Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 40 (1999), No. 2, 479-493.

Gröbner Bases of Certain Determinantal Ideals

Mátyás Domokos^{∗}

Mathematical Institute of the Hungarian Academy of Sciences P.O.B. 127, 1364 Budapest, Hungary e-mail: domokos@math-inst.hu

Abstract. Let μ_{1} < ⋯ < μ_{k} be non-negative integers. We prove that the ideal of the locus of points (x_{1}, …, x_{n}) in ℂ^{n} for which the n × k matrix (x_{i}^{μ_{j}}) has rank smaller than k is generated by the determinants of the k × k minors of this matrix, if μ_{1} = 0 or 1. Moreover, these generators form a universal Gröbner basis. Special cases of our result apply for ideals arising in the study of chromatic numbers of graphs or identities of matrices, and ideals of truncations of hyperplane arrangements related to pseudo-reflection groups. MSC 1991: 13C40 (primary), 13P10, 14M12 (secondary) Keywords: determinantal ideal, Gröbner basis, degree, Hilbert polynomial, subspace arrangement, reflection group

1. Introduction

We work over an algebraically closed field F of characteristic zero. Let μ = (μ_{1}, …, μ_{k}) be a strictly increasing sequence of non-negative integers with k ≥ 2, that is, 0 ≤ μ_{1} < ⋯ < μ_{k}. Consider the map

ϕ_{n,μ} : 𝔸^{n} → 𝔸^{nk}
(x_{1}, …, x_{n}) ↦ (x_{i}^{μ_{j}})

∗Supported by OTKA no. F023436 and the Bolyai Research Fellowship.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1999 Heldermann Verlag

from the n-dimensional affine space to the nk-dimensional affine space, where n ≥ k ≥ 2, which maps a point (x_{1}, …, x_{n}) to the n × k matrix whose (i, j) entry is x_{i}^{μ_{j}}. We study

X_{n,μ} = {x ∈ 𝔸^{n}| rk(ϕ_{n,μ}(x)) ≤ k − 1},

the locus of points in 𝔸^{n} whose image under ϕ_{n,μ} has rank smaller than k. One can also think of X_{n,μ} as the set of points x ∈ 𝔸^{n} with the property that there exists a polynomial (depending on x) f(t) = a_{1}t^{μ_{1}} + ⋯ + a_{k}t^{μ_{k}} ∈ F [t] such that all the coordinates of x are roots of f.

The aim of the present paper is to give generators (a Gröbner basis) of the ideal I(X_{n,μ}) of polynomials in F [t_{1}, …, t_{n}] vanishing on X_{n,μ}. We put

δ_{μ}(t_{1}, …, t_{k}) = det((t_{i}^{μ_{j}})_{1≤i,j≤k}),

so δ_{μ} is the determinant of the k × k matrix whose (i, j) entry is t_{i}^{μ_{j}}. Note that the nonnegative integers λ_{i} = μ_{i} − i + 1 form a non-decreasing sequence λ = (λ_{1}, …, λ_{k}), and the Schur function s_{λ} is usually defined by the equality

δ_{μ}(t_{1}, …, t_{k}) = s_{λ}(t_{1}, …, t_{k}) ∏_{1≤i<j≤k}(t_{j} − t_{i})

(see for example [8]). Since the points x ∈ X_{n,μ} are characterized by the property that all the k × k minors of ϕ_{n,μ}(x) are degenerate, the polynomials δ_{μ}(t_{i_{1}}, …, t_{i_{k}}) with 1 ≤ i_{1} < ⋯ < i_{k} ≤ n define X_{n,μ} set-theoretically. We shall show that they generate the ideal I(X_{n,μ}), unless δ_{μ} has multiple factors. So our first question is that when the polynomial δ_{μ} has multiple factors. This is answered by the following elementary lemma.

Lemma 1.1. The polynomial δ_{μ}(t_{1}, …, t_{k}) has multiple factors in F [t_{1}, …, t_{k}] if and only if μ_{1} ≥ 2.

Proof. The equality

δ_{μ}(t_{1}, …, t_{k}) = (t_{1} ⋯ t_{k})^{μ_{1}} det((t_{i}^{μ_{j}−μ_{1}})_{1≤i,j≤k})

shows that if δ_{μ} has no multiple factors, then μ_{1} ≤ 1.

To prove the converse direction we may assume that μ_{1} = 0, since in this case none of t_{1}, …, t_{k} is a divisor of δ_{μ}. We apply induction on k. The case k = 2 is obvious, because then we have δ_{μ}(t_{1}, t_{2}) = t_{2}^{μ_{2}} − t_{1}^{μ_{2}}, and this polynomial has no multiple factors in characteristic zero. Assume now that k ≥ 3, and

δ_{μ}(t_{1}, …, t_{k}) = f^{2} · g

for some f, g ∈ F [t_{1}, …, t_{k}], where f is non-constant. Since δ_{μ} is homogeneous, f and g are also homogeneous. Expand f and g according to the powers of t_{k} :

f = ∑_{i=0}^{r} a_{i}(t_{1}, …, t_{k−1})t_{k}^{i} and g = ∑_{j=0}^{s} b_{j}(t_{1}, …, t_{k−1})t_{k}^{j},

where a_{r} and b_{s} are non-zero. On expanding the determinant of (t_{i}^{μ_{j}}) according to its last row, we get

δ_{μ} = (−1)^{k+1}(t_{1} ⋯ t_{k−1})^{μ_{2}}δ_{μ′}(t_{1}, …, t_{k−1}) + ⋯ + t_{k}^{μ_{k}}δ_{μ′′}(t_{1}, …, t_{k−1}),

where μ′ = (0, μ_{3} − μ_{2}, …, μ_{k} − μ_{2}) and μ′′ = (μ_{1}, …, μ_{k−1}). It follows that δ_{μ′′} = a_{r}^{2}b_{s} and (−1)^{k+1}(t_{1} ⋯ t_{k−1})^{μ_{2}}δ_{μ′} = a_{0}^{2}b_{0}, and by the induction hypothesis a_{r} must be a non-zero constant and a_{0}^{2} is a divisor of (t_{1} ⋯ t_{k−1})^{μ_{2}}.

On the other hand, we may assume that f is fixed up to sign by the symmetric group Sym(k) acting on F [t_{1}, …, t_{k}] via permutation of the variables. Indeed, take an irreducible factor q of f, then q^{2} divides δ_{μ}. Now let q = q_{1}, …, q_{m} be the pairwise different images of q under the action of Sym(k), up to scalar multiplication. Since π(δ_{μ}) = sign(π)δ_{μ} holds for any π ∈ Sym(k), we have that π(q)^{2} divides δ_{μ}. The polynomials q_{1}, …, q_{m} are pairwise non-associate irreducibles, hence q_{1}^{2} ⋯ q_{m}^{2} is a divisor of δ_{μ}. So f = q_{1} ⋯ q_{m} has the required properties.

Summarizing, the leading term of f as a polynomial of t_{k} is ct_{k}^{r}, where r ≥ 1 and c is a non-zero constant. The symmetry of f implies that

f = c(t_{1}^{r} + ⋯ + t_{k}^{r}) + other terms,

hence

a_{0}(t_{1}, …, t_{k−1}) = c(t_{1}^{r} + ⋯ + t_{k−1}^{r}) + other terms,

which contradicts the fact that a_{0}^{2} is a divisor of (t_{1} ⋯ t_{k−1})^{μ_{2}}. □

Now we state the main result of the paper.

