Unspecified Journal Volume 00, Number 0, Pages 000–000 S ????-????(XX)0000-0

ON THE HILBERT IDEAL

MARA D. NEUSEL

Abstract. We prove the Hilbert number conjecture.

1. Introduction

Let 𝔽 be a field, G a finite group. Consider a faithful representation

ρ : G ↪ GL(n, 𝔽).

It induces a G-action on the vector space V = 𝔽^{n} by matrix multiplication and thus on the dual space V^{∗}. We extend this action additively and multiplicatively to the symmetric algebra on V^{∗}, denoted by 𝔽[V]. Once we choose a basis x_{1}, …, x_{n} for V^{∗} we can identify

𝔽[V] ≅ 𝔽[x_{1}, …, x_{n}]

as the ring of polynomial functions in n indeterminates over the field 𝔽. We are interested in the subalgebra

𝔽[V]^{G} ↪ 𝔽[V]

of G-invariant polynomials. By a classical result due to E. Noether the ring of invariants 𝔽[V]^{G} is finitely generated as an 𝔽-algebra. Furthermore, since our action is 𝔽-linear we can assume that the generators of 𝔽[V]^{G} are homogeneous polynomials. We define β(𝔽[V]^{G}) to be the maximal degree of an algebra generator in a minimal algebra generating set. This number is usually called the Noether number. If the group order, |G|, is invertible in 𝔽, the nonmodular case, then β(𝔽[V]^{G}) ≤ |G|, independent of the representation, see Theorem 2.3.3 in [8]. In the modular case, i.e., |G| = 0 ∈ 𝔽, the value of β(𝔽[V]^{G}) usually depends on the degree of the representation or the order of the field and the group order, see [3] for an overview on degree bounds, see [4] and [5] for recent results.

In this paper we consider not the ring of invariant polynomials but the Hilbert ideal

𝔥(V, G) = (\overline{𝔽[V]^{G}}) ⊆ 𝔽[V]

which is by definition generated by all homogeneous invariant polynomials of positive degree. The Hilbert number, denoted by β(𝔥(()V, G)) is the maximal degree of an ideal generator in a minimal ideal generating set. In [1] it has been conjectured that the Hilbert ideal is, in contrast to the ring of invariants, always generated by invariants of degree at most group order. Of course, in the nonmodular case this

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

Received by the editors December 3, 2007.

2000 Mathematics Subject Classification. Primary 13A50 Commutative Rings.

Key words and phrases. invariant theory of finite groups, degree bounds, Hilbert ideal, Hilbert number.

follows from Noether’s bound. Furthermore, this conjecure was proven for permutation representations in [2] and for indecomposable representations of ℤ/p in [9]. In this paper we prove this conjecture for all representations. While typing this manuscript I learned that the same conjecture has been proven simulteneously by Symonds, [10].

2. The Proof

Consider the induced 𝔽G-module 𝔽G ⊗_{𝔽} V and the canonical inclusion

V ↪ 𝔽G ⊗_{𝔽} V .

This induces a G-equivariant projection

η_{G} : 𝔽[𝔽G ⊗_{𝔽} V] ⟶ 𝔽[V]

of 𝔽-algebras. Upon restriction to the respective rings of invariants we obtain the classical Noether map

η_{G}^{G} : 𝔽[𝔽G ⊗_{𝔽} V]^{G} ⟶ 𝔽[V]^{G}.

This map remains surjective in the classical nonmodular case. However in the modular case it often fails to be surjective, see [6] and [7] for a thorough study of the Noether map.

In the nonmodular case the ring of invariants and hence the Hilbert ideal is generated by polynomials of degree at most group order. Thus we restrict our attention to the modular case and obtain a commutative diagram as follows.

