# Some Relations between Admissible Monomials for the Polynomial Algebra

Authors: Mbakiso Fix Mothebe; Lafras Uys
Journal: International Journal of Mathematics and Mathematical Sciences
 (2015)

---

## Abstract

LetP
(
n
)
=
F
2
[
x
1
,
…
,
x
n
] be the polynomial algebra in n variables x
i, of degree one, over the field F
2 of two elements. The mod-2 Steenrod algebra A acts on P
(
n
) according to well known rules. A major problem in algebraic topology is of determining A
+
P
(
n
), the image of the action of the positively graded part of A. We are interested in the related problem of determining a basis for the quotient vector space Q
(
n
)
=
P
(
n
)
/
A
+
P
(
n
). Q
(
n
) has been explicitly calculated for n
=
1,2
,
3,4 but problems remain for n
≥
5. Both P
(
n
)
=
⨁
d
≥
0
P
d
(
n
) and Q
(
n
) are graded, where P
d
(
n
) denotes the set of homogeneous polynomials of degree d. In this paper, we show that if u
=
x
1
m
1
⋯
x
n
-
1
m
n
-
1
∈
P
d
′
(
n
-
1
) is an admissible monomial (i.e., u meets a criterion to be in a certain basis for Q
(
n
-
1
)), then, for any pair of integers (j
,
λ), 1
≤
j
≤
n, and λ
≥
0, the monomial h
j
λ
u
=
x
1
m
1
⋯
x
j
-
1
m
j
-
1
x
j
2
λ
-
1
x
j
+
1
m
j
⋯
x
n
m
n
-
1
∈
P
d
′
+
(
2
λ
-
1
)
(
n
) is admissible. As an application we consider a few cases when n
=
5.


