained by a higher order discretization, but the formulae become more involved.

Relation to binary forms and concluding remarks
According to the Sylvester theorem [17] a general binary form p ∈ C[x, y] of odd degree k = 2m − 1 with complex coefficients can be written as This decomposition is determined up to permutation of linear factors and independent multiplication of each of them by a k-th root of unity.
In other words, we have the branched cover of order k m m! × m (C 2 ) → S 2m−1 C 2 and the deck group of this cover is S m × (Z k ) ×m . Note that in the real case, due to uniqueness of the odd root of unity, the corresponding cover over an open subset of the base × m (R 2 ) → S 2m−1 R 2 has the deck group S m .
With this approach the invariants of real binary forms are precisely the joint symmetric invariants studied in this paper, and for complex forms one has to additionally quotient by Z ×m k , which is equivalent to passing from a ij to a k ij and other invariant combinations (example for m = 4: a 3 12 a 2 13 a 2 14 a 2 23 a 2 24 a 3 34 ) and subsequent averaging by the map π. Other approaches to classification of binary forms, most importantly through differential invariants [11,3], can be related to this via symplectic discretization.
Remark. Note that the standard "root cover" C n+1 → S n C 2 : (a 0 , a 1 , . . . , a n ) → (p 0 , p 1 , . . . , p n ), has order n!, which is less than m 2m−1 m! for n = 2m − 1. Invariants of the binary forms with this approach correspond to SL(2, C)-invariant polynomials on the orbifold C n+1 /S n+1 and can be expressed both through a and p.
The above idea extends further to ternary and higher valence forms (see [4] for the differential invariants approach and [13] for that with joint differential invariants), with the Waring decompositions [1] and similar giving the cover, but here the group G will no longer be symplectic. We expect all the ideas of the present paper to generalize to the linear and affine actions of other reductive groups G.