𝔽[𝔽G ⊗_{𝔽} V] ↠^{η_{G}} 𝔽[V]
∪  ∪
𝔽[V]^{G}
∪
𝔽[𝔽G ⊗_{𝔽} V]^{G} ↠^{η_{G}^{G}} Im(η_{G}^{G})

Note that G acts by permutations on 𝔽G ⊗_{𝔽} V. Thus by Fleischmann’s result, [2], we have

β(𝔥(𝔽G ⊗_{𝔽} V, G)) ≤ |G|.

Therefore

(⋆)  β(\overline{Im(η_{G}^{G})}) ≤ |G|,

where \overline{Im(η_{G}^{G})} ⊆ 𝔽[V] denotes the ideal generated by the invariants of positive degree in the image of the Noether map.

Furthermore, the kernel of η_{G} is generated by linear forms. Moreover, Ker(η_{G}) ⊆ 𝔽[𝔽G⊗V] is a prime ideal of height n|G|−|G|. Thus we obtain the following chains of ideals

𝔽[𝔽G ⊗_{𝔽} V] ↠^{η_{G}} 𝔽[V]
∪  ∪
Ker(η_{G}) ⊆ η_{G}^{−1}(\overline{Im(η_{G}^{G})})  \overline{Im(η_{G}^{G})}
∪
𝔥(𝔽G ⊗ V, G)

Let I ⊆ 𝔥(V, G) ⊆ 𝔽[V] be the ideal generated by all G-invariant polynomials of positive degree at most group order. We want to show that I = 𝔥(V, G). By Inequality (⋆) we have

\overline{Im(η_{G}^{G})} ⊆ I ⊆ 𝔥(V, G) ⊆ 𝔽[V].

Thus we can extend the above diagram to the following

𝔽[𝔽G ⊗_{𝔽} V] ↠^{η_{G}} 𝔽[V]
∪  ∪
η_{G}^{−1}(𝔥(V, G))  𝔥(V, G)
∪  ∪
η_{G}^{−1}(I)  I
∪  ∪
Ker(η_{G}) ⊆ η_{G}^{−1}(\overline{Im(η_{G}^{G})})  \overline{Im(η_{G}^{G})}
∪
𝔥(𝔽G ⊗ V, G)

We collect some properties of the ideals involved in the following lemma.

Lemma 2.1. The ideals I ⊆ 𝔥(V, G) ⊆ 𝔽[V] as well as the ideals η_{G}^{−1}(I) ⊆ η_{G}^{−1}(𝔥(V, G)) ⊆ 𝔽[𝔽G ⊗ V] are closed under the action of the group G.

Proof. The first two ideals are by definition generated by invariant polynomials. Thus they are closed under the G-action. Since the map η_{G} is G-equivariant, the latter two are also closed under the G-action. □

Next we present a series a reduction arguments. The first one shows that it is enough to prove that the inverse images of I and 𝔥(V, G) in 𝔽[𝔽G ⊗ V] are equal.

Lemma 2.2. If η_{G}^{−1}(𝔥(V, G)) = η_{G}^{−1}(I) then 𝔥(V, G) = I.

Proof. By construction we have I ⊆ 𝔥(V, G). To prove the reverse inclusion let f ∈ 𝔥(V, G). Then there exists an element F ∈ η_{G}^{−1}(𝔥(V, G)) = η_{G}^{−1}(I) with

f = η_{G}(F) ∈ η_{G}(η_{G}^{−1}(𝔥(V, G))) = η_{G}(η_{G}^{−1}(I)) = I

as desired. □

The third lemma shows that it is enough to consider elements that map onto G-invariant polynomials under the Noether map.

Lemma 2.3. If F ∈ η_{G}^{−1}(I) for all F ∈ 𝔽[𝔽G ⊗ V] with η_{G}(F) = f ∈ 𝔽[V]^{G}, then η_{G}^{−1}(I) = η_{G}^{−1}(𝔥(V, G)).

