Gotzmann Monomials In Four Variables

It is a widely open problem to determine which monomials in the n-variable polynomial ring $K[x_1,...,x_n]$ over a field $K$ have the Gotzmann property, i.e. induce a Borel-stable Gotzmann monomial ideal. Since 2007, only the case $n \le 3$ was known. Here we solve the problem for the case $n = 4$. The solution involves a surprisingly intricate characterization.


Introduction
Let K be a field and let R n = K[x 1 , . . . , x n ] be the n-variable polynomial algebra over K endowed with its usual grading deg(x i ) = 1 for all i. We denote by S n ⊂ R n the set of all monomials u = x a1 1 · · · x an n in R n , and by S n,d ⊂ S n the subset of monomials of degree deg(u) = i a i = d.
A monomial ideal J ⊆ R n is said to be Borel-stable or strongly stable if for any monomial v ∈ J and any variable x j dividing v, one has x i v/x j ∈ J for all 1 ≤ i ≤ j. Given a monomial u ∈ S n , let u denote the smallest Borel-stable monomial ideal in R n containing u, and let B(u) denote the unique minimal system of monomial generators of u . Then B(u) may be described as the smallest set of monomials containing u and stable under the operations v → vx i /x j whenever x j divides v and i ≤ j.
Recall that a homogeneous ideal I ⊆ R n is a Gotzmann ideal if, from a certain degree on, its Hilbert function attains Macaulay's lower bound. See e.g. [4,7] for more details. Determining which homogeneous ideals are Gotzmann ideals is notoriously difficult. This will be illustrated in this paper, where our determination of all monomials u in S 4 such that the ideal u is a Gotzmann ideal involves a surprisingly complicated formula. We introduce the following definition. Definition 1.1. We say that a monomial u ∈ S n is a Gotzmann monomial if its associated Borel-stable monomial ideal u is a Gotzmann ideal.
Determining all Gotzmann monomials in S n is a widely open problem. Indeed, the current knowledge about it is limited to the case n ≤ 3. Specifically, for n ≤ 2 all monomials in S 1 or S 2 are Gotzmann, whereas for n = 3, the monomial x a 1 x b 2 x t 3 is Gotzmann in S 3 if and only if t ≥ b 2 . The latter result can be deduced from [13,Proposition 8]. A short proof using the general tools developed in this paper will be given in the last section. The above result for n = 3 illustrates a general property of Gotzmann monomials, proved in [4] using Gotzmann's persistence theorem.
(1) There exists k ∈ N such that ux k n is Gotzmann in S n . (2) If u is Gotzmann in S n , then so is ux n .
This reduces the determination of Gotzmann monomials in S n to the following question. Given u 0 ∈ S n−1 , what is the least exponent t ≥ 0 such that u 0 x t n is a Gotzmann monomial in S n ?
Our main result in this paper is the classification of all Gotzmann monomials in S 4 . It states that a monomial u = x a 1 x b 2 x c 3 x t 4 is a Gotzmann monomial in S 4 if and only if See Theorem 7.7. Interestingly, before achieving this rather intricate characterization, all the easy-to-perform computer-algebraic experiments we ran in order to get a clue at it were of no help. Only the conceptual tools developed below allowed us to formulate and prove this result. Completing the analogous task in S n for n ≥ 5 remains an open problem.
1.2. Contents. In Section 2, we recall or introduce basic notions such as lexsegments and lexintervals, the sets of gaps and cogaps of a monomial, the maxgen monomial of a set of monomials, and finally Gotzmann monomials and criteria in terms of gaps and cogaps to recognize them. In Section 3, we focus on properties of the gaps and cogaps of a monomial and how to compute them. In Section 4, we describe the lexicographic predecessor and successor of a monomial. Section 5 is devoted to the determination of the maxgen monomial of lexintervals. In Section 6, we show some specific behaviors of the first and last variables. Finally, in Section 7 we use all the material developed in the preceding sections to prove our main theorem on the characterization of Gotzmann monomials in four variables.