---

## Body

## 1. Introduction

Forn
≥
1 let P
(
n
) be the mod-2 cohomology group of the n-fold product of R
P
∞ with itself. Then P
(
n
) is the polynomial algebra(1)
P
n
=
F
2
x
1
,
…
,
x
nin n variables x
i, each of degree 1, over the field F
2 of two elements.The mod-2 Steenrod algebraA is the graded associative algebra generated over F
2 by symbols S
q
i for i
≥
0, called Steenrod squares subject to the Adem relations [1] and S
q
0
=
1. Let P
d
(
n
) denote the homogeneous polynomials of degree d. The action of the Steenrod squares S
q
i
:
P
d
(
n
)
→
P
d
+
i
(
n
) is determined by the formula(2)
S
q
i
u
=
u
,
i
=
0
,
u
2
,
deg
⁡
u
=
i
,
0
,
deg
⁡
u
<
i
,and the Cartan formula (3)
S
q
i
u
v
=
∑
r
=
0
i
S
q
r
u
S
q
i
-
r
v
.A polynomial u
∈
P
d
(
n
) is said to be hit if it is in the image of the action of A on P
(
n
), that is, if (4)
u
=
∑
i
>
0
S
q
i
u
i
,for some u
i
∈
P
(
n
) of degree d
-
i. Let A
+
P
(
n
) denote the subspace of all hit polynomials. The problem of determining A
+
P
(
n
) is called the hit problem and has been studied by several authors [2–4]. We are interested in the related problem of determining a basis for the quotient vector space (5)
Q
n
=
P
n
A
+
P
nwhich has also been studied by several authors [5–8]. Some of the motivation for studying these problems is mentioned in [9]. It stems from the Peterson conjecture proved in [4] and various other sources [10, 11].The following result is useful for determiningA-generators for P
(
n
). Let α
(
m
) denote the number of digits 1 in the binary expansion of m.In [4, Theorem 1], Wood proved the following.Theorem 1 (Wood [4]).
Letu
∈
P
(
n
) be a monomial of degree d. If α
(
n
+
d
)
>
n, then u is hit.Thus,Q
d
(
n
) is zero unless α
(
n
+
d
)
≤
n or, equivalently, unless d can be written in the form d
=
∑
i
=
1
n
‍
(
2
λ
i
-
1
), where λ
i
≥
0. Thus, Q
d
(
n
)
≠
0 only if P
d
(
n
) contains monomials v
=
x
1
2
λ
1
-
1
⋯
x
n
2
λ
n
-
1 called spikes. For convenience, we will assume that λ
1
≥
λ
2
≥
⋯
≥
λ
n
≥
0
. We, in addition, will consider a special one when λ
1
≥
λ
2
≥
⋯
≥
λ
s
≥
0 and λ
j
-
1
=
λ
j only if j
=
s or λ
j
+
1
=
0. In this case v is called aminimal spike.Q
(
n
) has been explicitly calculated by Peterson [8] for n
=
1,2, by Kameko in his thesis [5] for n
=
3, and independently by Kameko [6] and Sum [7] for n
=
4. In this work we will, unless otherwise stated, be concerned with a basis for Q
(
n
) consisting of “admissible monomials,” as defined below. Thus, when we write u
∈
Q
d
(
n
) we mean that u is an admissible monomial of degree d
.We define what it means for a monomialb
=
x
1
e
1
⋯
x
n
e
n
∈
P
(
n
) to be admissible. Write e
i
=
∑
j
≥
0
‍
α
j
(
e
i
)
2
j for the binary expansion of each exponent e
i. The expansions are then assembled into a matrix β
(
b
)
=
(
α
j
(
e
i
)
) of digits 0 or 1 with α
j
(
e
i
) in the (
i
,
j
)th position of the matrix.We then associate withb two sequences (6)
w
b
=
w
0
b
,
w
1
b
,
…
,
w
j
b
,
…
,
e
b
=
e
1
,
e
2
,
…
,
e
n
,where w
j
(
b
)
=
∑
i
=
1
n
‍
α
j
(
e
i
) for each j
≥
0. w
(
b
) is called theweight vector of the monomial b and e
(
b
) is called theexponent vector of the monomial b.Given two sequencesp
=
(
u
0
,
u
1
,
…
,
u
l
,
0
,
…
) and q
=
(
v
0
,
v
1
,
…
,
v
l
,
0
,
…
), we say p
<
q if there is a positive integer k such that u
i
=
v
i for all i
<
k and u
k
<
v
k. We are now in a position to define an order relation on monomials.Definition 2.
Leta
,
b be monomials in P
(
n
). We say that a
<