Proof. The inclusion ⊆ is true by construction. To prove the reverse inclusion let H ∈ η_{G}^{−1}(𝔥(V, G)) be an arbitrary element. Then

η_{G}(H) = ∑ α_{i}f_{i} ∈ 𝔥(V, G),

where f_{i} ∈ 𝔽[V]^{G} have positive degree and α_{i} ∈ 𝔽[V]. By assumption for each i there exists an F_{i} ∈ η_{G}^{−1}(I) such that η_{G}(F_{i}) = f_{i}. Thus

H = ∑ A_{i}F_{i} + K ∈ η_{G}^{−1}(I),

for some η_{G}(A_{i}) = α_{i} and K ∈ Ker(η_{G}). □

We write

𝔽G ⊗ V = span_{𝔽}\begin{cases}x_{1g_{1}} & … & x_{1g_{|G|}} \\ x_{2g_{1}} & … & x_{2g_{|G|}} \\ ⋮ & & ⋮ \\ x_{ng_{1}} & … & x_{ng_{|G|}} \end{cases}\}

for some enumeration of the group elements G = {g_{1}, …, g_{|G|}}. Withoutloss of generality we assume that g_{1} = 1 ∈ G is the identity element. Then the group G acts on the first index, and the Noether map is given by

η_{G}(x_{ig_{j}}) = g_{j}x_{i} ∈ 𝔽[V].

We choose the reverse lexicographic order x_{1g_{1}} < ⋯ < x_{1g_{|G|}} < x_{2g_{1}} < ⋯ < x_{ng_{|G|}}.

We note that for any polynomial f = ∑ a_{i}x_{1}^{i_{1}} ⋯ x_{n}^{i_{n}} ∈ 𝔽[V], where a_{i} ∈ 𝔽, there exists an inverse image

F = η_{G}^{−1}(f) = ∑ a_{i}x_{11}^{i_{1}} ⋯ x_{n1}^{i_{n}} ∈ 𝔽[𝔽G ⊗ V]

whose terms are monomials in the elements of the first column of (x_{ig_{j}})_{ig_{j}}.

Lemma 2.4. Let f ∈ 𝔥(V, G), and assume that the inverse image F of f consisting of terms in the first column is an element in η_{G}^{−1}(I). Then all inverse images of f lie in η_{G}^{−1}(I).

Proof. Since two elements F and F′ of η_{G}^{−1}(f) differ by an element in the kernel of η_{G} which in turn is contained in η_{G}^{−1}(I), the result follows. □

We want to prove our statement degree-wise by induction on the term order. The next result constitutes the induction start for every degree.

Proposition 2.5. Let F ∈ η_{G}^{−1}(𝔥(V, G)) have degree d. Let F be minimal with respect to the reverse lexicographic order. Assume that the ideals η_{G}^{−1}(I) and η_{G}^{−1}(𝔥(V, G)) coincide in degree less than d. Then F ∈ η_{G}^{−1}(I).

Proof. The unique minimal monomial in 𝔽[𝔽G ⊗ V] of degree d is x_{11}^{d}. If G is a p-group we assume without loss of generality that G fixes x_{1}, then x_{1} ∈ I and thus x_{11} ∈ η_{G}^{−1}(I).

Now, let G be an arbitrary group with p-Sylow subgroup P fixing x_{1}. Order the elements of G such that g_{2} ∈ P is a nontrivial element. Then η_{G}(x_{1g_{2}}) = g_{2}x_{1} = x_{1}. We have to consider two cases:

CASE: x_{11}^{d} ∉ η_{G}^{−1}(𝔥(V, G))

In this case we have

F = x_{11}^{d−1}(x_{11} − x_{1g_{2}}) ∈ Ker(η_{G}) ⊆ η_{G}^{−1}(𝔥(V, G)).

Its leading term is x_{11}^{d−1}x_{1g_{2}} and is thus the smallest leading term of an element of degree d in η_{G}^{−1}(𝔥(V, G)). Furthermore, η_{G}^{−1}(I) contains the kernel of η_{G} and hence it contains x_{11}^{d−1}(x_{11} − x_{1g_{2}}). Finally, note that F is the unique polynomial in η_{G}^{−1}(𝔥(V, G)) with this leading term.