Theorem 1.2. Let n ≥ k ≥ 2, and let μ = (μ_{1}, …, μ_{k}) be a strictly increasing sequence of non-negative integers with μ_{1} ≤ 1. Then the ideal of X_{n,μ} is generated by δ_{μ}(t_{i_{1}}, …, t_{i_{k}}) with 1 ≤ i_{1} < ⋯ < i_{k} ≤ n. Moreover, this generating set is a universal Gröbner basis of I(X_{n,μ}).

In order to explain the structure of the proof we introduce some terminology. Recall that for any homogeneous ideal I the factor algebra F [t_{1}, …, t_{n}]/I is naturally graded, and there exists a polynomial h_{I}(t) ∈ ℚ[t] called the Hilbert polynomial of I such that the dimension of the degree d homogeneous component of F [t_{1}, …, t_{n}]/I is h_{I}(d) for sufficiently large d. The leading term of h_{I}(t) is \frac{e(I)}{m!}t^{m}, where e(I) is a positive integer, called the degree of I, and m = dim(I) − 1. Obviously, for any homogeneous ideal I′ containing I we have dim(I′) ≤ dim(I), and in the case of equality also e(I′) ≤ e(I). We say that a homogeneous ideal I is critical, if for any homogeneous ideal I′ strictly containing I we have that dim(I′) < dim(I) or e(I′) < e(I) (see Lemma 3.1 for an algebraic characterization of critical homogeneous ideals). The strategy of the proof of Theorem 1.2 is based on the fact that the initial monomial ideal of I(X_{n,μ}) turns out to be critical. To verify the conjecture that a given critical ideal M generated by the initial monomials of certain polynomials vanishing on X_{n,μ} is indeed the initial ideal of I(X_{n,μ}), it is sufficient to show that the dimension and the degree of M is the same as for X_{n,μ}. In Section 2 we derive the necessary information on the geometry of X_{n,μ}. Section 3 is devoted to the combinatorial study of the ideals generated by the initial monomials of δ_{μ}(t_{i_{1}}, …, t_{i_{k}}), which we need to finish the proof of Theorem 1.2.

An interesting special case is μ = (0, 1, …, k), when X_{n,(0,1,...,k)} is the subspace arrangement consisting of points which have at most k different coordinates. The Kleitman-Lovász theorem [6] asserts that ∏_{1≤r<s≤k+1}(t_{j_{r}} − t_{j_{s}}) (with 1 ≤ j_{1} < ⋯ < j_{k+1} ≤ n) generate the ideal in question. The additional fact that the given generators form a universal Gröbner basis is due to de Loera [5]. The motivation of [6] comes from graph theory. An n-variable polynomial p_{G} is associated with any graph G having n nodes, and it is observed that the chromatic number of G is greater than k if and only if p_{G} is contained in I(X_{n,(0,1,...,k)}). So the result of [5] gives some information on graph colorings.

The description of I(X_{n,μ}) when μ = (0, 1, …, k) or μ = (0, 1, …, k − 2, k) has consequences also in the theory of polynomial identities and weak identities of matrices, as it is shown in [2].

In Section 4 we draw attention to other special cases of our result, which deal with subspace arrangements related to complex pseudo-reflection groups. More precisely, let G be a finite complex pseudo-reflection group, and consider the set of reflecting hyperplanes belonging to G. Take the union of k-dimensional subspaces which are intersections of reflecting hyperplanes. The vanishing ideal of this subspace arrangement is described by Theorem 1.2 for certain pseudo-reflection groups, including the Weyl groups of type A_{n}, B_{n}, and D_{n}.

We finish the introduction with some comments on the result. X_{n,μ} is the set of points in the n-dimensional affine space whose any k coordinates satisfy the alternating relation δ_{μ} = 0. Now let l(t_{1}, …, t_{k}) be any homogeneous alternating polynomial, that is,

l(t_{1}, …, t_{k}) = s(t_{1}, …, t_{k}) ∏_{1≤i<j≤k}(t_{i} − t_{j}),

where s(t_{1}, …, t_{k}) is a symmetric polynomial. Assume that l(t_{1}, …, t_{k}) has no multiple factors in F [t_{1}, …, t_{k}]. The following related problem is studied in [2]. We put

Z_{n,l} = {x ∈ 𝔸^{n}| x has at most k different coordinates and
l(x_{i_{1}}, …, x_{i_{k}}) = 0 for any 1 ≤ i_{1} < ⋯ < i_{k} ≤ n},

and let denote δ(t_{1}, …, t_{k+1}) = ∏_{1≤r<s≤k+1}(t_{s} − t_{r}). We show in [2] that

I(Z_{n,l}) = ⟨l(t_{i_{1}}, …, t_{i_{k}}), δ(t_{1}, t_{j_{1}}, …, t_{j_{k}})|1 ≤ i_{1} < ⋯ < i_{k} ≤ n, 2 ≤ j_{1} < ⋯ < j_{k} ≤ n⟩,

and this generating set is a Gröbner basis with respect to the lexicographic monomial order induced by t_{1} ≺ ⋯ ≺ t_{n}. The motivation to study I(Z_{n,l}) in [2] came from the theory of polynomial identities of matrices, therefore the condition “x has at most k different coordinates” was natural. However, one can also ask whether the ideal of

X_{n,l} = {x ∈ 𝔸^{n}| l(x_{i_{1}}, …, x_{i_{k}}) = 0 for any 1 ≤ i_{1} < ⋯ < i_{k} ≤ n}

is generated by the polynomials l(t_{i_{1}}, …, t_{i_{k}}) (with 1 ≤ i_{1} < ⋯ < i_{k} ≤ n). The answer to this question is no in general, as the following example shows, so it has some interest that we have affirmative answer in the determinantal case.

Let n = 3 and let l = (t_{1} − t_{2})(t_{1}^{3} + t_{2}^{3}). We claim that X_{3,l} does not contain a point with pairwise different coordinates. Indeed, assume in the contrary that (x_{1}, x_{2}, x_{3}) is such a point, its coordinates are necessarily non-zero. Then \frac{x_{2}}{x_{1}} , \frac{x_{3}}{x_{1}} , \frac{x_{3}}{x_{2}} must all be roots of the polynomial 1 + t^{3}. The roots of t^{3} + 1 are ρ, ρ^{5} and −1, where ρ is a primitive 6-th root of unity. But the ratio of any two elements of {ρ, ρ^{5}, −1} is not contained in this set, a contradiction. Therefore, the polynomial (t_{1} − t_{2})(t_{1} − t_{3})(t_{2} − t_{3}) lies in I(X_{3,l}), but it is of degree 3, hence is clearly not contained in ⟨l(t_{1}, t_{2}), l(t_{1}, t_{3}), l(t_{2}, t_{3})⟩.

On the other hand, ∏_{1≤r<s≤k+1}(t_{j_{r}} −t_{j_{s}}) is not always contained in I(X_{n,l}). For example, X_{3,t_{1}^{3}−t_{2}^{3}} contains the point (1, ρ, ρ^{2}), where ρ is a primitive third root of unity.