b if one of the following holds: (1)
w
(
a
)
<
w
(
b
),
(2)
w
(
a
)
=
w
(
b
) and e
(
a
)
<
e
(
b
).Note that the order relation on the set of sequences is the lexicographical one.Following Kameko [5] we define the following.Definition 3.
A monomialb
∈
P
(
n
) is said to beinadmissible if there exist monomials b
1
,
b
2
,
…
,
b
r
∈
P
(
n
) with b
j
<
b for each j, 1
≤
j
≤
r, such that (7)
b
≡
∑
j
=
1
r
b
j
mod
⁡
A
+
P
n
.
bis said to beadmissible if it is not inadmissible.Clearly the set of all admissible monomials inP
(
n
) form a basis for Q
(
n
)
.Letu
=
x
1
m
1
⋯
x
n
-
1
m
n
-
1
∈
P
(
n
-
1
) be a monomial of degree d
′
. Given any pair of integers (
j
,
λ
), 1
≤
j
≤
n, λ
≥
0, we will write h
j
λ
(
u
) for the monomial x
1
m
1
⋯
x
j
-
1
m
j
-
1
x
j
2
λ
-
1
x
j
+
1
m
j
⋯
x
n
m
n
-
1
∈
P
d
′
+
(
2
λ
-
1
)
(
n
)
. Our main result is the following.Theorem 4.
Letu
∈
P
(
n
-
1
) be a monomial of degree d
′, where α
(
d
′
+
n
-
1
)
≤
n
-
1. If u is admissible, then, for each pair of integers (
j
,
λ
), 1
≤
j
≤
n, λ
≥
0, h
j
λ
(
u
) is admissible.Our proof of Theorem4 is deferred until Section 3.As a corollary to Theorem4, suppose that λ is fixed so that d
′ is also fixed. Let x
1
2
λ
1
-
1
⋯
x
n
-
2
2
λ
n
-
2
-
1 be the minimal spike of degree d
′. If λ
>
λ
1, then consider the following.Corollary 5.
Q
d
(
n
) has a subspace isomorphic to ⨁
i
=
1
n
Q
d
′
(
n
-
1
)
.As our main application of the theorem we consider a few cases whenn
=
5. The relevant result in this case is Theorem 6 stated below. To explain Table 1 we recall that, given an integer d such that α
(
n
+
d
)
≤
n, we let(8)
C
n
,
d
=
λ
≥
0
∣
α
d
-
2
λ
-
1
+
n
-
1
≤
n
-
1
.Then given any explicit admissible monomial basis for Q
(
n
-
1
) one may compute L
B
(
n
,
d
), the dimension of the subspace of Q
d
(
n
) generated by all monomials of the form h
j
λ
(
u
)
∈
P
d
(
n
)
,
λ
∈
C
(
n
,
d
) and u
∈
Q
d
-
(
2
λ
-
1
)
(
n
-
1
)
. In general L
B
(
n
,
d
)
≤
d
i
m
(
Q
d
(
n
)
)
.Table 1
d
dim(Q
d
(
5
)
)
Reference
L
B
(
5
,
d
)
1
5
5
2
10
10
3
25
25
4
45
[12]
45
5
46
46
6
74
74
7
110
110
8
174
[13]
174
9
191
191
10
280
275
11
315
[14]
313
12
190
[12]
190
13
—
246
14
—
295
15
432
[15]
384
16
—
414
17
—
538
18
—
673
19
—
778
20
—
591
21
—
741
22
—
780
23
—
916
24
—
908
25
—
974
26
1024
[16]
985
27
315
313
28
480
[12]
480
29
—
450
30
—
763In [7] Sum gives an explicit admissible monomial basis for Q
(
4
)
. In this paper we make use of his results to compute L
B
(
5
,
d
), 1
≤
d
≤
30, and compare these values with d
i
m
Q
d
(
5
) in the given range. The results are given in Table 1. The table is incomplete as not all values for d
i
m
Q
d
(
5
) are known in the given range. While in general it is true that L
B
(
5
,
d
)
≤
d
i
m
(
Q
d
(
5
)
), there are cases, even nontrivial, where equality holds. This is demonstrated with the aid of known results for d
i
m
(
Q
d
(
5
)
) (cited in Table 1) but we will show later that the same conclusions can be reached independently.Theorem 6.
Table1 gives lower bounds, L
B
(
5
,
d
), for the dimension of Q
d
(
5
), 1
≤
d
≤
30.While this approach remains to be explored, in general these test results suffice for our purpose in this paper and we hope to make a more general account in subsequent work. We are thus only required to prove Theorem4.Our work is organized as follows. In Section2, we recall some results on admissible monomials and hit monomials in P
(
n
). In Section 3, we prove Theorem 4. We conclude with an application of the theorem in Section 4.