CASE: x_{11}^{d} ∈ η_{G}^{−1}(𝔥(V, G))

Then η_{G}(x_{11}^{d}) = x_{1}^{d} ∈ 𝔥(V, G). Thus if d ≤ |G| it follows that x_{1}^{d} ∈ I by definition of I, and hence x_{11} ∈ η_{G}^{−1}(I). Otherwise we have

x_{1}^{d} = ∑_{i} α_{i}f_{i}

for some G-invariant polynomials f_{i}. If the degree of the α_{i}’s is positive, we have f_{i} ∈ I by assumption, and thus

x_{11}^{d} = ∑_{i} A_{i}F_{i} + K ∈ η_{G}^{−1}(I)

for η_{G}(F_{i}) = f_{i}, η_{G}(A_{i}) = α_{i}, and an element K in the kernel. Finally assume we have

x_{1}^{d} = f_{0} + ∑_{i} α_{i}f_{i}.

Then the relative transfer gives

Tr_{P}^{G}(x_{1}^{d}) = |G : P |f_{0} + ∑_{i} Tr_{P}^{G}(α_{i})f_{i}.

Since d > |G| we can apply Lemma 2.2 in [2] and obtain

Tr_{P}^{G}(x_{1}^{d}) ∈ I.

Hence

f_{0} = \frac{1}{|G : P |}(Tr_{P}^{G}(x_{1}^{d}) − ∑_{i} Tr_{P}^{G}(α_{i})f_{i}) ∈ I.

So, finally we get

x_{11}^{d} ∈ η_{G}^{−1}(x_{1}^{d}) = η_{G}^{−1}(f_{0} + ∑_{i} α_{i}f_{i}) ⊆ η_{G}^{−1}(I)

as desired. □

We need one more preparation:

Lemma 2.6. Let M, M′ ∈ 𝔽[𝔽G ⊗ V] be monomials of the same degree. Assume that M consists of elements in the first column, and assume that M > M′ in the reverse lexicographic order. Then for all g ∈ G we have that gM > gM′.

Proof. M is a monomial in the first column, say

M = x_{1,1}^{i_{1}} ⋯ x_{n,1}^{i_{n}}.

Thus

gM = x_{1,g}^{i_{1}} ⋯ x_{n,g}^{i_{n}}

is a monomial in the gth column.

We turn to the monomial M′. Since M′ is smaller it must look like

M′ = x_{n,1}^{i_{n}} … x_{j+1,1}^{i_{j+1}}x_{j,1}^{k_{j}}N

where N is a monomial avoiding the columns j through n and k_{j} < i_{j}. Thus gM′ looks like

gM′ = x_{n,g}^{i_{n}} … x_{j+1,g}^{i_{j+1}}x_{j,g}^{k_{j}}gN

where gN still avoids columns j through n. Thus gM′ < gM. □

The following result proves the Hilbert ideal conjecture for p-groups in characteristic p.

Theorem 2.7. Let P be a p-group, and assume that ρ(P) ⊆ GL(n, 𝔽) is in lower triangular form, i.e., gx_{i} ∈ span_{𝔽}{x_{1}, …, x_{n}}. Then

𝔥(V, P) = I.

Proof. By Lemma 2.2 it is enough to show that η_{G}^{−1}(𝔥(()V, G)) = η_{G}^{−1}(I). The inclusion ” ⊃ ” is valid by construction. To show the reverse inclusion we proceed by double induction on degree and term order.

Let F ∈ η_{P}^{−1}(𝔥(V, P)) be an element of degree d. If d ≤ |G| there is nothing to show. Thus we assume that the degree of F is strictly larger than the group order.