Background and basic notions
2.1. Lexsegments and lexintervals. Recall the definition of the lexicographic order on S n,d . Let u, v ∈ S n,d . Write u = x a = x a1 1 · · · x an n with a = (a 1 , . . . , a n ) ∈ N n , and similarly v = x b with b ∈ N n . By definition, we have u > lex v if and only if the leftmost nonzero coordinate of a − b is positive. Equivalently, let Then u > lex v if and only if the leftmost nonzero coordinate of (i 1 − j 1 , . . . , i d − j d ) is negative. For simplicity, we shall omit the subscript and write ≥ instead of ≥ lex .
We shall need below the following well-known equivalence.
where e i ∈ N n is the basis vector with a 1 at the ith coordinate and 0 elsewhere. The statement follows since The following notation will be used throughout.
Notation 2.2. For u ∈ S n,d , we denote by L(u) the lexsegment determined by u, i.e. L(u) = {v ∈ S n,d | v ≥ u}. More generally, for u 1 , u 2 ∈ S n,d such that u 1 ≥ u 2 , we denote by L(u 1 , u 2 ) the lexinterval of intermediate monomials, namely Definition 2.4. A monomial ideal I ⊆ R n is said to be Borel-stable if its set of monomials I ∩ S n is a Borel-stable set.
Let B ⊆ S n,d . We define and denote the shade 1 of B by For i ≥ 2, the i-th shade of B is defined recursively by Shad i (B) = Shad(Shad i−1 (B)). 1 Shad should stand for shade as in Combinatorial set theory [1], and not for "shadow" as written in [4,7]. The shadow of B actually corresponds to the set of all monomials u/x j with u ∈ B and x j dividing u.

2.3.
The maxgen monomial revisited. Given B ⊆ S n,d with d ≥ 1, we now describe maxgen(B) in a slightly more useful way. First some preliminaries.
Notation 2.15. Let u ∈ S n be a monomial distinct from 1. We denote by Thus λ(u) divides u, and it is the "last", or lexicographically smallest, variable with this property. This yields a function Proof. Directly follows from the definitions.
Thus, maxgen(B) may be viewed as the maximal index generating function of all monomials in B.
We shall sometimes tacitly use the following easy observation.

Gaps and cogaps.
Notation 2.18. Let u ∈ S n . We denote by B(u) the smallest Borel-stable subset of S n containing u.
Observe that if u ∈ S n,d , then B(u) ⊆ S n,d . Proof. Let v ∈ B(u). Then v is obtained from u by repeated operations of the form u ′ → x i u ′ /x j where u ′ ∈ B(u), x j divides u ′ and i ≤ j. Since x i u ′ /x j ≥ u ′ at each such step, it follows that v ≥ u, whence v ∈ L(u).
For our present purposes, it is of particular interest to consider the set difference L(u) \ B(u). The following concept first arose in [4].
Definition 2.20. Let u ∈ S n,d . We set gaps(u) = L(u) \ B(u), and we call gaps of u the elements of this set.
Notation 2.21. Let u ∈ S n,d . We denote byũ ∈ S n,d the unique monomial such that Since B(u) and B(u) lex have the same cardinality by definition, we have Moreover, since B(u) ⊆ L(u) by the above lemma, we have x d 1 ≥ũ ≥ u and so L(ũ) ⊆ L(u). Here is an illustration of the situation: Since |L(ũ)| = |B(u)|, we have This motivates our definition of cogaps(u), a lexinterval with the same cardinality as gaps(u).
By construction, we have | gaps(u)| = | cogaps(u)| and two partitions of L(u), namely: Example 2.23. Let n = 4, d = 2 and u = x 2 x 3 . Then A word of caution regarding L(u) and B(u) is needed here. Remark 2.24. For the lexsegment determined by u ∈ S n , one should write L n (u) rather than L(u). Indeed, let m < n be positive integers. Then S m ⊂ S n canonically. Let now u ∈ S m . Then L m (u) = L n (u) in general. For instance, with u = x 2 x 3 as above, we have Consequently, one should also write gaps n (u) rather than gaps(u). However, we shall systematically omit the index n since it will be fixed in any given discussion below. On the other hand, the set B(u) is independent of n.
is a Gotzmann set.
Remark 2.26. Note that Gotzmann monomials in S n may no longer be Gotzmann Our determination of Gotzmann monomials in S 3 and S 4 will use the following general characterization.
Proof. It follows from Definition 2.25 and Lemma 2.14 that u is a Gotzmann monomial if and only if maxgen(B(u)) = maxgen(B(u) lex ).
Thus, from now on, our task will be to develop tools to compute or determine gaps(u), cogaps(u) and their respective maxgen monomials, so as to be able to apply the characterization of Gotzmann monomials provided by Theorem 2.27.