Another possibility to generalize Theorem 1.2 is to consider X_{n,μ,m}, the locus of points in 𝔸^{n} whose image under ϕ_{n,μ} has rank smaller than m. (When m = k, the algebraic set X_{n,μ,k} coincides with X_{n,μ}.) Though the m × m minors of ϕ_{n,μ}(t_{1}, …, t_{n}) define X_{n,μ,m} settheoretically, they do not generate its ideal in general. For example, take μ = (0, 1, 3, 6), m = 3, and n = 3. The 3 × 3 minors of ϕ_{3,μ}(t_{1}, t_{2}, t_{3}) have no multiple factors in this case, however, they do not generate a radical ideal. Indeed, we claim that (t_{1}t_{2} + t_{1}t_{3} + t_{2}t_{3})(t_{1} − t_{2})(t_{1} − t_{3})(t_{2} − t_{3}) vanishes on X_{3,(0,1,3,6),3}. Since this polynomial is not divisible by (t_{1}+t_{2}+t_{3})(t_{1}−t_{2})(t_{1}−t_{3})(t_{2}−t_{3}), and all the 3×3 minors not from the first three columns are of degree greater than 5, this polynomial is not contained in the ideal generated by the 3 × 3 minors of ϕ_{3,μ}(t_{1}, t_{2}, t_{3}). To show the claim take a point x on X_{3,(0,1,3,6),3} with three different coordinates. Then the first two columns of ϕ_{3,(0,1,3,6)}(x_{1}, x_{2}, x_{3}) are linearly independent, so their span contains the third and fourth columns. Therefore there are a, b, c, d ∈ ℂ such that x_{i}^{3} = ax_{i} + b, and x_{i}^{6} = cx_{i} + d for i = 1, 2, 3, hence (ax_{i} + b)^{2} = cx_{i} + d, implying that a = 0. So x_{1}, x_{2}, x_{3} are the third roots of b, and the claim follows. Similarly, the 3 × 3 minors of ϕ_{n,(0,1,3,6)}(t_{1}, …, t_{n}) do not generate a radical ideal for any n ≥ 3.

2. On the geometry of X_{n,μ}

In this section we analyze a bit the geometry of X_{n,μ}. We assume here that the base field is F = ℂ, the field of complex numbers. We pass to the (n − 1)-dimensional projective space ℙ^{n−1}, whose points are the lines in 𝔸^{n} containing the origin. A point of ℙ^{n−1} is usually written as (x_{1} : ⋯: x_{n}), by which we mean the line spanned by the non-zero (x_{1}, …, x_{n}) ∈ 𝔸^{n}. The coordinate ring F [t_{1}, …, t_{n}] of 𝔸^{n} becomes the homogeneous coordinate ring of ℙ^{n−1}. Obviously, X_{n,μ} is a cone over the origin, hence the one-dimensional subspaces contained in X_{n,μ} form a projective algebraic subset of ℙ^{n−1}, we denote it by Y_{n,μ}. The ideal I(Y_{n,μ}) of polynomials in F [t_{1}, …, t_{n}] vanishing on Y_{n,μ} is the same as I(X_{n,μ}).

Throughout this section we use the following notation:

π : ℙ^{n−1} \ {p} → ℙ^{n−2}
(x_{1} : ⋯: x_{n}) ↦ (x_{1} : ⋯: x_{n−1})

is the projection from p = (0 : ⋯: 0 : 1) ∈ ℙ^{n−1}.

First we show that Y_{n,μ} is unmixed of dimension k−2. For the proof we need to introduce a slightly more general class of varieties. As earlier, let n, k ≥ 2 be integers, and μ = (μ_{1}, …, μ_{k}) with 0 ≤ μ_{1} < ⋯ < μ_{k}. For any positive integer m less than or equal to n + 1 and k we put

Y_{n,μ,m} = {(x_{1} : ⋯: x_{n})| rk((x_{i}^{μ_{j}})_{i=1,...,n}^{j=1,...,k}) ≤ m − 1}.

Note that in the special case m = k we get Y_{n,μ,k} = Y_{n,μ}.

Proposition 2.1. Any irreducible component of Y_{n,μ} is of dimension k − 2.

Proof. In the special case m = 2 it is easy to see that Y_{n,μ,2} is a finite subset of ℙ^{n−1}. When n = m − 1, then obviously Y_{m−1,μ,m} = ℙ^{m−2}.

Now assume that n ≥ m ≥ 3, and by induction on m and n we show that dim(Y_{n.μ,m}) ≤ m − 2. It is clear that π(Y_{n,μ,m} \ {p}) is contained in Y_{n−1,μ,m}. By the induction hypothesis dim(Y_{n−1,μ,m−1}) ≤ m−3. Since adding one row to a matrix increases its rank at most by one, the whole π^{−1}(Y_{n−1,μ,m−1}) ∪ {p} belongs to Y_{n,μ,m}, and all of its components have dimension bounded by (m − 3) + 1 = m − 2. Now take an irreducible component Y of Y_{n,μ,m} such that π(Y) is not contained in Y_{n−1,μ,m−1}. Denote by Z the Zariski closure of π(Y) in Y_{n−1,μ,m}. Then Z is an irreducible projective variety, and Z \ Y_{n−1,μ,m−1} is a Zariski dense open subset in it. Consider the restriction

res_{Y} π : Y \ {p} → Z

of the regular map π. For any z ∈ Z \Y_{n−1,μ,m−1} there exists 1 ≤ i_{1} < ⋯ < i_{m−1} ≤ n−1 and 1 ≤ j_{1} < ⋯ < j_{m} ≤ k such that some (m−1)×(m−1) minor of the (m−1)×m matrix (z_{i_{r}}^{j_{s}}) is non-degenerate. Then if (z_{1} : ⋯: z_{n−1} : z_{n}) is contained in (res_{Y} π)^{−1}(z), then z_{n} must be a root of the non-zero polynomial δ_{(μ_{j_{1}},...,μ_{jm})}(z_{i_{1}}, …, z_{i_{m−1}}, t) ∈ ℂ[t]. So Z contains a Zariski dense open subset over which res_{Y} π has finite fibers, implying that dim(Y) = dim(Z) (see for example [9, 3.13]). By induction on n we know that m − 2 ≥ dim(Y_{n−1,μ,m}) ≥ dim(Z).

We have shown that any irreducible component of Y_{n,μ,m} has dimension not exceeding m − 2. On the other hand, a result from [7] shows that any irreducible component of Y_{n,μ} has dimension not smaller than k − 2 (see also [3, Exercise 10.9]). □

Remark. Y_{n,μ,m} need not be unmixed in general. For example, Y_{3,(0,1,3,4),3} is the union of Y_{3,(0,1,2)} (which is the union of the three lines V (t_{i} − t_{j}), 1 ≤ i < j ≤ 3) and the two points {(1 : ρ : ρ^{2}), (1 : ρ^{2} : ρ)}, where ρ is a primitive third root of unity.

It is clear that π maps Y_{n,μ} \ {p} into Y_{n−1,μ}. Denote by

ψ = res_{Y_{n,μ}} π : Y_{n,μ} \ {p} → Y_{n−1,μ}

the restriction of π to Y_{n,μ}.