## 2. Preliminaries

In this section we recall some results in Kameko [17] and Singer [3] on admissible monomials and hit monomials in P
(
n
).The following theorem has been used to great effect by Kameko and Sum in computing a basis forQ
(
3
) and Q
(
4
), respectively.Theorem 7 (Kameko [5] and Sum [18]).
Leta
,
b be monomials in P
(
n
) such that w
j
(
a
)
=
0 for j
>
r
>
0. If b is inadmissible, then a
b
2
r is also inadmissible.Up to permutation of representatives weight order provides a total order relation amongst spikes in a given degree.It is easy to show that a spikev
=
x
1
2
λ
1
-
1
⋯
x
n
2
λ
n
-
1
∈
P
d
(
n
) is a minimal spike if its weight order is minimal with respect to other spikes of degree d. We say v is amaximal spike if its weight order is maximal with respect to other spikes of degree d. In [17, Theorem 4.2] Kameko proved the following.Theorem 8 (Kameko).
Letd be a positive integer and let v be the minimal spike of degree 2
d
+
n. Define a linear mapping, f
:
P
d
(
n
)
→
P
2
d
+
n
(
n
), by (9)
f
x
1
m
1
⋯
x
n
m
n
=
x
1
2
m
1
+
1
⋯
x
n
2
m
n
+
1
.If w
0
(
v
)
=
n, then f induces an isomorphism f
∗
:
Q
d
(
n
)
→
Q
2
d
+
n
(
n
).From Wood’s theorem and the above result of Kameko the problem of determiningA-generators for P
(
n
) is reduced to the cases for which w
0
(
v
)
≤
n
-
1 whenever v is a minimal spike of a given degree d
.We recall the following result of Singer on hit polynomials inP
(
n
). In [3, Theorem 1.2], Singer proved the following.Theorem 9 (Singer).
Letb
∈
P
(
n
) be a monomial of degree d, where α
(
n
+
d
)
≤
n. Let v be the minimal spike of degree d. If w
(
b
)
<
w
(
v
), then b is hit.We will require the following stronger version of Theorem9. Let b be a monomial of degree d. For l
>
0 define d
l
(
b
) to be the integer d
l
(
b
)
=
∑
j
≥
l
‍
w
j
(
b
)
2
j
-
l.In [2, Theorem 1.2], Silverman proved the following.Theorem 10 (Silverman).
Letb
∈
P
(
n
) be a monomial of degree d, where α
(
n
+
d
)
≤
n. Let v be the minimal spike of degree d. If d
l
(
b
)
>
d
l
(
v
) for some l
≥
1, then b is hit.Finally we note that for any elementS
q
k
∈
A
+ and any polynomial u
∈
P
(
n
) we have(10)
S
q
k
2
λ
u
2
λ
=
S
q
k
u
2
λfor a given λ
≥
0
.