By Lemma 2.4 we can assume that F lives in the first column and by Lemma 2.3 we can assume that η_{P} (F) is invariant.

Since the induction start is proven in Proposition 2.5 we assume that for all elements of smaller degree or same degree and lower in term order we have shown that it is in η_{G}^{−1}(I). We write F as a sum of monomials

F = M_{0} + M_{1} + ⋯ + M_{k}

and without loss of generality we assume that M_{0} > M_{1} > ⋯ > M_{k}.

Let r = max{1, …, n} such that the variable x_{r1} appears in F.

CASE: M_{0} = x_{r1}^{d} We consider the top orbit Chern class of x_{r}

c_{top}(x_{r}) ∈ 𝔽[V]^{P}

which is a polynomial of degree, say, t with leading term x_{r}^{d}. Note that t cannot exceed the group order. We find an inverse image C ∈ η_{P}^{−1}(c_{top}(x_{r})) in the first column

C ∈ 𝔽[x_{11}, x_{21}, …, x_{r1}].

We note that this polynomial is by construction in η_{P}^{−1}(I). Then

F − C^{d−t}

lies in η_{P}^{−1}(𝔥(V, P)) and has lower leading term than F. Thus by induction

F − C^{d−t} ∈ η_{P}^{−1}(I)

and thus F ∈ η_{P}^{−1}(I) as desired.

CASE: Assume that M_{0} = x_{r1}^{i}N_{0}, where N_{0} has positive degree and x_{r1} does not divide N_{0}, M_{1}, …, M_{k}.

Since η_{P} (F) is invariant we have that gF − F ∈ Ker(η_{P}) ⊆ η_{P}^{−1}(I) for all g ∈ P. Since x_{r1} appears solely in M_{0}, we have that x_{r}|η_{P} (M_{0}) = m_{0} but it does not divide any of the other η_{P} (M_{i}) for i = 1, …, k. Since η_{P} (gF − F) = gη_{P} (F) − η_{P} (F) = 0 the terms with the same x_{r}-degree must cancel. Thus

η_{P} (gM_{0} − M_{0}) = g(x_{r})^{i}gn_{0} − x_{r}^{i}n_{0} = x_{r}^{i}(gn_{0} − n_{0}) + R,

where the remainder R has lower x_{r}-degree and η_{G}(N_{0}) = n_{0}. Thus gn_{0} − n_{0} = 0 and n_{0} is an invariant of strictly smaller degree than m_{0}. Thus N_{0} ∈ η_{P}^{−1}(I), therefore M_{0} ∈ η_{P}^{−1}(I). Hence

F − M_{0} ∈ η_{P}^{−1}(𝔥(V, P))

has strictly smaller term-order, and is by induction in η_{P}^{−1}(I). Thus F ∈ η_{P}^{−1}(I). CASE: x_{r1} divides M_{0}, …, M_{j} but none of the other terms of F Write

M_{0} = x_{r1}^{i_{1}}N_{0}, …, M_{j} = x_{r1}^{i_{j}}N_{j}

If i_{1} > max{i_{2}, …, i_{j}}, then the preceding argument goes through without change. Thus we assume that i_{1} = i_{2} = ⋯ = i_{l} for some 1 < l ≤ j. Since M_{0} > M_{1} > ⋯ > M_{l} there exists a largest index s, 1 ≤ s ≤ r − 1, such that the monomials differ. Hence we can write

M_{0} = x_{r1}^{i_{r}}x_{r−1,1}^{i_{r−1}} ⋯ x_{s+1,1}^{i_{s+1,1}}x_{s1}^{i_{s,1}}N_{0}, …, M_{l} = x_{r1}^{i_{r}}x_{r−1,1}^{i_{r−1}} ⋯ x_{s+1,1}^{i_{s+1,1}}x_{s1}^{j_{s,1}}N_{l}.