Proposition 2.2. Assume that n > k ≥ 3 and μ_{1} ≤ 1. Then there exists a Zariski dense open subset U of Y_{n−1,μ} such that for any y ∈ U the fiber ψ^{−1}(y) consists of μ_{k} different points.

Proof. We put K = F [t_{1}, …, t_{k−1}]. Consider f(t) = δ_{μ}(t_{1}, …, t_{k−1}, t) as a polynomial of t over the ring K. It is of degree μ_{k}, and its leading coefficient is δ_{μ′}(t_{1}, …, t_{k−1}), where μ′ is the sequence obtained by removing μ_{k} from μ. Consider the discriminant D_{μ}(t_{1}, …, t_{k−1}) of f(t), so D_{μ} is an element of K. We claim that D_{μ} is non-zero. Indeed, suppose in the contrary that D_{μ} = 0. It means that f(t) has a multiple root in the algebraic closure of the quotient field L of K. Therefore f(t) has multiple factors in L(t), and it follows easily that f(t) has multiple factors in K[t], contradicting to Lemma 1.1.

Take an irreducible component Y of Y_{n−1,μ}, and assume that for some 1 ≤ i_{1} < ⋯ < i_{k−1} ≤ n − 1, the polynomial δ_{μ′}(t_{i_{1}}, …, t_{i_{k−1}}) does not vanish identically on Y. Then the image of Y under the projection

τ : Y \ V (t_{i_{1}}, …, t_{i_{k−1}}) → ℙ^{k−2}
(x_{1} : ⋯: x_{n−1}) ↦ (x_{i_{1}} : ⋯: x_{i_{k−1}})

is not contained in V (δ_{μ′}). Since all the coordinates of a point in the fiber of τ over any (x_{1} : ⋯: x_{k−1}) contained in the set τ(Y) \ V (δ_{μ′}) must be roots of the non-zero polynomial δ_{μ}(x_{i_{1}}, …, x_{i_{k−1}}, t), the map τ : Y \ V (t_{i_{1}}, …, t_{i_{k−1}}) → \overline{τ(Y)} has finite fibers over a Zariski dense open subset of the irreducible variety \overline{τ(Y)}. It follows that dim(\overline{τ(Y)}) = dim(Y) = k−2, so τ is a dominating morphism into ℙ^{k−2}. Hence by the previous paragraph D_{μ}(t_{i_{1}}, …, t_{i_{k−1}}) is not identically zero on Y.

Now remove from Y_{n−1,μ} the (k − 3)-dimensional algebraic subset Y_{n−1,μ′}, and from all irreducible components Y of Y_{n−1,μ} remove the proper subvarieties Y ∩ V (δ_{μ′}(t_{i_{1}}, …, t_{i_{k−1}})) and Y ∩ V (D_{μ}(t_{i_{1}}, …, t_{i_{k−1}})) for all 1 ≤ i_{1} < ⋯ < i_{k−1} ≤ n − 1 such that δ_{μ′}(t_{i_{1}}, …, t_{i_{k−1}}) is not identically zero on Y. In this way we get a Zariski dense open subset U of Y_{n−1,μ}, and we claim that this U satisfies the conditions of the proposition.

Indeed, take any y = (x_{1} : ⋯: x_{n−1}) ∈ U. It is not contained in Y_{n−1,μ′}, hence there exist some 1 ≤ j_{1} < ⋯ < j_{k−1} ≤ n − 1 with δ_{μ′}(x_{j_{1}}, …, x_{j_{k−1}}) ≠ 0. Hence the j_{1}, …, j_{k−1} rows of ϕ_{n−1,μ}(y) are linearly independent, and all the other rows of ϕ_{n−1,μ}(y) are contained in their span. So x = (y : x_{n}) is contained in Y_{n,μ} if and only if the last row of ϕ_{n,μ}(x) is contained in the span of its j_{1}, …, j_{k−1} rows, that is, if x_{n} is a root of δ_{μ}(x_{j_{1}}, …, x_{j_{k−1}}, t). But the discriminant of this degree μ_{k} polynomial is non-zero by the construction of U, hence it has μ_{k} different roots, which yield the μ_{k} points in the fiber of ψ over y. □

Let Y be an irreducible projective subvariety of ℙ^{n} of dimension r, let x be a point in ℙ^{n}, and let L be an (n − r)-dimensional linear subspace of ℙ^{n} intersecting Y in finitely many points, one of them being y. We refer to [9] for the definitions of the following basic notions of algebraic geometry: the degree deg(Y) of Y, the multiplicity mult_{x}(Y) of the point x on Y, and the intersection multiplicity i(y; Y ∩ L) of Y and L at the point y. If Y is not necessarily irreducible, but unmixed, that is, its irreducible components Y_{1}, …, Y_{m} all have dimension r, then one can define deg(Y) = ∑_{j=1}^{m} deg(Y_{j}), mult_{x}(Y) = ∑_{j=1}^{m} mult_{x}(Y_{j}), i(y; Y ∩ L) = ∑_{j=1}^{m} i(y; Y_{j} ∩ L). Recall that dim(Y) = dim(I(Y)) − 1 and deg(Y) = e(I(Y)). These definitions apply for Y_{n,μ}, and we have the following recursive formula for its degree.

Proposition 2.3. Assume that n > k ≥ 3 and μ_{1} ≤ 1. Then we have the formula

deg(Y_{n,μ}) = mult_{p}(Y_{n,μ}) + μ_{k} deg(Y_{n−1,μ}).

Proof. Let M be an (n − k)-dimensional linear subvariety of ℙ^{n−2}, and denote by L = π^{−1}(M) ∪ {p} its preimage under the projection π, so L is an (n − k + 1)-dimensional linear subspace of ℙ^{n−1} containing p. It is well-known (see for example [9, Chapter 5]), that if M is sufficiently general, then the following conditions hold:

(1) M intersects Y_{n−1,μ} in deg(Y_{n−1,μ}) points, and all of them are contained in the Zariski dense open subset U taken from Proposition 2.2;

(2) i(p; Y_{n,μ} ∩ L) = mult_{p}(Y_{n,μ});

(3) L intersects Y_{n,μ} transversally in finitely many points besides p, that is, i(x; Y_{n,μ}∩L) = 1 for any x ≠ p from the finite set L ∩ Y_{n,μ}.

Assume that M and L satisfy the above conditions. Then by [9, 5.3] and the choice of L we have

deg(Y_{n,μ}) = ∑_{x∈Y_{n,μ}∩L} i(x; Y_{n,μ} ∩ L) = i(p; Y_{n,μ} ∩ L) + ∑_{x∈Y_{n,μ}∩L\{p}} 1
= mult_{p}(Y_{n,μ}) + |Y_{n,μ} ∩ L \ {p}|.