## 3. Proof of Theorem4

In this section we prove Theorem4. Our main observation in this work is made in terms of pairs of integers j
,
λ, 1
≤
j
≤
n, λ
≥
0, which determine a monomial h
j
λ
(
u
)
∈
P
d
′
+
(
2
λ
-
1
)
(
n
) once a monomial u
∈
P
d
′
(
n
-
1
) is given. We first note that if u
=
x
1
e
1
⋯
x
n
-
1
e
n
-
1, then h
n
λ
(
u
)
=
x
1
e
1
⋯
x
n
-
1
e
n
-
1
x
n
2
λ
-
1 while, for any other value of j,  h
j
λ
(
u
) is a permutation of h
n
λ
(
u
) that replaces the exponent of x
j by 2
λ
-
1 and those for x
j
+
i, n
-
j
≥
i
≥
1, by e
j
+
i
-
1. We may therefore use permutation notation in place of j
. Let(11)
S
n
=
σ
∈
S
n
∣
σ
r
<
σ
s
i
f
r
<
s
<
n
.Then S
n
=
n and for each integer j, 1
≤
j
≤
n, there exists σ
∈
S
(
n
) such that h
j
λ
(
x
1
e
1
⋯
x
n
-
1
e
n
-
1
)
=
x
σ
(
1
)
e
1
⋯
x
σ
(
n
-
1
)
e
n
-
1
x
σ
(
n
)
2
λ
-
1 whenever x
1
e
1
⋯
x
n
-
1
e
n
-
1
∈
P
d
′
(
n
-
1
)
. For convenience of presentation we will write h
σ
λ
(
x
1
e
1
⋯
x
n
-
1
e
n
-
1
) in place of h
j
λ
(
x
1
e
1
⋯
x
n
-
1
e
n
-
1
)
.Proof of Theorem4.
Suppose thatu
=
x
1
e
1
⋯
x
n
-
1
e
n
-
1
∈
P
(
n
-
1
) is a monomial of degree d
′, where α
(
d
′
+
n
-
1
)
≤
n
-
1. Let λ
≥
0 be an integer. We must show that, for any permutation σ
∈
S
(
n
), h
σ
λ
(
u
) is admissible whenever u is admissible. Put d
=
d
′
+
2
λ
-
1.We first note that, for eachσ
∈
S
(
n
), every monomial in h
σ
λ
(
P
(
n
-
1
)
) is of the form (12)
x
σ
1
m
1
⋯
x
σ
n
-
1
m
n
-
1
x
σ
n
2
λ
-
1
.Thus, if S
q
k
∈
A
+ and w is a monomial in P
d
-
k
(
n
) for which S
q
k
(
w
) contains a term belonging to h
σ
λ
(
P
d
′
(
n
-
1
)
), then w must be of the form (13)
x
σ
1
e
1
⋯
x
σ
n
-
1
e
n
-
1
x
σ
n
2
λ
-
1
,for some x
1
e
1
⋯
x
n
-
1
e
n
-
1
∈
P
d
′
-
k
(
n
-
1
). This must be the case since for any pair of positive integers λ, i there exists no integer t such that S
q
i
(
x
t
)
=
x
2
λ
-
1. By the Cartan formula S
q
k
(
w
) may be written in the form(14)
S
q
k
x
σ
1
e
1
⋯
x
σ
n
-
1
e
n
-
1
x
σ
n
2
λ
-
1
+
∑
t
≥
1
S
q
k
-
t
x
σ
1
e
1
⋯
x
σ
n
-
1
e
n
-
1
S
q
t
x
σ
n
2
λ
-
1
.For a given permutationσ
∈
S
(
n
) write (15)
P
d
n
=
P
d
,
σ
n
⨁
h
σ
λ
P
d
′
n
-
1
,where P
(
d
,
σ
)
(
n
) is the complement of h
σ
λ
(
P
d
′
(
n
-
1
)
) in P
d
(
n
) and let π
σ denote the projection(16)
π
σ
:
P
d
n
⟶
h
σ
λ
P
d
′
n
-
1of P
d
(
n
) onto h
σ
λ
(
P
d
′
(
n
-
1
)
) with respect to this splitting.The assignment,(17)
h
σ
λ
:
S
q
k
x
1
e
1
⋯
x
n
-
1
e
n
-
1
⟼
S
q
k
x
σ
1
e
1
⋯
x
σ
n
-
1
e
n
-
1
x
σ
n
2
λ
-
1
,gives a bijection between(18)
A
+
P
n
-
1
∩
P
d
′
n
-
1
,
π
σ
A
+
P
n
∩
P
d
n
.Thus, for each σ
∈
S
(
n
), we can think of h
σ
λ as having an inverse(19)
g
σ
λ
:
π
σ
P
d
n
⟶
P
d
′
n
-
1
.Now, letu
∈
Q
d
′
(
n
-
1
). Then, h
σ
λ
(
u
) is an element of h
σ
λ
(
P
d
′
(
n
-
1
)
). Thus, any hit polynomial that has h
σ
λ
(
u
) as a term is a sum of polynomial expressions of the form (14). Proceeding by contraposition, suppose that h
σ
λ
(
u
) is inadmissible; that is, it is not an element of Q
d
(
n
). Then we can find a polynomial in which modulo image of the action, h
σ
λ
(
u
), is a sum of monomials of the form x
σ
(
1
)
m
1
⋯
x
σ
(
n
-
1
)
m
n
-
1
x
σ
(
n
)
2
λ
-
1 plus error terms S
q
k
-
t
x
σ
(
1
)
e
1
⋯
x
σ
(
n
-
1
)
e
n
-
1
S
q
t
x
σ
(
n
)
2
λ
-
1 all of order lower than that of h
σ
λ
(
u
). Since in this event the error terms are of no consequence (in terms of determining whether u is admissible or not) we may as well restrict ourselves to that part of the polynomial which consists of terms belonging to π
σ
(
P
d
(
n
)
), that is, to the image of h
σ
λ. But the mapping h
σ
λ is injective and preserves the order of monomials, so by considering its inverse g
σ
λ we see that u must also be inadmissible.Proof of Corollary5.
We must show that ifλ is fixed and is greater than λ
1, then for any permutation σ
∈
S
(
n
) we have u
∈
Q
d
′
(
n
-
1
) if and only if h
σ
λ
(
u
)
∈
Q
d
(
n
), or, by virtue of Theorem 4, that we have u admissible whenever h
σ
λ
(
u
) is admissible. But any hit polynomial which has h
σ
λ
(
u
) as a term is generated by polynomial expressions of the form (14) so it suffices to show that an error term in each such polynomial expression,(20)
S
q
k
-
t
x
σ
1
e
1
⋯
x
σ
n
-
1
e
n
-
1
S
q
t
x
σ
n
2
λ
-
1
,is either hit or of lower weight order than that of h
σ
λ
(
u
)
. But if v
=
x
1
2
λ
1
-
1
⋯
x
n
-
2
2
λ
n
-
2
-
1 is the minimal spike of degree d
′, λ
>
λ
1, and b is a term in a polynomial expression of the form (20), then d
l
(
b
)
>
d
l
(
v
) for l
=
λ
1 and so by Theorem 10
b is hit. The proof is then completed by noting that if λ
>
λ
1, then the subspace of P
d
(
n
) spanned by ⋃
σ
‍
h
σ
λ
(
P
d
′
(
n
-
1
)
) is a direct summand ⨁
σ
h
σ
λ
(
P
d
′
(
n
-
1
)
) of P
d
(
n
) isomorphic to ⨁
j
=
1
n
(
P
d
′
(
n
-
1
)
) and that the respective subspace of Q
d
(
n
) with basis, the union over all σ
∈
S
(
n
) of the sets {
h
σ
λ
(
u
)
∣
u
∈
Q
d
′
(
n
-
1
)
} inherits this splitting.


