SEPARATING INVARIANTS FOR ARBITRARY LINEAR ACTIONS OF THE ADDITIVE GROUP

EMILIE DUFRESNE, JONATHAN ELMER, AND MÜFİT SEZER

Abstract. We consider an arbitrary representation of the additive group 𝔾_{a} over a field of characteristic zero and give an explicit description of a finite separating set in the corresponding ring of invariants.

1. Introduction

The problem of distinguishing the orbits of an action of a group G on a vector space V is one of the most fundamental in mathematics, and some of the most widely studied questions in mathematics are merely special cases of this problem. For example, if we take G to be the group GL_{n}(𝕜) acting by conjugation on the vector space of n × n matrices over a field 𝕜, then this is the problem of classifying square matrices up to conjugacy. If we take G to be the group SL_{2}(𝕜) and V to be the nth symmetric power, S^{n}(W) of the natural representation W, then this is the problem of classifying binary forms of degree n over 𝕜 up to equivalence.

The classical approach to solving these problems is to construct “invariant polynomials”. These are polynomial functions V → 𝕜 which are constant on the Gorbits. One can also view these as the G-fixed points 𝕜[V]^{G} of the 𝕜-algebra 𝕜[V] of polynomial functions from V to 𝕜, where G acts on 𝕜[V] via

g · f(v) = f(g^{−1} · v)

for v ∈ V, g ∈ G and f ∈ 𝕜[V]. From this point of view it is clear that 𝕜[V]^{G} is a subalgebra of 𝕜[V]. A natural approach to the orbit problem is then to try to find algebra generators.

Invariant theory can be considered to be the study of the subalgebras 𝕜[V]^{G} ⊆ 𝕜[V]. The problem of finding algebra generators has been studied rather extensively over the past 200 years, but we are still a very long way from being able to write down algebra generators in the general case. For example, in the case of SL_{2}(𝕜) acting on S^{n}(W), a complete set of algebra generators is known only for n ≤ 10, and the number of generators required appears to grow very quickly with n. While the list of groups and representations for which a complete set of generating invariants is known is very small, the problem has been solved algorithmically for reductive algebraic groups acting on an algebraic variety ([10, 11, 2]) and for certain nonreductive algebraic groups ([18], [4]). Many of these algorithms rely on Gröbner basis calculations, which have a tendency to explode in higher dimensions. For this reason, using full sets of generating invariants to separate orbits is rarely a realistic proposition.

Date: November 7, 2012.

2010 Mathematics Subject Classification. 13A50.

Key words and phrases. Additive group, locally nilpotent derivation, invariant theory, separating set, degree bounds.

It has been known for a number of years that one can sometimes obtain as much information about the orbits of a group using a smaller subset of 𝕜[V]^{G}; for a very simple example, see [3, Example 2.3.9]. With this in mind, a new trend in invariant theory has emerged, based around the following definition:

Definition 1 (Derksen and Kemper [3, Definition 2.3.8]). A separating set for the ring of invariants 𝕜[V]^{G} is a subset S ⊂ 𝕜[V]^{G} with the following property: given v, w ∈ V, if there exists an invariant f such that f(v) ≠ f(w), then there also exists s ∈ S such that s(v) ≠ s(w).

Separating sets have, in many respects, “nicer” properties than generating sets. As a first example, it is well known that if G is finite and the characteristic of 𝕜 does not divide |G|, then 𝕜[V]^{G} is generated by elements of degree ≤ |G| [7, 8], but this is not necessarily true in the modular case [16]. On the other hand, the analogue for separating invariants holds in arbitary characteristic [3, Theorem 3.9.13]. Second, Nagata famously showed that if G is not reductive, then 𝕜[V]^{G} is not always finitely generated [13]. On the other hand, regardless of whether 𝕜[V]^{G} is finitely generated, it must contain a finite separating set [3, Theorem 2.3.15]. Unfortunately, this existence proof is non-constructive. No algorithm is known for computing finite separating sets of invariants for non-reductive groups.

In this paper, we describe a finite separating set for any finite dimensional representation of the additive group 𝔾_{a} over a field 𝕜 of characteristic zero, extending the results of Elmer and Kohls for the indecomposable representations (see [6]). Accordingly, from now on, 𝕜 denotes a field of characteristic zero and 𝔾_{a} its additive group. The group 𝔾_{a} is in some sense the simplest of all non-reductive groups. We describe briefly its representation theory. In each dimension there is exactly one indecomposable representation. Following the classical convention, we let V_{n} denote the indecomposable representation of dimension n + 1. We have V_{n} ≅ V_{n}^{∗}. There is a basis x_{0}, …, x_{n} for V_{n}^{∗} such that the action of 𝔾_{a} on V_{n}^{∗} is given by

α · x_{i} = ∑_{j=0}^{i} \frac{α^{j}}{j!} x_{i−j} for α ∈ 𝔾_{a}, 0 ≤ i ≤ n.

In this case, we say that 𝔾_{a} acts basically with respect to the basis {x_{0}, …, x_{n}}. Note that 𝔾_{a} acts on V_{n}^{∗} via upper triangular and on V_{n} via lower triangular matrices. We note that V_{n}^{∗} is isomorphic to the nth symmetric power S^{n}(V_{1}^{∗}) of V_{1}^{∗}; if 𝔾_{a} acts basically on V_{1}^{∗} with respect to the basis {x_{0}, x_{1}}, then it acts basically on S^{n}(V_{1}^{∗}) with respect to the basis \{\frac{1}{j!}x_{0}^{n−j}x_{1}^{j} 0 ≤ j ≤ n\}.