Clearly, we have Y_{n,μ} ∩ L \ {p} = ψ^{−1}(M ∩ Y_{n−1,μ}), and by Proposition 2.2 and the choice of M this set contains μ_{k} deg(Y_{n−1,μ}) points. □

Now we shall determine the tangent cone of Y_{n,μ} at p. Consider the affine space 𝔸^{n−1} = ℙ^{n−1} \ V (t_{n}) with affine coordinates (\frac{t_{1}}{t_{n}} , …, \frac{t_{n−1}}{t_{n}}) = (u_{1}, …, u_{n−1}). The point p corresponds to the origin, and denote by X the affine part Y_{n,μ} ∩ 𝔸^{n−1} of Y_{n,μ}. Any non-zero polynomial f ∈ F [u_{1}, …, u_{n−1}] can be decomposed as a sum f = f_{s} + f_{s+1} + ⋯ + f_{m}, where f_{i} is homogeneous of degree i, and f_{s} is non-zero. The homogeneous component f_{s} is called the leading term of f at 0, and is denoted by lead_{0}(f). Recall that the tangent cone T C_{0}X at 0 is the common zero locus of the leading terms of the elements in the ideal of X, so T C_{0}X = V (lead_{0}(f)|f ∈ I(X)) (see for example [9, 5.10]). The multiplicity mult_{0}(X) = mult_{p}(Y_{n,μ}) equals the degree of ⟨lead_{0}(f)|f ∈ I(X)⟩.

Proposition 2.4. The tangent cone of X at 0 is the algebraic set X_{n−1,μ′}, where μ′ is the sequence (μ_{1}, …, μ_{k−1}).

Proof. Let μ̂_{i} denote the sequence of k − 1 increasing integers obtained by removing the i-th member of μ, for example, μ̂_{k} = μ′. For any 1 ≤ i_{1} < ⋯ < i_{k−1} ≤ n − 1 the polynomial δ_{μ}(t_{i_{1}}, …, t_{i_{k−1}}, t_{n}) is contained in I(Y_{n,μ}), and on passing to 𝔸^{n−1} = ℙ^{n−1} \V (t_{n}) it corresponds to

δμ̂_{k}(u_{i_{1}}, …, u_{i_{k−1}}) − δμ̂_{k−1}(u_{i_{1}}, …, u_{i_{k−1}}) ± ⋯ + (−1)^{k+1}δμ̂_{1}(u_{i_{1}}, …, u_{i_{k−1}}),

as one can see from the expansion of the determinant of (t_{i}^{μ_{j}})_{1≤i,j≤k} according to its last row. The leading terms δ_{μ′}(u_{i_{1}}, …, u_{i_{k−1}}) (with 1 ≤ i_{1} < ⋯ < i_{k−1} ≤ n − 1) are contained in I(T C_{0}X), and their common zero locus is X_{n−1,μ′} by definition, implying that X_{n−1,μ′} contains T C_{0}X.

Now we turn to the proof of the reverse inclusion. Let x = (x_{1}, …, x_{n−1}) ∈ X_{n−1,μ′} be a non-zero point. We claim that the line {λx|λ ∈ ℂ} belongs to T C_{0}X. We use the following geometric characterization of the tangent cone [9, 5.7 b]: the projectivized tangent cone ℙT C_{0}X is the set of limiting positions of lines \overline{0q} as q ∈ X \ \{0\} approaches 0.

Clearly, some (z_{1}, …, z_{n−1}) belongs to X if and only if (z_{1}, …, z_{n−1}, 1) belongs to X_{n,μ}. If the rank of the matrix (x_{i}^{μ_{j}})_{i=1,...,n−1}^{j=1,...,k} is at most k − 2, then the whole line \overline{0x} lies on X, so it belongs to T C_{0}X.

From now on we suppose that rk((x_{i}^{μ_{j}})_{i=1,...,n−1}^{j=1,...,k}) = k − 1, say the first k − 1 rows of this matrix are linearly independent. So in the (k−1)×k matrix (x_{i}^{μ_{j}})_{i=1,...,k−1}^{j=1,...,k} the (k−1)×(k−1) minor obtained by removing the last column is degenerate, but there exists some 1 ≤ m ≤ k − 1 such that the minor obtained by removing the m-th column is non-degenerate.

For any y, z ∈ 𝔸^{n−1} let d(y, z) denote the usual Euclidean distance of y and z.

Take a y = (y_{1}, …, y_{n−1}) ∈ X_{n−1,μ}. If y is sufficiently close to x, then the minor obtained by removing the m-th column of (y_{i}^{μ_{j}})_{i=1,...,k−1}^{j=1,...,k} is non-degenerate. Some point λy on \overline{0y} belongs to X if and only if (1, …, 1) is contained in the row space of the matrix ((λy_{i})^{μ_{j}})_{i=1,...,n−1}^{j=1,...,k}, that is, if λ is a root of the polynomial δ_{μ}(ty_{1}, …, ty_{k−1}, 1) ∈ ℂ[t]. We have the equalities

δ_{μ}(ty_{1}, …, ty_{k−1}, 1) = t^{μ_{1}+···+μ_{k−1}}(δμ̂_{k}(y_{1}, …, y_{k−1}) − t^{μ_{k}−μ_{k−1}}δμ̂_{k−1}(y_{1}, …, y_{k−1}) ± ⋯ + (−1)^{k+1}t^{μ_{k}−μ_{1}}δμ̂_{1}(y_{1}, …, y_{k−1})) = t^{μ_{1}+···+μ_{k−1}}g(y_{1}, …, y_{k−1})(t).

We may choose y on X_{n−1,μ} arbitrarily close to x satisfying also δ_{μ′}(y_{1}, …, y_{k−1}) ≠ 0 (i.e. the constant term of g(y_{1}, …, y_{k−1})(t) is non-zero). Indeed, let Z be an irreducible component of X_{n−1,μ} containing x. The polynomial δμ̂_{m}(u_{1}, …, u_{k−1}) is not identically zero on Z (since it does not vanish in x), hence the projection

τ : 𝔸^{n−1} → 𝔸^{k−1}
(z_{1}, …, z_{n−1}) ↦ (z_{1}, …, z_{k−1})

has finite fibers over the Zariski dense open subset \overline{τ(Z)} \ V (δμ̂_{m}), hence by Proposition 2.1 dim(\overline{τ(Z)}) = dim(Z) = k − 1, implying that res_{Z} τ : Z → 𝔸^{k−1} is a dominating morphism, so δ_{μ′}(u_{1}, …, u_{k−1}) is not identically zero on Z (we have used already this argument in the proof of Proposition 2.2). The proper algebraic subset Z ∩ V (δ_{μ′}(u_{1}, …, u_{k−1})) of Z does not contain a neighborhood (in the Euclidean topology) of x, implying our claim.

Zero is a root of the non-constant polynomial g(x_{1}, …, x_{k−1})(t) ∈ ℂ[t]. Since the coefficients of the polynomial g(y_{1}, …, y_{k−1})(t) depend continuously on y, if y is close enough to x, then the polynomial has a root close to zero. More precisely, for any natural n there exists y^{(n)} on X_{n−1,μ} satisfying the following conditions:

(1) d(x, y^{(n)}) < \frac{1}{n};

(2) δμ̂_{m}(y_{1}, …, y_{k−1}) ≠ 0;

(3) δ_{μ′}(y_{1}, …, y_{k−1}) ≠ 0;

(4) g(y_{1}^{(n)}, …, y_{k−1}^{(n)})(t) has a non-zero root λ_{n} with d(0, λ_{n}y^{(n)}) < \frac{1}{n}.