Similar to the above we obtain that

x_{s1}^{i_{s,1}}N_{0} + ⋯ + x_{s1}^{j_{s,1}}N_{l}

maps under η_{P} to an invariant. By construction it has lower degree than F and is thus in η_{P}^{−1}(I). Here we can apply the argument of the preceding case to find that N_{0} ∈ η_{P}^{−1}(I). □

Finally, we are prepared to prove the Hilbert number conjecture for all groups and representations.

Theorem 2.8. Let ρ : G ↪ GL(n, 𝔽) be a faithful representation of a finite group over an arbitrary field 𝔽. Then

𝔥(V, G) = I

and hence the Hilbert number is bounded above by the group order

β(𝔥(V, G)) = β(I) ≤ |G|.

Proof. By Lemma 2.2 it is enough to show that η_{G}^{−1}(𝔥(()V, G)) = η_{G}^{−1}(I). The inclusion ” ⊃ ” is valid by construction. To show the reverse inclusion we proceed as above by double induction on degree and term order.

Let F ∈ η_{G}^{−1}(𝔥(V, G)).If degree if F is at most group order then F ∈ η_{G}^{−1}(I) by construction. Thus we assume that deg(F) = d > |G|.

The polynomial of minimal leading term of degree d in η_{G}^{−1}(𝔥(V, G)) is contained in η_{G}^{−1}(I) by Proposition 2.5.

Thus assume that the leading term of F is not minimal, and all polynomials of degree less than d or of degree d with smaller leading term than F are in η_{G}^{−1}(I).

We denote the leading term of F by LT(F). We note that by Lemmata 2.3 and 2.4 we may assume that F is a polynomial in the first column mapping under η_{G} onto an invariant polynomial.

We note also that the leading term of gF is strictly larger than F for all g ≠ 1, because F lives in the first column, while gF lives in the gth column.

Since F maps to an invariant polynomial under η_{G} we have that gF − F ∈ Ker(η_{G}) ⊆ η_{G}^{−1}(I). Thus

LT(gF − F) = LT(gF) ∈ LT(Ker(η_{G})) ⊆ LT(η_{G}^{−1}(I)),

where LT(I) denotes the ideal of leading terms of the ideal I.

Hence there exists an element H ∈ η_{G}^{−1}(I) such that LT(H) = LT(gF). If H = gF we are done by Lemma 2.3.

Otherwise, consider gF − H ∈ η_{G}^{−1}(𝔥(V, G)) with LT(gF − H) < LT(gF).

If LT(gF − H) < LT(F), then gF − H ∈ η_{G}^{−1}(I) by induction on the term order, and thus gF ∈ η_{G}^{−1}(I). Hence F ∈ η_{G}^{−1}(I). Otherwise, we consider the polynomial (gF − F) − H which is an element in η_{G}^{−1}(I).

If LT(gF −F −H) > LT(F), we pick an element K_{1} ∈ η_{G}^{−1}(I) such that LT(K_{1}) = LT(gF − F − H). Then

LT(gF − F − H − K_{1}) < LT(gF − F − H)

and we can proceed inductively and find K_{1}, …, K_{l} ∈ η_{G}^{−1}(I) such that

LT(gF − F − H − K_{1} − ⋯ − K_{l}) ≤ LT(F).

Thus we may assume without loss of generality that LT(gF − F − H) ≤ LT(F). If there exists an element H such that LT(gF − F − H) = LT(F), we have

LT(gF − F − H − F) < LT(gF − F − H) = LT(F).

By construction gF − F − H − F ∈ η_{G}^{−1}(𝔥(V, G)) and thus by induction on the term order in η_{G}^{−1}(I). Since H as well as gF − F are in η_{G}^{−1}(I) we conclude that F ∈ η_{G}^{−1}(I).

Finally we have to take care of the case LT(gF − F − H) < LT(F) for all H ∈ η_{G}^{−1}(I) and all g ∈ G.