For any finite dimensional representation W of 𝔾_{a}, there is a multiset of nonnegative integers n := {n_{1}, n_{2}, …, n_{k}} such that W ≅ V_{n_{1}} ⊕ V_{n_{2}} ⊕ … ⊕ V_{n_{k}} as representations of 𝔾_{a}. For shorthand, we let V_{(n)} denote the latter and identify 𝕜[V_{(n)}] with 𝕜[x_{i,j} | 0 ≤ i ≤ n_{j}, 1 ≤ j ≤ k]. For convenience, we will assume that n_{1}, n_{2}, …, n_{k} are ordered so that n_{j} is even for 1 ≤ j ≤ l and odd for l + 1 ≤ j ≤ k, and further assume that n_{j} ≡ 2 mod 4 for 1 ≤ j ≤ l′ and n_{j} ≡ 0 mod 4 for l′ + 1 ≤ j ≤ l. As the problem of computing separating sets for indecomposable linear 𝔾_{a}-actions was considered in [6], we assume throughout that k ≥ 2.

The main result of this paper is as follows:

Theorem 1. Let V be a finite dimensional representation of 𝔾_{a}, with dim(V) = n. Then there exists a separating set S ⊂ 𝕜[V]^{𝔾_{a}} with the following properties:

(1) S consists of invariants of degree at most 2n − 1.

(2) The size of S is quadratic in n.

(3) S consists of invariants which involve variables coming from at most 2 indecomposable summands.

This result will be proved in section 2. We also discuss and compare the number and degrees of elements in S with those of generating invariants in known cases. It should be noted that, while we can describe explicitly a separating set for any ring of invariants of a linear 𝔾_{a}-action, generating sets are known only in small dimensions. Section 3 explains the interest in the third property.

This work was carried out during a visit of the first author to Bilkent University funded by Tüba-Gebip and a later visit of the second author to Universität Basel. The authors would like to thank Hanspeter Kraft for making this visit possible.

2. Separating sets

Let V_{n} be the indecomposable representation of 𝔾_{a} of dimension n + 1 and suppose 𝔾_{a} acts basically with respect to the basis {x_{0}, …, x_{n}} of V_{n}^{∗}. The action of 𝔾_{a} is given by the formula

α · f = exp(αD_{n})f for α ∈ 𝔾_{a}, f ∈ 𝕜[V_{n}],

where D_{n} is the Weitzenböck derivation

D_{n} = x_{0} \frac{∂}{∂x_{1}} + ⋯ + x_{n−1} \frac{∂}{∂x_{n}} .

The algebra of invariants 𝕜[V_{n}]^{𝔾_{a}} is precisely the kernel of the derivation D_{n}. More generally, the ring of invariants 𝕜[V_{(n)}]^{𝔾_{a}} coincides with the kernel of the derivation

D_{(n)} := ∑_{j=1}^{k} x_{0,j} \frac{∂}{∂x_{1,j}} + ⋯ + x_{n_{j}−1,j} \frac{∂}{∂x_{n_{j},j}} .

Let n = (n_{1}, n_{2}, …, n_{k}) and n′ = (n′_{1}, n′_{2}, …, n′_{k}) be two vectors in ℕ^{k} with n_{j} ≥ n′_{j} for 1 ≤ j ≤ k. Define the linear map Π_{n′,n} : V_{(n′)} → V_{(n)} to be the map induced by the linear maps V_{n′_{j}} → V_{n_{j}},

(a_{0,j}, …, a_{n′_{j},j}) ↦ (0, …, 0, a_{0,j}, …, a_{n′_{j},j}).

The map Π_{n′,n} is 𝔾_{a}-equivariant, and so we have Π_{n,n′}^{∗}(𝕜[V_{(n)}]^{𝔾_{a}}) ⊆ 𝕜[V_{(n′)}]^{𝔾_{a}}, where Π_{n,n′}^{∗} is the corresponding algebra map. For a vector n = (n_{1}, n_{2}, …, n_{k}) set ⌊n/2⌋ = (⌊n_{1}/2⌋, ⌊n_{2}/2⌋, …, ⌊n_{k}/2⌋), where the symbol ⌊x⌋ denotes the largest integer less than or equal to x. We let n denote the dimension of V_{(n)}. Note that n = ∑_{j=1}^{k}(n_{j} + 1).

Proposition 2. Assume the convention of section 1. Then we have

Π_{n,⌊n/2⌋}^{∗}(𝕜[V_{(n)}]^{𝔾_{a}}) ⊆ 𝕜[x_{0,j} | 1 ≤ j ≤ l].

Moreover, Π_{n,⌊n/2⌋}^{∗}(𝕜[V_{(n)}]^{𝔾_{a}}) is contained in the ring of invariants of the cyclic group of order two acting on 𝕜[x_{0,j} | 1 ≤ j ≤ l] as multiplication by −1 on x_{0,j} for 1 ≤ j ≤ l′ and trivially on the remaining variables.

Proof. The proof essentially carries over from the indecomposable case (see [6, Proposition 3.1]). The isomorphisms V_{n_{j}}^{∗} ≅ S^{n_{j}}(V_{1}^{∗}) extend the 𝔾_{a}-action on 𝕜[V] to a SL_{2}(𝕜)-action when we identify V_{1}^{∗} with the natural representation of SL_{2}(𝕜). A well known theorem of Roberts [17] states that the 𝔾_{a}-equivariant linear map

\begin{aligned}Φ : & V_{(n)} & ⟶ & V_{(n)} ⊕ V_{1} \\ & v & ⟼ & (v, (0, 1)) \end{aligned}

induces an isomorphism Φ^{∗} : 𝕜[V_{(n)} ⊕ V_{1}]^{SL_{2}(𝕜)} → 𝕜[V_{(n)}]^{𝔾_{a}}. The elements μ_{α} and τ of SL_{2}(𝕜) acting on V_{1}^{∗} via

μ_{α} = ( \begin{matrix}α & 0 \\ 0 & α^{−1} \end{matrix} ) for α ∈ 𝕜 \ \{0\}, and τ = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )

act on V_{(n)} ⊕ V_{1} as follows:

\begin{matrix}μ_{α} · (…, a_{i,j}, …, b_{0}, b_{1}) = (…, α^{2i−n_{j}}a_{i,j}, …, α^{−1}b_{0}, αb_{1}) \\ τ · (…, a_{i,j}, …, b_{0}, b_{1}) = (…, (−1)^{i} \frac{i!}{(n_{j}−i)!}a_{n_{j}−i,j}, …, b_{1}, −b_{0}). \end{matrix}

Let f ∈ 𝕜[V_{(n)}]^{𝔾_{a}} and pick h ∈ 𝕜[V_{(n)} ⊕ V_{1}]^{SL_{2}(𝕜)} such that Φ^{∗}(h) = f. Then h is fixed by μ_{α} and so, for all α ∈ 𝕜 \ {0},

f(…, a_{i,j}, …) = h(…, a_{i,j}, …, 0, 1) = h(…, α^{2i−n_{j}}a_{i,j}, …, 0, α).