Some results on gaps
Let u ∈ S n,d . Recall that B(u) ⊆ L(u) and that gaps(u) = L(u) \ B(u). We first describe the gaps of u in an equivalent way.
The existence of index s with the given property then follows from the hypothesis v ∈ L(u). The existence of index t > s with its property then directly follows from the hypothesis v / ∈ B(u).
We need yet another notation which will be used to give a structural description of gaps(u).
Observe that pre k (u) may be characterized as follows: it is the unique monomial v of degree k dividing u and satisfying max(v) ≤ min(u/v). Definition 3.3. For any v ∈ S n,k , we define subsets A 1 (v), A 2 (v) ⊂ S n,k as follows: Then , is a gap of u by construction and Lemma 3.1.
In the notation of Lemma 3.1, let s be such that i s < j s , and let t > s be the least index satisfying j t > i t . Set k = t − 1. Then by construction, the factor x j1 · · · x j k of degree k of v belongs to B(u k ), since i α ≤ j α for all α < t by minimality of t, and in fact belongs to Proof. The number of monomials w 2 ∈ S n of degree d−k in the variables x i k+1 +1 , . . . , x n is equal to the number of monomials of the same degree in S n−i k+1 , i.e. to S n−i k+1 ,d−k .
This prompts us to find good formulas for |B(w)| for any monomial w. Here is an inductive approach.
Proposition 3.6. Let w ∈ S n and m = max(w). Let r ≥ 1 be the largest exponent such that x r m divides w.
Proof. Directly follows from the above partition of B(w).
This corollary reduces the computation of |B(w)| for monomials in S n to that for monomials in S n−1 .

Predecessors and successors
Definition 4.1. Let u, v ∈ S n,d such that u > v. We say that u covers v if there are no intermediate monomials between them, i.e. if for any w ∈ S n,d such that u ≥ w ≥ v, we have w = u or w = v. In that case, we say that u is the predecessor of v, that v is the successor of u, and we write Since x d 1 and x d n are the largest and smallest elements in S n,d , respectively, the predecessor of x d 1 and the successor of x d n are undefined. Note that, for all u ∈ S n,d with u / ∈ {x d 1 , x d n }, we have succ(pred(u)) = pred(succ(u)) = u. Proposition 4.2. Let u ∈ S n and g = | gaps(u)|. Thenũ is the gth predecessor of u, i.e.ũ = pred g (u).
Lexintervals ending at a monomial u are made up of iterated predecessors of u. This motivates the following notation.
That is, pred r (u) is the set of r predecessors of u, including u itself. This set is well defined since r ≤ |L(u)| by hypothesis. Of course pred r (u) is a lexinterval, since We may now reinterpret the lexinterval cogaps(u) in terms of the above concept. Proof. By Definition 2.22, we have | cogaps(u)| = | gaps(u)| = g, and cogaps(u) = L * (ũ, u) whereũ = pred g (u). The stated equality follows from (3).
As we shall need to determine pred i (u) for any given u ∈ S n,d , we need an explicit description of pred(u). We start with the description of the successor of a monomial in S n,d distinct from x d n .  The next corollary compares max(pred(u)) with max(u). There are only two possible outcomes, linked to whether λ(u) 2 divides u or not; recall that λ(u) always divides u by construction.
Example 4.8. The lexicographically smallest monomial u ∈ S 4,d such that