These conditions imply that z^{(n)} = λ_{n}y^{(n)} lie on X \ {0}, they tend to 0 as n goes to infinity, and the limit of the secant lines \overline{0z^{(n)}} is the line \overline{0x}. □

3. Monomial ideals

A summary about Gröbner bases can be found for example in [3, Chapter 15]. To speak of Gröbner bases one has to fix an admissible order ≺ of the monomials in F [t_{1}, …, t_{n}]. For any f ∈ F [t_{1}, …, t_{n}] denote by In_{≺}(f) the initial monomial of f, that is, the monomial of f with non-zero coefficient which is maximal with respect to ≺. The initial ideal of an ideal I is

In_{≺}(I) = ⟨In_{≺}(f)|f ∈ I⟩,

the ideal generated by the initial monomials of the elements of I. The polynomials f_{1},...,f_{m} ∈ I form a Gröbner basis of I with respect to ≺ if In_{≺}(I) is generated by In_{≺}(f_{1}), …, In_{≺}(f_{m}). In this case I is generated by f_{1}, …, f_{m}. Any ideal has a finite Gröbner basis. We shall use the fact that the residue classes of monomials not contained in In_{≺}(I) form an F -linear basis of the factor ring F [t_{1}, …, t_{n}]/I. In particular, the Hilbert polynomial of a homogeneous ideal I coincides with the Hilbert polynomial of its initial ideal. A subset of I is called a universal Gröbner basis if it is a Gröbner basis with respect to any admissible monomial order.

Critical homogeneous ideals (the property ‘critical’ was defined in the Introduction after Theorem 1.2) have the following algebraic characterization.

Lemma 3.1. A homogeneous ideal is critical if and only if all of its associated primes have the same dimension.

Proof. Let I = Q_{1} ∩ … ∩ Q_{q} be an irredundant primary decomposition of a homogeneous ideal. Then \sqrt{Q}_{1}, …, \sqrt{Q}_{q} are the associated primes of I. We may assume that the first m of them have maximal dimension. It is well-known that e(I) = e(Q_{1}) + ⋯ + e(Q_{m}).

If m < q, then Q_{1} ∩ … ∩ Q_{m} is a homogeneous ideal that strictly contains I and has the same degree as I, so I is not critical.

Conversely, assume that m = q. Let f be any homogeneous polynomial not contained in I. Then f is not contained in some primary component of I, say f ∉ Q_{1}. By assumption we have dim(\sqrt{Q}_{1}) = dim(I) = k. Therefore we can find k variables, say t_{1}, …, t_{k}, whose images are algebraically independent in the domain F [t_{1}, …, t_{n}]/\sqrt{Q}_{1}. Since Q_{1} is primary, it follows that f · h(t_{1}, …, t_{k}) ∉ Q_{1} ⊇ I for any polynomial h. Hence for sufficiently large d the degree d homogeneous component of ⟨f, I⟩/I has dimension at least \binom{d−deg(f)+k−1}{k−1}. This shows that the coefficient of t^{k−1} in h_{⟨f,I⟩}(t) is strictly smaller than in h_{I}(t) for any homogeneous f ∉ I, that is, I is critical. □

Now we turn to the investigation of our concrete ideal.

Proposition 3.2. (i) If D = {δ_{μ}(t_{i_{1}}, …, t_{i_{k}})|1 ≤ i_{1} < ⋯ < i_{k} ≤ n} is a Gröbner basis of ⟨D⟩ for any admissible monomial order satisfying t_{1} ≺ ⋯ ≺ t_{n}, then D is a universal Gröbner basis of ⟨D⟩.

(ii) Let ≺ be an admissible monomial order with t_{1} ≺ ⋯ ≺ t_{k}. Then the initial monomial of δ_{μ}(t_{1}, …, t_{k}) = t_{1}^{μ_{1}} ⋯ t_{k}^{μ_{k}}.

Proof. (i) See the proof of [5, Lemma 2.1]. The key fact is that D is stabilized by any permutation of the variables. On the other hand, any admissible monomial order can be obtained from an order with t_{1} ≺ ⋯ ≺ t_{n} using a permutation of the variables.

(ii) Any monomial of δ_{μ} is of the form t_{π(1)}^{μ_{1}} ⋯ t_{π(k)}^{μ_{k}} for some π ∈ Sym(k), so the statement is obvious. □

Thus we may restrict to monomial orders with t_{1} ≺ ⋯ ≺ t_{n}, and then the ideal generated by the initial monomials of the elements in D is

M_{n,μ} = ⟨t_{i_{1}}^{μ_{1}} ⋯ t_{i_{k}}^{μ_{k}}|1 ≤ i_{1} < ⋯ < i_{k} ≤ n⟩.

For any 1 ≤ i_{1} < ⋯ < i_{n−k+1} ≤ n we put

Q_{n,μ}(i_{1}, …, i_{n−k+1}) = ⟨t_{i_{1}}^{μ_{i_{1}}}, …, t_{i_{s}}^{μ_{is−s+1}}, …, t_{i_{n−k+1}}^{μ_{i_{n−k+1}−n+k}}⟩.

Proposition 3.3. The irredundant primary decomposition of M_{n,μ} is

M_{n,μ} = ⋂_{1≤i_{1}<···<i_{n−k+1}≤n} Q_{n,μ}(i_{1}, …, i_{n−k+1})

(if μ_{1} = 0, then the intersection runs over 2 ≤ i_{1} < ⋯ < i_{n−k+1} ≤ n).

Proof. Clearly, Q_{n,μ}(i_{1}, …, i_{n−k+1}) is a primary ideal with radical ⟨t_{i_{1}}, …, t_{i_{n−k+1}}⟩.

⊆: Consider a monomial w = t_{j_{1}}^{μ_{1}} ⋯ t_{j_{k}}^{μ_{k}} ∈ M_{n,μ}. For any 1 ≤ i_{1} < ⋯ < i_{n−k+1} ≤ n the intersection {j_{1}, …, j_{k}} ∩ {i_{1}, …, i_{n−k+1}} is non-empty, let j_{r} = i_{s} be the element of the intersection with r, s maximal. We have the inequality i_{s} ≤ n−(k−r)−(n−k+1−s) = r+s−1, implying that w ∈ ⟨t_{j_{r}}^{μ_{r}}⟩ ⊆ ⟨t_{i_{s}}^{μ_{is−s+1}}⟩ ⊆ Q_{n,μ}(i_{1}, …, i_{n−k+1}).