Thus, for all α ∈ 𝕜 \ {0}, we have

(Π_{n,⌊n/2⌋}^{∗}f)(…, a_{i,j}, …) = f(…, 0, …, a_{0,j}, …, a_{⌊n_{j}/2⌋,j}, …)

= h(…, 0, …, α^{n_{j}−2⌊n_{j}/2⌋}a_{0,j}, …, α^{n_{j}}a_{⌊n_{j}/2⌋,j}, …, 0, α).

Since this is a polynomial equation in α and 𝕜 is an infinite field, the equality must also hold for α = 0, in which case we have:

(Π_{n,⌊n/2⌋}^{∗}f)(…, a_{i,j}, …) = h(…, 0, …, a_{0,j}, 0, …, 0, 0),

where 2|n_{j}, proving the first statement.

To prove the second assertion, we use that h is also fixed by τ. We then have

(Π_{n,⌊n/2⌋}^{∗}f)(…, a_{i,j}, …) = h(…, 0, …, a_{0,j}, 0, …, 0, 0)
= h(…, 0, (−1)^{^{n_{j}}/2}a_{0,j}, 0, …, 0, 0),

ending the proof. □

Let f, g be two polynomials in 𝕜[V_{(n)} ⊕ V_{1}]^{SL_{2}(𝕜)}. Assume that the total degrees of these polynomials in the variables y_{0}, y_{1} are d_{1} and d_{2}, respectively, where we identify 𝕜[V_{1}] with 𝕜[y_{0}, y_{1}]. Then for r ≤ min(d_{1}, d_{2}), the polynomial

∑_{q=0}^{r}(−1)^{q}\binom{r}{q}\frac{∂^{r}f}{∂y_{0}^{r−q}∂y_{1}^{q}} \frac{∂^{r}g}{∂y_{0}^{q}∂y_{1}^{r−q}}

also lies in 𝕜[V_{(n)} ⊕ V_{1}]^{SL_{2}(𝕜)} (see, for example, [15, p. 88]). This polynomial is called the rth transvectant of f and g and is denoted by ⟨f, g⟩^{r}. Together with Roberts’ isomorphism this process produces a new invariant in 𝕜[V_{(n)}]^{𝔾_{a}} from a given pair as follows. Let f_{1}, f_{2} ∈ 𝕜[V_{(n)}]^{𝔾_{a}}, and let d_{1} and d_{2} denote the total degrees in y_{0}, y_{1} of Φ^{∗}^{−1}(f_{1}) and Φ^{∗}^{−1}(f_{2}), respectively. For r ≤ min(d_{1}, d_{2}) the rth semitransvectant of f_{1} and f_{2} is defined by

[f_{1}, f_{2}]^{r} := Φ^{∗}(⟨Φ^{∗}^{−1}(f_{1}), Φ^{∗}^{−1}(f_{2})⟩^{r}).

A crucial part of our separating set consists of semitransvectants of two polynomials each depending on only one summand. For these invariants, the inverse of Roberts’ isomorphism is given in terms of a derivation. For 1 ≤ j ≤ k, set

Δ_{j} = ∑_{i=0}^{n_{j}}(n_{j} − i)(i + 1)x_{i+1,j} \frac{∂}{∂x_{i,j}} .

Let f be in 𝕜[x_{0,j}, x_{1,j}, …, x_{n_{j},j}]^{𝔾_{a}} for some 1 ≤ j ≤ k. Then f is called isobaric of weight m, if all of the monomials x_{0,j}^{e_{0}}x_{1,j}^{e_{1}} ⋯ x_{n_{j},j}^{e_{n_{j}}} in f satisfy m = ∑_{i=0}^{n_{j}}(n_{j} −2i)e_{i}. For an isobaric f ∈ 𝕜[x_{0,j}, x_{1,j}, …, x_{n_{j},j}]^{𝔾_{a}} of weight m, the inverse of Roberts’ isomorphism is given by

Φ^{∗}^{−1}(f) = ∑_{i=0}^{m}(−1)^{i} \frac{Δ_{j}^{i}(f)}{i!}y_{0}^{i}y_{1}^{m−i},

see [9, p. 43]. For 1 ≤ j_{1} ≠ j_{2} ≤ l′, let N denote the least common multiple of n_{j_{1}} and n_{j_{2}}. We define w_{j_{1},j_{2}} := [x_{0,j_{1}}^{N/n_{j_{1}}} , x_{0,j_{2}}^{N/n_{j_{2}}} ]^{N}.

Proposition 3. Let 1 ≤ j_{1} ≠ j_{2} ≤ l′. There exists a non-zero scalar d such that

Π_{n,⌊n/2⌋}^{∗}(w_{j_{1},j_{2}}) = dx_{0,j_{1}}^{N/n_{j_{1}}} x_{0,j_{2}}^{N/n_{j_{2}}} .