5.
The maxgen monomial of lexintervals 5.1. The function µ n . We now introduce a function of two monomials u 1 ≥ u 2 in S n,d which will later be used to give an equivalent description of cogaps. Recall the notation Definition 5.1. For u 1 , u 2 ∈ S n,d such that u 1 ≥ u 2 , we define µ n (u 2 , u 1 ) ∈ S n to be the maxgen monomial of the lexinterval L * (u 1 , u 2 ), i.e. µ n (u 2 , u 1 ) = maxgen(L * (u 1 , u 2 )).
Equivalently, recalling that maxgen collects the last variables of a set of monomials and takes their product: Note that by construction, the last variable of u 1 is not taken into account in µ n (u 2 , u 1 ).
Remark 5.2. As in Remark 2.24, we have S n,d ⊂ S n+1,d canonically. Now if u 1 , u 2 ∈ S n,d , then µ n (u 2 , u 1 ) and µ n+1 (u 2 , u 1 ) differ in general. However, when the number n of variables involved is clear from the context, we shall simply write µ(u 2 , u 1 ) for µ n (u 2 , u 1 ).
Proof. Directly follows from the definition.
Notation 5.4. For monomials u 1 ≥ u 2 in S n,d , we shall occasionally denote the equality µ(u 2 , u 1 ) = w as follows:

Lemma 5.3 then amounts to arrow composition
For instance, starting from x 2 3 and taking successive predecessors in S 3 , one has By arrow composition, this may be summarized as In particular, with r = | gaps(u)|, this yields the following tool in view of effectively applying Theorem 2.27.
In order to apply this lemma, we need some control on min µ(u 2 , u 1 ). This is provided by the next proposition. First a lemma. For v < u 1 , the above lemma implies min v ≥ min u 1 . Hence max v > min u 1 , for otherwise we would have max v = min v = min u 1 , implying v = x d i for some i. But from v = x d i < u 1 , it follows that min u 1 < i, contradicting min v = min u 1 . Having established max v > min u 1 for all v < u 1 , it follows that min µ(u 2 , u 1 ) > min u 1 , as stated.
Here are straightforward applications.
In view of a general statement, the following intermediate formula will be useful.
Proposition 5.13. For all 2 ≤ m ≤ n and all k ≥ 1, we have Proof. By induction on k. For k = 1, it is clear that µ(x m , x m−1 ) = x m , since the predecessor of x m is x m−1 . Assume now k ≥ 2 and that formula (4) holds for k − 1, Thus, in order to establish (4), we only need to show In S n , the predecessor of x k m is x m−1 x k−1 n . Hence we have But it follows from Corollary 5.11 that By Lemma 5.3, we have . Moreover, by (6) and (7) again, we have Hence . This proves (5) and hence the claimed formula (4).
Here is the promised general statement. As usual, by convention, an empty product equals 1, as occurs below for m = n.
Theorem 5.14. For all 2 ≤ m ≤ n, for all k ≥ 1, and for all v ∈ S n such that max v ≤ m − 1, we have Proof. By Corollary 5.11, we have . Hence, it suffices to establish (9) for v = 1. We proceed by induction on k and descending induction on m. For k = 1 and any m ≥ 1, we have µ(x m , x m−1 ) = x m , and this plainly coincides with the right-hand side of (9) since 1−1+i i+1 = 0 for all i ≥ 1. Similarly, for m = n and any k ≥ 1, we have Equivalently, in formula: µ(x k n , x k n−1 ) = x k n . Assume now that (9) holds for some m such that n ≥ m ≥ 2 and some k ≥ 2. We now show that (9) also holds for m − 1. By arrow composition, we have Exchanging the names of the indices i, j in the last product, we get Finally, substituting i by i − 1 in the last product yields Hence (9) also holds for m − 1, as claimed. This concludes the proof of the theorem.