⊇: Let w = t_{1}^{α_{1}} ⋯ t_{n}^{α_{n}} be a monomial contained in the intersection on the right-hand side. We define recursively j_{k} > ⋯ > j_{1} as follows. Since w ∈ Q_{n,μ}(k, k + 1, …, n) = ⟨t_{k}^{μ_{k}}, t_{k+1}^{μ_{k}}, …, t_{n}^{μ_{k}}⟩, there exists some j with α_{j} ≥ μ_{k} and k ≤ j ≤ n. We put j_{k} for the maximal such j. Assume that we have already defined j_{k} > ⋯ > j_{k−r+1} ≥ k − r + 1 (1 ≤ r < k). Then w is contained in Q_{n,μ}(i_{1}, …, i_{n−k+1}), where {i_{1}, …, i_{n−k+1}} ∪ {j_{k}, j_{k−1}, …, j_{k−r+1}} = {k − r, k − r + 1, …, n}. Therefore there exists some j, k − r ≤ j < j_{k−r+1}, with α_{j} ≥ μ_{k−r}, and we put j_{k−r} for the maximal such j. Now w is divisible by t_{j_{1}}^{μ_{1}} ⋯ t_{j_{k}}^{μ_{k}} by construction, hence w ∈ M_{n,μ}. □

Corollary 3.4. For any n ≥ k ≥ 2 the monomial ideal M_{n,μ} is (k − 1)-dimensional, critical, and its degree equals to the (n + 1 − k)-th complete symmetric function of μ_{1}, …, μ_{k}. In particular, we have the recursion e(M_{n,μ}) = e(M_{n−1,μ′}) + μ_{k}e(M_{n−1,μ}), where n > k ≥ 3 and μ′ = (μ_{1}, …, μ_{k−1}).

Proof. By Proposition 3.3 any associated prime of M_{n,μ} is of the form ⟨t_{i_{1}}, …, t_{i_{n−k+1}}⟩, so (k − 1)-dimensional, and hence by Lemma 3.1 M_{n,μ} is critical. The primary decomposition also shows that

e(M_{n,μ}) = ∑ deg(Q_{n,μ}(i_{1}, …, i_{n−k+1})) = ∑_{1≤i_{1}<···<i_{n−k+1}≤n} μ_{i_{1}}μ_{i_{2}−1} ⋯ μ_{i_{n−k+1}−n+k}. □

Proof of Theorem 1.2. Since the polynomial δ_{μ} has rational coefficients, then if the theorem holds over ℂ, it holds over any algebraically closed field of characteristic zero. Hence we may assume that F = ℂ, and we can use the results of Section 2.

Let ≺ be an admissible monomial order with t_{1} ≺ ⋯ ≺ t_{n}. Since the polynomials δ_{μ}(t_{i_{1}}, …, t_{i_{k}}) vanish on Y_{n,μ}, we have In_{≺}(I(Y_{n,μ})) ⊇ M_{n,μ}, and our aim is to show that equality holds here. By Corollary 3.4 M_{n,μ} is critical, therefore it is sufficient to show that dim(In_{≺}(I(Y_{n,μ}))) = dim(M_{n,μ}) and e(In_{≺}(I(Y_{n,μ}))) = e(M_{n,μ}). The Hilbert polynomials of I(Y_{n,μ}) and In_{≺}(I(Y_{n,μ})) coincide, hence we have dim(In_{≺}(I(Y_{n,μ}))) = dim(Y_{n,μ})+1 = k−1 = dim(M_{n,μ}) and e(In_{≺}(I(Y_{n,μ}))) = deg(Y_{n,μ}).

By double induction on k and n we prove that e(M_{n,μ}) = deg(Y_{n,μ}).

Consider first the case n = k ≥ 2. Then e(M_{k,μ}) = μ_{1} + ⋯ + μ_{k} by Corollary 3.4. The algebraic set Y_{k,μ} is a hypersurface defined by δ_{μ}(t_{1}, …, t_{k}), and since this polynomial has no multiple factors by Lemma 1.1, we have deg(Y_{k,μ}) = deg(δ_{μ}) = μ_{1} + ⋯ + μ_{k}.

Assume next that n ≥ k = 2. If μ_{1} = 0, then Y_{n,(0,μ_{2})} consists of the points (1 : x_{2} : ⋯: x_{n}) whose any coordinate is a μ_{2}-th root of unity, and there are μ_{2}^{n−1} such points. If μ_{1} = 1, then Y_{n,(1,μ_{2})} consists of the points whose any coordinate is either zero or a (μ_{2} − 1)-th root of unity, and there are \frac{μ_{2}^{n}−1}{μ_{2}−1} such points. In both cases we have deg(Y_{n,μ}) = \frac{μ_{2}^{n}−μ_{1}^{n}}{μ_{2}−μ_{1}} = e(M_{n,μ}).

Finally, assume that n > k ≥ 3. By Corollary 3.4 and the induction hypothesis we have

e(M_{n,μ}) = e(M_{n−1,μ′}) + μ_{k}e(M_{n−1,μ}) = deg(Y_{n−1,μ′}) + μ_{k} deg(Y_{n−1,μ}),  (3.1)

and by Proposition 2.3 we have

deg(Y_{n,μ}) = mult_{p}(Y_{n,μ}) + μ_{k} deg(Y_{n−1,μ}).  (3.2)

We use the notation of Proposition 2.4. The multiplicity mult_{p}(Y_{n,μ}) = mult_{0}(X) equals (k−3)! times the leading coefficient of the Hilbert polynomial of the ideal ⟨lead_{0}(f)|f ∈ I(X)⟩ (see for example [4, p.258]). The common zero set of ⟨lead_{0}(f)|f ∈ I(X)⟩ is the tangent cone T C_{0}X, which by Proposition 2.4 coincides with X_{n−1,μ′}. On the other hand, we saw in the proof of Proposition 2.4 that

⟨lead_{0}(f)|f ∈ I(X)⟩ ⫆ ⟨δ_{μ′}(t_{i_{1}}, …, t_{i_{k−1}})|1 ≤ i_{1} < ⋯ < i_{k−1} ≤ n − 1⟩,

and by the induction hypothesis this latter ideal is exactly the ideal of X_{n−1,μ′}. It follows that F [u_{1}, …, u_{n−1}]/⟨lead_{0}(f)|f ∈ I(X)⟩ is the coordinate ring of X_{n−1,μ′}, and so

mult_{p}(Y_{n,μ}) = deg(Y_{n−1,μ′}).  (3.3)

Comparing (3.1), (3.2), and (3.3) we obtain the desired equality e(M_{n,μ}) = deg(Y_{n,μ}).

So we proved that In_{≺}(I(Y_{n,μ})) = M_{n,μ}, and by Proposition 3.2 this implies the universal Gröbner basis property. □

Corollary 3.4 and Theorem 1.2 have the following corollary:

Corollary 3.5. The degree of X_{n,μ} equals the (n + 1 − k)-th complete symmetric function of μ_{1}, …, μ_{k}, that is,

deg(X_{n,μ}) = ∑_{j_{1}+···+j_{k}=n−k+1} μ_{1}^{j_{1}} ⋯ μ_{k}^{j_{k}}.  □

Remark. Since M_{n,μ} is critical, to verify the equality In_{≺}(I(X_{n,μ})) = M_{n,μ} it was sufficient to show that the dimension and the degree of M_{n,μ} computed by combinatorial arguments is what was predicted by the geometry of X_{n,μ}. So it is a natural question for which affine cones it is true that the initial ideal of their vanishing ideal is critical. Let X be an affine cone over the origin. Obviously, if In_{≺}(I(X)) is critical then I(X) is also critical, that is X must be unmixed. The converse is not true, the unmixedness of X does not imply that In_{≺}(I(X)) is critical, as the following example shows. We put