Proof. Let 0 ≤ q ≤ N be an integer. Since the weight of the invariant x_{0,j_{1}}^{N/n_{j_{1}}} is N, the formula for Φ^{∗}^{−1} in the previous paragraph yields

\frac{∂^{N}Φ^{∗}^{−1}(x_{0,j_{1}}^{N/n_{j_{1}}} )}{∂y_{0}^{N−q}∂y_{1}^{q}} = ∑_{i=N−q}^{N−q}(−1)^{i} \frac{Δ_{j_{1}}^{i}(x_{0,j_{1}}^{N/n_{j_{1}}} )}{i!} \frac{i!}{(i − N + q)!} \frac{(N − i)!}{(N − i − q)!}y_{0}^{i−N+q}y_{1}^{N−i−q}
= (−1)^{N−q}q!Δ_{j_{1}}^{N−q}(x_{0,j_{1}}^{N/n_{j_{1}}} ).

Similarly, we have

\frac{∂^{N}Φ^{∗}^{−1}(x_{0,j_{2}}^{N/n_{j_{2}}} )}{∂y_{0}^{q}∂y_{1}^{N−q}} = ∑_{i=q}^{q}(−1)^{i} \frac{Δ_{j_{2}}^{i}(x_{0,j_{2}}^{N/n_{j_{2}}} )}{i!} \frac{i!}{(i − q)!} \frac{(N − i)!}{(q − i)!}y_{0}^{i−q}y_{1}^{q−i}
= (−1)^{q}(N − q)!Δ_{j_{2}}^{q}(x_{0,j_{2}}^{N/n_{j_{2}}} ).

Using that Φ^{∗} is an algebra homomorphism, we get

w_{j_{1},j_{2}} = ∑_{q=0}^{N}(−1)^{q}N!Δ_{j_{1}}^{N−q}(x_{0,j_{1}}^{N/n_{j_{1}}} )Δ_{j_{2}}^{q}(x_{0,j_{2}}^{N/n_{j_{2}}} ).

Since both j_{1} and j_{2} are congruent to two modulo four, we have Π_{n,⌊n/2⌋}^{∗}(x_{i,j}) = 0 if i < n_{j}/2, and Π_{n,⌊n/2⌋}^{∗}(x_{i,j}) = x_{i−^{n_{j}}/2,j} if i − n_{j}/2 ≥ 0 for j = j_{1}, j_{2}. Therefore to compute Π_{n,⌊n/2⌋}^{∗}(w_{j_{1},j_{2}}), it suffices to consider w_{j_{1},j_{2}} modulo the ideal of 𝕜[V_{(n)}] generated by x_{0,j_{1}}, …, x_{^{n_{j_{1}}}/2−1,j_{1}}, x_{0,j_{2}}, …, x_{^{n_{j_{2}}}/2−1,j_{2}}. Call this ideal I.

A monomial x_{0,j_{1}}^{e_{0}}x_{1,j_{1}}^{e_{1}} ⋯ x_{n_{j_{1}} ,j_{1}}^{e_{n_{j_{1}}}} in 𝕜[x_{0,j_{1}}, …, x_{n_{j_{1}} ,j_{1}}] is said to have j_{1}-weight p if p = ∑_{i=0}^{n_{j_{1}}} ie_{i}. Let m be a monomial with j_{1}-weight p and m′ be any other monomial appearing in Δ_{j_{1}}(m). Then m and m′ have the same degree and the j_{1}-weight of m′ is p + 1. It follows that the j_{1}-weight of any monomial appearing in Δ_{j_{1}}^{i}(x_{0,j_{1}}^{N/n_{j_{1}}} ) is i. But the smallest possible j_{1}-weight of a monomial of degree N/n_{j_{1}} in 𝕜[x_{^{n_{j_{1}}}/2,j_{1}}, …, x_{n_{j_{1}} ,j_{1}}] is N/2. Hence all monomials of degree N/n_{j_{1}} of j_{1}-weight less than N/2 lie in I. It follows that Δ_{j_{1}}^{i}(x_{0,j_{1}}^{N/n_{j_{1}}} ) ∈ I for i < N/2. Similarly, Δ_{j_{2}}^{i}(x_{0,j_{2}}^{N/n_{j_{2}}} ) ∈ I for i < N/2. Therefore we have

w_{j_{1},j_{2}} ≡ (−1)N!Δ_{j_{1}}^{^{N}/2}(x_{0,j_{1}}^{N/n_{j_{1}}} )Δ_{j_{2}}^{^{N}/2}(x_{0,j_{2}}^{N/n_{j_{2}}} ) mod I.

Furthermore, we claim that Δ_{j_{1}}^{^{N}/2}(x_{0,j_{1}}^{N/n_{j_{1}}} ) is equivalent to a non-zero multiple of

x_{^{n_{j_{1}}}/2,j_{1}}^{^{N}/n_{j_{1}}} modulo I. To see this, first note that the j_{1}-weight of monomials appear-

ing in Δ_{j_{1}}^{^{N}/2}(x_{0,j_{1}}^{N/n_{j_{1}}} ) is \frac{N}{2} . But x_{^{n_{j_{1}}}/2,j_{1}}^{N/n_{j_{1}}} is the only monomial of degree N/n_{j_{1}} in

𝕜[x_{^{n_{j_{1}}}/2,j_{1}}, …, x_{n_{j_{1}} ,j_{1}}] with j_{1}-weight N/2. Thus it suffices to show that x_{^{n_{j_{1}}}/2,j_{1}}^{^{N}/n_{j_{1}}}

appears with a non-zero coefficient in Δ_{j_{1}}^{^{N}/2}(x_{0,j_{1}}^{^{N}/n_{j_{1}}} ). This follows because for an arbitrary monomial m ∈ 𝕜[x_{0,j_{1}}, …, x_{n_{j_{1}} ,j_{1}}], any monomial that appears in Δ_{j_{1}}(m) has positive coefficients, and x_{^{n_{j_{1}}}/2,j_{1}}^{^{N}/n_{j_{1}}} can be obtained from x_{0,j_{1}}^{^{N}/n_{j_{1}}} in N/2 steps by replacing a variable u with another variable appearing in Δ_{j_{1}}(u) at each step. This establishes the claim. A similar argument shows that Δ_{j_{2}}^{^{N}/2}(x_{0,j_{2}}^{^{N}/n_{j_{2}}} ) is equivalent to

a non-zero multiple of x_{^{n_{j_{2}}}/2,j_{2}}^{^{N}/n_{j_{2}}} modulo I. The assertion of the proposition now fol-

lows because Π_{n,⌊n/2⌋}^{∗} is an algebra homomorphism and Π_{n,⌊n/2⌋}^{∗}(x_{^{n_{j}}/2,j}^{^{N}/n_{j}}) = x_{0,j}^{^{N}/n_{j}} for j = j_{1}, j_{2}. □