Some maxgen computations.
The following result uses the sets A 1 (v), A 2 (v) introduced in Definition 3.3. It will be needed in view of applying Theorem 2.27.
Proof. Consider the description of gaps given in Proposition 3.4. For any monomial w = w 1 w 2 ∈ A 1 (u k )A 2 (u/u k ) with w 1 ∈ A 1 (u k ) and w 2 ∈ A 2 (u/u k ), we have max(w) = max(w 2 ), since max(w 1 ) < min(w 2 ) by construction. Therefore, for all k we have Since A 2 (u/u k ) is the set of all monomials of degree deg(u/u k ) in the variables x i with min(u/u k ) + 1 ≤ i ≤ n, we will be able to determine maxgen(A 2 (u/u k )) if we can determine maxgen(S n,d ) for any n, d. Let us proceed to do just that. We start with a recurrence formula. Proof. Obviously, we have This follows from writing any u ∈ S n,d as u = vx i n with v ∈ S n−1,d−i . Hence We conclude the proof by applying the well known formula Corollary 5.17. For all n, d, we have Proof. Use above induction formula. Proof. Directly follows from the preceding corollary by properly translating indices.
We may now inject this information into Proposition 5.15. This yields the following result.
Theorem 5.19. Let u = x i1 · · · x i d with i 1 ≤ · · · ≤ i d . Then where the internal product is set to 1 if i k+1 = n.
Proof. The proof follows from Proposition 5.15 together with the above corollary.
Moreover, A 2 (u/u k ) is the set of all monomials of degree deg(u/u k ) in the variables x i with min(u/u k )+1 ≤ i ≤ n. Therefore, in order to determine maxgen(A 2 (u/u k )), it remains to apply Corollary 5.18, using l = i k+1 + 1 since u/u k = x i k+1 . . . x i d and so min(u/u k ) = i k+1 .

On the first and last variables
For the determination of Gotzmann monomials in S n , both variables x 1 and x n behave in some specific ways. This section describes how. 6.1. Neutrality of x 1 . Our purpose here is to show that a monomial u ∈ S n is Gotzmann if and only if x 1 u is. We start with some intermediate results.
Without loss of generality, we may assume u > v. Hence there exists an index 1 ≤ k ≤ d such that Therefore i 1 ≤ j 1 . Assume x 1 divides v. This is equivalent to j 1 = 1. Since 1 ≤ i 1 ≤ j 1 , then i 1 = 1 whence x 1 divides u and we are done.
In particular, the lemma implies that multiplying any lexsegment by x 1 again yields a lexsegment.
Proof. We have |B lex | = |B| = |x 1 B|. Applying this to the set x 1 B yields Now B lex is a lexsegment, whence x 1 B lex also is by Lemma 6.2 and the comment following it. Moreover, x 1 B lex has the same cardinality as the lexsegment (x 1 B) lex by (10). Whence these two lexsegments coincide.
since v ≥ u and v cannot belong to B(u) for otherwise x 1 v would belong to B(x 1 u).
Hence v ′ ∈ x 1 gaps(u). Conversely, let v ∈ gaps(u). Then v > u, whence Theorem 6.5. Let u ∈ S n . Then u is Gotzmann if and only if x 1 u is Gotzmann.
Proof. First some preliminary steps.

Indeed, by applying transformations of the form
, with x j dividing v and 1 ≤ i < j, the variable x 1 is not affected since j ≥ 2. Whence the claimed equality.