X = {(x, x, y, y), (x, y, x, y), (x, y, y, x)|x, y ∈ ℂ} ⊂ 𝔸^{4},

so X is the union of three 2-dimensional linear subspaces, and h_{X}(t) = 3t + some constant. The polynomials (t_{i_{1}}−t_{i_{2}})(t_{i_{1}}−t_{i_{3}})(t_{i_{2}}−t_{i_{3}}) and (t_{i_{4}}−t_{i_{1}})(t_{i_{4}}−t_{i_{2}})(t_{i_{4}}−t_{i_{3}}) with {i_{1}, i_{2}, i_{3}, i_{4}} = {1, 2, 3, 4} vanish on X. Let ≺ be any admissible order with t_{1} ≺ t_{2} ≺ t_{3} ≺ t_{4}. Then

In_{≺}(I(X)) ⊇ ⟨t_{2}t_{3}^{2}, t_{2}t_{4}^{2}, t_{3}t_{4}^{2}, t_{2}^{2}t_{4}, t_{3}^{2}t_{4}, t_{2}t_{3}t_{4}, t_{4}^{3}⟩

(t_{2}^{2}t_{4} is the initial monomial of (t_{1} − t_{2})(t_{1} − t_{3})(t_{1} − t_{4}) + (t_{2} − t_{1})(t_{2} − t_{3})(t_{2} − t_{4})). Those degree d monomials outside In_{≺}(I(X)) which are divisible by t_{4} are all contained in the set {t_{1}^{d−2}t_{4}^{2}, t_{1}^{d−2}t_{2}t_{4}, t_{1}^{d−2}t_{3}t_{4}, t_{1}^{d−1}t_{4}}, hence

#{degree d monomials ∉ In_{≺}(I(X))} − #{degree d monomials ∉ (In_{≺}(I(X)), t_{4})} ≤ 4.

It follows that e(In_{≺}(I(X)), t_{4}) = e(In_{≺}(I(X))), though t_{4} is clearly not contained in the ideal In_{≺}(I(X)).

4. Subspace arrangements related to reflection groups

A finite collection H of (n − 1)-dimensional linear subspaces of ℂ^{n} is called a complex hyperplane arrangement. For any 1 ≤ k ≤ n − 1 denote by H(k) the collection of k-dimensional linear subspaces which are intersections of elements of H. It is called the (n − k)-truncation of H, following the terminology of [1, 5.1.(iv)]. An interesting example is the so called braid arrangement, consisting of the hyperplanes V (t_{i} − t_{j}) with 1 ≤ i < j ≤ n. The union of the elements of the (n − k)-truncation of the braid arrangement is

{x ∈ ℂ^{n}|x has at most k different coordinates},

and the special case (μ_{1}, …, μ_{k+1}) = (0, 1, 2, …, k) of Theorem 1.2 gives a Gröbner basis of the vanishing ideal of this subspace arrangement. The reflections with respect to the hyperplanes in the braid arrangement generate the standard representation of the symmetric group Sym(n). There are other interesting hyperplane arrangements belonging to the other Weyl groups, or more generally, to the complex pseudo-reflection groups. Recall that a linear transformation of ℂ^{n} is called a pseudo-reflection, if it is of finite order and it fixes a hyperplane. A subgroup of Gl_{n}(ℂ) generated by pseudo-reflections is called a pseudoreflection group.

Now let G be a finite pseudo-reflection group, and denote by H_{G} the set of the reflecting hyperplanes. The problem to describe a Gröbner basis of the vanishing ideal of the Gorbit of a k-dimensional subspace which is the intersection of reflecting hyperplanes (orbit arrangement) was raised in [1, 13.5]. Note that the union of these orbit arrangements is the (n − k)-truncation of H_{G}. We would like to point out that the corresponding special cases of Theorem 1.2 give Gröbner bases of the vanishing ideal of the (n − k)-truncation of H_{G} for all but finitely many irreducible pseudo-reflection groups G.

The finite irreducible unitary pseudo-reflection groups were classified in [10]. We discussed already the infinite series of the symmetric groups Sym(n). (To get an irreducible representation one has to restrict the standard representation onto the hyperplane V (t_{1} + ⋯ + t_{n}).) For the dihedral groups (the symmetry groups of regular m-gons) our question is not interesting, because then n = 2 and the only truncations of H_{G} are H_{G} itself and {0}. Besides finitely many exceptional groups the remaining groups in the list are the groups G(m, q, n), where m ≥ 2 and q divides m, defined as follows:

Consider Sym(n) as a subgroup of Gl_{n}(ℂ) consisting of permutation matrices, and denote by D(m, q, n) the group of diagonal matrices whose diagonal entries are m-th roots of unity, and whose determinant is an m/q-th root of unity. Obviously, Sym(n) normalizes D(m, q, n), so they generate in Gl_{n}(ℂ) their semi-direct product G(m, q, n) = D(m, q, n) ⋊ Sym(n). Case I: G = G(m, q, n), m ≥ 2, q ≠ m. In this case the reflecting hyperplanes are

H_{G} = {V (t_{i} − ζt_{j}), V (t_{l})|1 ≤ i < j ≤ n, l = 1, …, n, ζ^{m} = 1}.

We claim that for any 1 ≤ k ≤ n−1 the union of the linear subspaces in the (n−k)-truncation of H_{G} is X_{n,μ} with μ = (1, m + 1, 2m + 1, …, km + 1). By definition we have

X_{n,μ} = {x ∈ ℂ^{n}|x_{i_{1}} ⋯ x_{i_{k+1}} ∏_{1≤r<s≤k+1}(x_{i_{s}}^{m} − x_{i_{r}}^{m}) = 0 for any 1 ≤ i_{1} < ⋯ < i_{k+1} ≤ n}.

On the other hand, take an element L of H_{G}(k). There are k ”free coordinates” on L, say t_{1}, …, t_{k}, and L is defined by n − k linear equations l_{k+1} = ⋯ = l_{n} = 0, where l_{j} = t_{j} or l_{j} = t_{j} − ζ_{j}t_{i_{j}} with ζ_{j}^{m} = 1 and i_{j} ∈ {1, …, k}. It is easy to see that the union of all such subspaces is

{x ∈ ℂ^{n}| among any k + 1 coordinates of x there is a 0

or there are two whose m-th powers are equal} = X_{n,μ}

Note that in the special case m = 2 we get the Weyl group G(2, 1, n) associated with the Dynkin diagram B_{n}. (In this case the subspace arrangement is defined over the real numbers, and the conclusion of Theorem 1.2 clearly holds with F = ℝ.) Case II: G = G(m, m, n), m ≥ 2. In this case we have

H_{G} = {V (t_{i} − ζt_{j})|1 ≤ i < j ≤ n, ζ^{m} = 1},

and similarly to the above case for any 1 ≤ k ≤ n − 1 we have ∪H_{G}(k) = X_{n,μ} with μ = (0, m, 2m, …, km). In the special case m = 2 the group G(2, 2, n) is the Weyl group belonging to the Dynkin diagram D_{n}.

So the special cases μ = (1, m + 1, 2m + 1, …, km + 1) and μ = (0, m, 2m, …, km) of Theorem 1.2 give Gröbner bases of the vanishing ideals of (n−k)-truncations of the reflection arrangements belonging to the pseudo-reflection groups G(m, q, n).

Acknowledgements. The author thanks Endre Szabó and Sándor Kovács for helpful discussions.

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Received July 8, 1998; revised version September 22, 1998