We introduce some invariants which will play a key role in the construction of our separating set, as they did in the construction of separating sets for the indecomposable representations, see [6]. For 1 ≤ j ≤ k and 1 ≤ i ≤ ⌊n_{j}/2⌋ define

f_{i,j} := ∑_{q=0}^{i−1}(−1)^{q}x_{q,j}x_{2i−q,j} + \frac{1}{2}(−1)^{i}x_{i,j}^{2}

and f_{0,j} = x_{0,j}. Also, for 1 ≤ i ≤ ⌊\frac{n_{j}−1}{2}⌋ set

s_{i,j} := ∑_{q=0}^{i}(−1)^{q} \frac{2i + 1 − 2q}{2} x_{q,j}x_{2i+1−q,j}

and s_{0,j} = x_{1,j}. Note that we have D_{(n)}(s_{i,j}) = f_{i,j} for 0 ≤ i ≤ ⌊\frac{n_{j}−1}{2}⌋. An element f in 𝕜[V_{(n)}] is called a local slice if D_{(n)}(f) ∈ 𝕜[V_{(n)}]^{𝔾_{a}}. For a non-zero element f ∈ 𝕜[V_{(n)}], let ν(f) denote the maximum integer d such that D_{(n)}^{d}(f) ≠ 0. For a local slice s and an arbitrary polynomial f define

ϵ_{s}(f) := ∑_{q=0}^{ν(f)} \frac{(−1)^{q}}{q!} (D_{(n)}^{q}f)s^{q}(D_{(n)}s)^{ν(f)−q}.

We remark that ϵ_{s}(f) ∈ 𝕜[V_{(n)}]^{𝔾_{a}}. Furthermore, for l′ + 1 ≤ j ≤ l, we define z_{j} := [x_{0,j}, f_{n_{j}/4,j}]^{n_{j}}. We can now make our main result precise:

Theorem 4. Let T denote the union of the following set of polynomials in 𝕜[V_{(n)}]^{𝔾_{a}}. (1) f_{i,j} for 1 ≤ j ≤ k and 0 ≤ i ≤ ⌊n_{j}/2⌋.

(2) ϵ_{s_{i_{2},j_{2}}} (x_{i_{1},j_{1}}) for 1 ≤ j_{1} < j_{2} ≤ k, ⌊\frac{n_{j_{1}} −1}{2}⌋ < i_{1} ≤ n_{j_{1}} and 0 ≤ i_{2} ≤

⌊\frac{n_{j_{2}} −1}{2}⌋.

(3) ϵ_{s_{i_{2},j}}(x_{i_{1},j}) for 1 ≤ j ≤ k, 0 ≤ i_{2} ≤ ⌊\frac{n_{j}−1}{2}⌋, i_{2} ≤ i_{1} ≤ n_{j}.

(4) ϵ_{s_{i_{2},j_{2}}} (x_{i_{1},j_{1}}) for 1 ≤ j_{2} < j_{1} ≤ k, 0 ≤ i_{1} ≤ n_{j_{1}}, 0 ≤ i_{2} ≤ ⌊\frac{n_{j_{2}} −1}{2}⌋.

(5) w_{j_{1},j_{2}} for 1 ≤ j_{1} ≠ j_{2} ≤ l′.

(6) z_{j} for l′ + 1 ≤ j ≤ l.

Then T is a separating set for 𝕜[V_{(n)}]^{𝔾_{a}}.

Proof. We first show that the invariants labelled (1)-(4) above separate any pair of vectors that do not simultaneously lie in V_{V_{(n)}}(x_{i_{2},j_{2}} | 1 ≤ j_{2} ≤ k, 0 ≤ i_{2} ≤ ⌊\frac{n_{j_{2}} −1}{2}⌋). If v_{1} = (a_{i,j}) and v_{2} = (b_{i,j}) are any two such vectors, then there exists 1 ≤ j′ ≤ k such that, for some 0 ≤ i′ ≤ ⌊\frac{n_{j′} −1}{2}⌋, a_{i′,j′} and b_{i′,j′} are not simultaneously zero. We assume that i′ and j′ are minimal among such indices, that is, that we have

(1) a_{i,j′} = b_{i,j′} = 0 for i < i′.

(2) a_{i,j} = b_{i,j} = 0 for j < j′ and 0 ≤ i ≤ ⌊\frac{n_{j}−1}{2}⌋.

If exactly one of a_{i′,j′} and b_{i′,j′} is zero, then f_{i′,j′} separates v_{1} and v_{2}. Otherwise the value of any invariant at v_{1} and v_{2} is determined by the set {f_{i′,j′}, ϵ_{s_{i′,j′}} (x_{i_{1},j_{1}}) | 0 ≤ i_{1} ≤ n_{j_{1}}, 1 ≤ j_{1} ≤ k}. Indeed, as D_{(n)}s_{i′,j′} = f_{i′,j′}, the “Slice Theorem” [18, 2.1] implies that

𝕜[V_{(n)}]_{f_{i′,j′}}^{𝔾_{a}} = 𝕜[ϵ_{s_{i′,j′}} (x_{i_{1},j_{1}}) | 0 ≤ i_{1} ≤ n_{j_{1}}, 1 ≤ j_{1} ≤ k]_{f_{i′,j′}} .