Recall that F is a polynomial in the first column and that η_{G}(F) = f is invariant. We write it as a sum of monomials

F = M_{0} + M_{1} + ⋯ + M_{k}

and assume without loss of generality that M_{0} > M_{1} > ⋯ > M_{k} in the term-order. Then

gF = gM_{0} + gM_{1} + ⋯ + gM_{k}

has leading term gM_{0} by Lemma 2.6. Since the leading term of gF − F − H is strictly smaller than the leading term of F we have

H = gM_{0} − M_{0} + ∑_{K_{i}<M_{0}} K_{i} ∈ η_{G}^{−1}(I),

where the K_{i}’s are monomials smaller than M_{0}.

We choose an element g ∈ G of maximal order s. We order the group elements such that

x_{i1} < x_{ig} < x_{ig^{2}} < ⋯ < x_{ig^{s−1}}.

We consider

g^{s−1}F = g^{s−1}M_{0} + ⋯ + g^{s−1}M_{k}

and note that

η_{G}(g^{s−1}F) = g^{s−1}η_{G}(F) = g^{s−1}f = f = η_{G}(F).

Thus there exists a polynomial H̃ ∈ η_{G}^{−1}(I) with the same leading term as g^{s−1}F. Moreover,

H̃ = g^{s−1}M_{0} − M_{0} + ∑_{K_{i}<M_{0}} K_{i},

because LT(g^{s−1}F − H̃ − F) < LT(F) by assumption. Thus

g^{s−1}H̃ = g^{s−2}M_{0} − g^{s−1}M_{0} + ∑_{K_{i}<M_{0}} g^{s−1}K_{i}

has leading term g^{s−1}M_{0}, the same leading term as g^{s−1}F. Hence

\begin{matrix}g^{s−1}F − (−g^{s−1}H) & − F = \\ g^{s−1}M_{1} + ⋯ + g^{s−1}M_{k} & + ∑_{K_{i}<M_{0}} g^{s−1}K_{i} + g^{s−2}M_{0} − M_{0} − ⋯ − M_{k}. \end{matrix}

By assumption this polynomial must be smaller than F. However, M_{0} < g^{s−2}M_{0} unless g^{s−2} = 1. Thus the element g must have order 2. Since g was an element of maximal order in G, it follows that G is an elementary abelian 2-group. Thus if the characteristic of the ground field is not two, we are in the nonmodular case, and the result follows from Noether’s bound. If the characteristic is two, then the result follows from the preceding Theorem 2.7. □

References

1. Harm Derksen and Gregor Kemper, Computational Invariant Theory, Encyclopaedia of Mathematical Sciences Vol. 130, Springer Verlag, Berlin 2002.

2. Peter Fleischmann, On Invariant Theory of Finite Groups, Centre de Recherches Mathématiques Proceedings and Lecture Notes Vol 35 (2004), 43-69.

3. Mara D. Neusel, Degree Bounds - An Invitation to Postmodern Invariant Theory -, Topology and its Applications 154 (2007), 792-814.

4. Mara D. Neusel, Degree Bounds and the Regular Representation, preprint, 2007.

5. Mara D. Neusel and Müfit Sezer, The Invariants of Modular Indecomposable Representations of ℤ_{p^{2}}, Mathematische Annalen, to appear.

6. Mara D. Neusel and Müfit Sezer, The Noether Map I, Forum Mathematicum, to appear.

7. Mara D. Neusel and Müfit Sezer, The Noether Map II, Proceedings of the AMS 135 (2007), 2347-2354.

8. Mara D. Neusel and Larry Smith, Invariant Theory of Finite Groups, Math. Surveys and Monographs Vol. 94, AMS, Providence RI 2002.

9. Müfit Sezer, A Note on the Hilbert Ideal of a Cyclic Group of Prime Order, Journal of Algebra, to appear.

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Department of Mathematics and Statistics, MS 1042, Texas Tech University, Lubbock, Texas 79409

  E-mail address: Mara.D.Neusel@ttu.edu