6.2.
On gaps(ux n ). For use in the next section, we shall need to control maxgen(gaps(ux n )). Definition 6.6. Let u ∈ S n . For all i ≤ n, denote Note that gaps(u, 1) is empty, for x d 1 cannot be a gap since it obviously belongs to B(u) for all u ∈ S n,d .
Theorem 6.7. Let u ∈ S n . Then for all 1 ≤ j ≤ n, we have Here is an equivalent formulation.
Proof. Let d = deg(u), and write u = x i1 · · · x i d with 1 ≤ i 1 ≤ · · · ≤ i d ≤ n. We may assume deg(w) = d+1, for otherwise w cannot be a gap of ux n . Set max(w) = m. Let v = w/λ(w), and write v = x j1 · · · x j d with 1 ≤ j 1 ≤ · · · ≤ j d ≤ m. By Lemma 3.4, v is a gap of u if and only if there exist indices 1 ≤ s < t ≤ d such that j s < i s and j t > i t . If these conditions are met, then since w = vx m with m ≥ max(v), then automatically w is a gap of ux n , still by Lemma 3.4. Conversely, if w is a gap of ux n , and since max(w) ≤ max(ux n ) = n, then the index t ≤ d + 1 given by Lemma 3.4 necessarily satisfies t ≤ d. Hence v is a gap of u. Proof. By Theorem 6.7, we have The statement now follows from the definition of the maxgen monomial.
7. Gotzmann monomials in S 2 , S 3 , S 4 This section contains the main result of this paper, namely the characterization of Gotzmann monomials in S n for n = 4. This is achieved in Theorem 7.7. The strategy is as follows. Let u = x a1 1 · · · x an−1 n−1 x t n ∈ S n . We may assume a 1 = 0 by Theorem 6.5, according to which u is Gotzmann in S n if and only if u/x a1 1 is. We first compute w 1 = maxgen(gaps(u)) using Theorem 5.19. The degree g of w 1 gives the numbers of gaps of u. We then focus on cogaps(u) = pred g (u) and, more precisely, compute its maxgen monomial w 2 = maxgen(pred g (u)). Finally, requiring w 1 = w 2 gives necessary and sufficient conditions on the exponent t of x n for u to be a Gotzmann monomial.
Before turning to the case n = 4, we start by reviewing the known cases n = 2 and 3.
7.1. The case n = 2. This is easy. Indeed, every monomial u = x a 1 x t 2 is Gotzmann in S 2 . For in this case, the sets B(u) and L(u) coincide, whence B(u) lex = B(u) and so B(u) is a Gotzmann set by Lemma 2.12.
Hence for 1 ≤ k ≤ d, we have i k = 2 if k ≤ b, and i k = 3 otherwise. By Theorem 5.19, we have maxgen(gaps(u 0 )) = We now compute the involved exponents. We have |B(x k 2 )| = k + 1, as follows from the set equality B( of Corollary 3.7, the above formula for |B(x k 2 )| and some straightforward computations.
Inserting these exponent values into the above formula for maxgen(gaps(u 0 )), we get maxgen(gaps(u 0 )) = b−1 k=1 and some straightforward computations, we get Proposition 7.5. We have Proof. Starting from x b 2 x c 3 x t 4 and taking t + 1 successive predecessors, Proposition 4.6 yields From x b+1 2 x c+t−1

4
, we must still reach x ). This can be done using Lemma 7.4 to (r, s, i) = (b + 1, c + t − 1, b 2 − 1). We obtain Hence, combining (17) and (18) using arrow composition, we get It remains to show that the exponent of x 4 in the monomial of the above right-hand side is equal to h(t). Indeed, we have the desired equality with h(t) follows.
Remark 7.6. By Propositions 7.2 and 7.5, for t ≥ 0 we have In particular, f (t) − h(t) is a positive constant. This will be used below.
Here is our main result.