On the other hand, if i_{1} < i′ and j_{1} = j′ or if 0 ≤ i_{1} ≤ ⌊\frac{n_{j_{1}} −1}{2}⌋ and j_{1} < j′, then ϵ_{s_{i′,j′}} (x_{i_{1},j_{1}}) vanishes at v_{1} and v_{2} . It follows that the set

f_{i′,j′} ∪ \{ϵ_{s_{i′,j′}} (x_{i_{1},j_{1}}) | ⌊\frac{n_{j_{1}} − 1}{2}⌋ < i_{1} ≤ n_{j_{1}}, j_{1} < j′\} ∪ \{ϵ_{s_{i′,j′}} (x_{i_{1},j′}) | i′ ≤ i_{1} ≤ n_{j′}\}
∪ {ϵ_{s_{i′,j′}} (x_{i_{1},j_{1}}) | j′ < j_{1}, 0 ≤ i_{1} ≤ n_{j_{1}}}

separates v_{1} and v_{2} whenever they are separated by some invariant.

It remains to show that T is a separating set on the zero set of the ideal I := (x_{i_{2},j_{2}} | 1 ≤ j_{2} ≤ k, 0 ≤ i_{2} ≤ ⌊\frac{n_{j_{2}} −1}{2}⌋). Note that 𝕜[V_{(n)}]/I ≅ Π_{n,⌊n/2⌋}^{∗}(𝕜[V_{(n)}]) = 𝕜[V_{(⌊n/2⌋)}]. Thus, finding a set which separates on V_{V_{(n)}}(I) is equivalent to finding a subset E ⊆ 𝕜[V_{(n)}]^{𝔾_{a}} such that Π_{n,⌊n/2⌋}^{∗}(E) separates the same points of V_{(⌊n/2⌋)} as Π_{n,⌊n/2⌋}^{∗}(𝕜[V_{(n)}]^{𝔾_{a}}). By Proposition 2, Π_{n,⌊n/2⌋}^{∗}(𝕜[V_{(n)}]^{𝔾_{a}}) ⊆ 𝕜[x_{0,j} | 1 ≤ j ≤ l]^{C_{2}}, where the cyclic group of order two C_{2} acts as multiplication by −1 on the first l′ variables and trivially on the remaining variables.

Consider the subset B ⊆ Π_{n,⌊n/2⌋}^{∗}(T) formed by the following:

• Π_{n,⌊n/2⌋}^{∗}(f_{⌊n_{j}/2⌋,j}) = x_{0,j}^{2}, for 1 ≤ j ≤ l,

• Π_{n,⌊n/2⌋}^{∗}(w_{j_{1},j_{2}}) = dx_{0,j_{1}}^{N/n_{j_{1}}} x_{0,j_{2}}^{N/n_{j_{2}}} for 1 ≤ j_{1} ≠ j_{2} ≤ l′, where d ≠ 0 and N

is the least common multiple of n_{j_{1}} and n_{j_{1}}.

• Π_{n,⌊n/2⌋}^{∗}(z_{j}) = x_{0,j}^{3} for l′ + 1 ≤ j ≤ l, see [6, Lemma 5.4].

Showing that B is a separating set for 𝕜[x_{0,j} | 1 ≤ j ≤ l]^{C_{2}} will end the proof. More precisely, we show that value of the generators of 𝕜[x_{0,j} | 1 ≤ j ≤ l]^{C_{2}} is entirely determined by the value of the elements of B. The ring of invariants is given by

𝕜[x_{0,j} | 1 ≤ j ≤ l]^{C_{2}} = 𝕜[x_{0,j_{1}}x_{0,j_{2}}, x_{0,j} | 1 ≤ j_{1} ≤ j_{2} ≤ l′, l′ + 1 ≤ j ≤ l].

Suppose 1 ≤ j_{1} ≠ j_{2} ≤ l′. Note that N/n_{j_{1}} and N/n_{j_{2}} are odd integers. On points where either x_{0,j_{1}}^{2} or x_{0,j_{2}}^{2} is zero, so is x_{0,j_{1}}x_{0,j_{2}}. Otherwise, we have

x_{0,j_{1}}x_{0,j_{2}} = \frac{dx_{0,j_{1}}^{N/n_{j_{1}}} x_{0,j_{2}}^{N/n_{j_{2}}}}{d(x_{0,j_{1}}^{2})^{1/2(N/n_{j_{1}} −1)}(x_{0,j_{2}}^{2})^{1/2(N/n_{j_{2}} −1)}} .

Now suppose l′ + 1 ≤ j ≤ l. On points where x_{0,j}^{2} is zero, so is x_{0,j}, and otherwise, x_{0,j} = x_{0,j}^{3}/x_{0,j}^{2}. Therefore B ⊆ Π_{n,⌊n/2⌋}^{∗}(T) is a separating set. □

Theorem 1 is an easy consequence of Theorem 4:

Proof of Theorem 1 . The degree of the each invariant f_{i,j} is two, and the invariants z_{j} all have degree three. The degree of w_{j_{1},j_{2}} is N/n_{j_{1}} + N/n_{j_{2}}, where N = lcm(n_{j_{1}}, n_{j_{2}}). Since 1 ≤ j_{1}, j_{2} ≤ l′, we have N ≤ (n_{j_{1}}n_{j_{2}})/2 and so the degree of w_{j_{1},j_{2}} is at most (n_{j_{1}} +n_{j_{2}})/2. Finally, the degree of ε_{s_{i_{2},j_{2}}} (x_{i_{1},j_{1}}) is deg(s_{i_{2},j_{2}})i_{1}+1 which is less than or equal to 2n_{j_{1}} +1 which is in turn at most 2n−1, since n_{j} ≤ n−1 for all 1 ≤ j ≤ k. It then follows that the degree of each invariant in T is at most 2n − 1, as claimed.

The number of invariants of the form f_{i,j} in our separating set is

∑_{j=1}^{k} ⌊\frac{n_{j}}{2} ⌋ + 1 ≤ \frac{n + k}{2} .

Since 1 ≤ k ≤ n, this is linear in n. Note at this point that for each j_{1}, j_{2}, i_{2} we have ϵ_{s_{i_{2},j_{2}}} (x_{0,j_{1}}) = f_{0,j_{1}}, so we have already counted these elements. The number of further invariants in T of the form ε_{s_{i_{2},j_{2}}} (x_{i_{1},j_{1}}) is

∑_{j_{2}=1}^{k} ∑_{j_{1}=j_{2}+1}^{k}n_{j_{1}}⌊\frac{n_{j_{2}} + 1}{2}⌋ + ∑_{j=1}^{k} \begin{matrix}⌊ & \frac{n_{j} −1}{2} & ⌋ \\ & ∑ \\ & i_{2}=0 \end{matrix} (n_{j} − i_{2} + 1) − 1 + ∑_{j_{2}=1}^{k} ∑_{j_{1}=1}^{j_{2}−1}⌊\frac{n_{j_{1}} + 2}{2}⌋ ⌊\frac{n_{j_{2}} + 1}{2}⌋ .

Here the three terms correspond to the invariants labeled (4),(3), and (2) in our definition of T. Using that for any half-integer x we have x −^{1}/2 ≤ ⌊x⌋ ≤ x (which we also used to derive the third term above), the first term is bounded above by

\frac{1}{2} ∑_{j_{2}=1}^{k} ∑_{j_{1}=j_{2}+1}^{k} n_{j_{1}}(n_{j_{2}} + 1)

≤ \frac{1}{4} (( ∑_{j_{1}=1}^{k} n_{j_{1}} ) ( ∑_{j_{2}=1}^{k} n_{j_{2}} ) − ∑_{j=1}^{k} n_{j}^{2}) + \frac{k − 1}{2} ∑_{j=1}^{k} n_{j}

= \frac{1}{4}(n − k)(n + k − 2) − \frac{1}{4} ∑_{j=1}^{k} n_{j}^{2}.

For the same reason, the second term is bounded above by

∑_{j=1}^{k} \frac{1}{2}(n_{j} + 1)^{2} − ∑_{j=1}^{k} \frac{1}{2} \frac{(n_{j} − 2)}{2} \frac{n_{j}}{2} = \frac{3}{8} ∑_{j=1}^{k} n_{j}^{2} + linear terms.

The third term is bounded above by

∑_{j_{2}=1}^{k} ∑_{j_{1}=1}^{j_{2}−1} \frac{(n_{j_{1}} + 2)(n_{j_{2}} + 1)}{4}

= ∑_{j_{2}=1}^{k} ∑_{j_{1}=1}^{j_{2}−1} \frac{(n_{j_{1}} + 1)(n_{j_{2}} + 1)}{4} + ∑_{j_{2}=1}^{k} ∑_{j_{1}=1}^{j_{2}−1} \frac{n_{j_{2}} + 1}{4}

≤ \frac{1}{8} ∑_{j_{1}=1}^{k} ∑_{j_{2}=1}^{k}(n_{j_{1}} +1)(n_{j_{2}} +1)− \frac{1}{8} ∑_{j=1}^{k}(n_{j} +1)^{2} + \frac{1}{4} ∑_{j_{1}=1}^{k} ∑_{j_{2}=1}^{k}(n_{j_{2}} +1)− \frac{1}{4} ∑_{j=1}^{k}(n_{j_{2}} +1)

= \frac{1}{8}n^{2} − \frac{1}{8} ∑_{j=1}^{k} n_{j}^{2} + \frac{1}{4}nk + linear terms.

Moreover, there are \frac{1}{2}l′(l′ − 1) invariants of the form w_{j_{1},j_{2}}, and l − l′ of the form z_{j}. Ignoring linear terms, the size of T is therefore bounded above by

\frac{1}{4}nk + \frac{3}{8}n^{2} − \frac{1}{4}k^{2} + \frac{1}{2}l′^{2}

which is indeed quadratic in n as claimed, since l′ ≤ k and k ≤ n. Note that when k = 1 we get a separating set of size approximately \frac{3}{8}n^{2}, which coincides with the size of the separating set found in [6]. Indeed, our separating set specializes to the separating set found in [6] when k = 1.

The following tables show the exact size of T for certain representations V of 𝔾_{a}. It also shows the size of a minimal generating set c_{n} of 𝕜[V_{(n)}]^{𝔾_{a}}, when this is known. The data for the numbers c_{n} was taken from Andries Brouwer’s website [1]. Note that nV_{k} is taken to mean the direct sum of n copies of V_{k}. The generators of nV_{1} which coincide with our separating set T were first conjectured by Nowicki [14], and first proved by Khoury [12]. The case nV_{2} was recently solved by Wehlau [19].

\begin{matrix}V & 2V_{2} & 3V_{2} & 4V_{2} & nV_{2} & V_{3} & 2V_{3} & 3V_{3} & 4V_{3} & 5V_{3} & nV_{3} \\ |T | & 10 & 21 & 36 & 2n^{2} + n & 7 & 24 & 51 & 108 & 135 & 5n^{2} + 2n \\ |c_{n}| & 6 & 13 & 24 & \frac{1}{6}n(n^{2} + 3n + 8) & 4 & 26 & 97 & 280 & 689 & ? \end{matrix}

\begin{matrix}V & V_{4} & 2V_{4} & 3V_{4} & 4V_{4} & 5V_{4} & nV_{4} & V_{5} & 2V_{5} & nV_{5} & nV_{6} \\ |T | & 11 & 35 & 75 & 128 & 195 & 7n^{2} + 4n & 16 & 56 & 12n^{2} + 4n & \frac{1}{2}(31n^{2} + 9n) \\ |c_{n}| & 5 & 28 & 103 & 305 & ? & ? & 23 & ? & ? & ? \end{matrix}

\begin{matrix}V & V_{1}⊕V_{2} & V_{1}⊕V_{3} & V_{1}⊕V_{4} & V_{1}⊕V_{5} & V_{2}⊕V_{3} & V_{2}⊕ V_{4} & V_{2} ⊕ V_{5} & V_{3}⊕V_{4} & V_{3}⊕V_{5} \\ |T | & 7 & 12 & 17 & 23 & 15 & 21 & 29 & 30 & 39 \\ |c_{n}| & 5 & 13 & 20 & 94 & 15 & 18 & 92 & 63 & ? \end{matrix}

To prove that T contains only invariants depending on at most two summands, simply observe that the invariants f_{i,j} and z_{j} are non-zero only on the summand V_{n_{j}} of V_{(n)}, while ε_{s_{i_{2},j_{2}}} (x_{i_{1},j_{1}}) and w_{j_{1},j_{2}} are non-zero on only on V_{n_{j_{1}}} and V_{n_{j_{2}}} .

□

3. A note on Helly dimension

In [5], the authors define the Helly dimension of an algebraic group as follows:

Definition 2 (see [5, Definition 1.1]). The Helly dimension κ(G) of an algebraic group is the minimal natural number d such that any finite system of closed cosets in G with empty intersection, has a subsystem consisting of at most d cosets with empty intersection. We define κ(G) := ∞, if there are no such natural numbers.

They go on to show that if 𝕜 is a field of characteristic zero, and G acts on the affine 𝕜-variety X := Π_{i=1}^{k}X_{i}, then there exists a dense G-stable open subset U of X and a set S ⊂ 𝕜[X]^{G} of invariants each depending on at most κ(G) indecomposable factors of X such that S is a separating set on U [5, Theorem 4.1]. It is easy to see that the Helly dimension of 𝔾_{a} is two: in characteristic zero, the additive group does not have any proper nontrivial closed subgroups. That is, its only proper subgroup is {0}, and the only possible cosets are singletons. In particular, it follows from their work that for any product of 𝔾_{a}-varieties, we should be able to find an ideal I of 𝕜[X]^{𝔾_{a}} and a set S ⊂ 𝕜[X]^{𝔾_{a}} of invariants each depending on at most two factors, such that S is a separating set on the open set X \ V(I). We recover this result for representations of 𝔾_{a} in the first part of the proof of Theorem 4. In fact, one could easily prove the same result directly for a product of arbitrary 𝔾_{a}-varieties by applying the “Slice Theorem” with a local slice depending on just one factor.

For X a product of G-varieties, Domokos and Szabo also consider the quantities

σ(G, X) := min{d | ∃S ⊂ 𝕜[X]^{G}, a separating set depending on d factors of X},

and δ(G, X), defined as the minimum natural number d such that given x ∈ X with Gx closed in X, there exists a set {j_{1}, j_{2}, …, j_{d}} such that the projection y of x onto the subvariety Y := Π_{i=1}^{d}X_{j_{i}} has Gy closed in Y with the same dimension as Gx. The supremum of these quantities over all possible product varieties are denoted by σ(G) and δ(G), respectively. They remark that for any unipotent group G, δ(G) ≤ dim(G) [5, Section 5], and in particular δ(𝔾_{a}) = 1. Finally, they show that for any reductive group G, we have [5, Lemma 5.9]

σ(G) ≤ κ(G) + δ(G).

We do not know whether this inequality holds for non-reductive groups. If it did, it would follow that, given any affine 𝔾_{a}-variety X, we could find a separating subset of 𝕜[X]^{G} depending on at most 3 indecomposable factors of X. Theorem 4(3) shows that, provided 𝔾_{a} acts linearly, two factors suffices. It would be interesting to know whether this holds for products of arbitrary affine 𝔾_{a}-varieties.

References

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[4] Harm Derksen and Gregor Kemper. Computing invariants of algebraic groups in arbitrary characteristic. Adv. Math., 217(5):2089–2129, 2008.

[5] Mátyás Domokos and Endre Szabó. Helly dimension of algebraic groups. J. Lond. Math. Soc. (2), 84(1):19–34, 2011.

[6] Jonathan Elmer and Martin Kohls. Separating invariants for the basic 𝔾_{a}-actions. Proc. Amer. Math. Soc., 140(1):135–146, 2012.

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[9] David Hilbert. Theory of algebraic invariants. Cambridge University Press, Cambridge, 1993. Translated from the German and with a preface by Reinhard C. Laubenbacher, Edited and with an introduction by Bernd Sturmfels.

[10] Gregor Kemper. Calculating invariant rings of finite groups over arbitrary fields. J. Symbolic Comput., 21(3):351–366, 1996.

[11] Gregor Kemper. Computing invariants of reductive groups in positive characteristic. Transform. Groups, 8(2):159–176, 2003.

[12] Joseph Khoury. A Groebner basis approach to solve a conjecture of Nowicki. J. Symbolic Comput., 43(12):908–922, 2008.

[13] Masayoshi Nagata. On the 14-th problem of Hilbert. Amer. J. Math., 81:766–772, 1959.

[14] Andrzej Nowicki. Polynomial derivations and their rings of constants. Uniwersytet Mikołaja Kopernika, Toruń, 1994.

[15] Peter J. Olver. Classical invariant theory, volume 44 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1999.

[16] David R. Richman. On vector invariants over finite fields. Adv. Math., 81(1):30–65, 1990.

[17] Michael Roberts. On the Covariants of a Binary Quantic of the n^{th} Degree. The Quarterly Journal of Pure and Applied Mathematics, 4:168–178, 1861.

[18] Arno van den Essen. An algorithm to compute the invariant ring of a G_{a}-action on an affine variety. J. Symbolic Comput., 16(6):551–555, 1993.

[19] David Wehlau. Weitzenböck derivations of nilpotency 3. Forum Math., to appear, 2011.

Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Switzerland

E-mail address: emilie.dufresne@unibas.ch

Department of Mathematics, University of Aberdeen, King’s College, Aberdeen AB24 3UE, Scotland

E-mail address: j.elmer@abdn.ac.uk

Department of Mathematics, Bilkent University, Ankara 06800, Turkey

E-mail address: sezer@fen.bilkent.edu.tr