## 4. Application of Theorem4

We have already applied Theorem4 to derive the table of Theorem 6. The table can be extended to all d
≥
0 but in this work we have limited ourselves to d
≤
30. The purpose of this restriction is to show that, besides the cases, 1
≤
d
≤
9, we also have d
i
m
(
Q
d
(
5
)
)
=
L
B
(
5
,
d
), when d
=
12 and d
=
28. We also compute d
i
m
(
Q
d
(
5
)
) for d
=
10 and d
=
11
.To prove all this we require some preliminary observations. For eachr
,
1
≤
r
≤
n, let (21)
X
r
=
S
p
a
n
x
1
m
1
⋯
x
r
m
r
∈
P
r
∣
m
1
m
2
⋯
m
r
≠
0
.Then X
(
r
) is an A-submodule of P
(
r
). Let (22)
W
r
=
X
r
A
+
X
r
.Then we have a direct sum decomposition (23)
Q
n
≅
⨁
r
=
1
n
⨁
k
=
1
n
r
W
r
.Thus, we have the following.For any integerd
>
0,(24)
d
i
m
Q
d
n
=
∑
r
=
1
n
n
r
d
i
m
W
d
r
.If we were to computed
i
m
(
Q
d
(
n
)
) by an inductive procedure on the number of variables, then we would only be required to compute d
i
m
(
W
d
(
n
)
) in proceeding from the case n
-
1 to n. Apart from spikes all subsequent references to monomials will mean those belonging to X
(
n
) and, by virtue of Singer’s Theorem 9, we will assume they are of weight order greater than or equal to that of the minimal spike. We note also that any variation between d
i
m
(
Q
d
(
n
)
) and L
B
(
n
,
d
) is dependant only on X
d
(
n
), since, for any permutation σ
∈
S
(
n
), whenever u
∈
Q
d
(
n
-
1
), then choosing λ
=
0, we have h
σ
λ
(
u
)
∈
Q
d
(
n
).Thatd
i
m
(
Q
d
(
5
)
)
=
L
B
(
5
,
d
), 1
≤
d
≤
9, d
=
12 and d
=
28 will be consequent from the following results.Letd
≥
0 and suppose that α
(
d
+
n
)
≤
n. Suppose there is a spike v
=
x
1
2
λ
1
-
1
⋯
x
n
-
1
2
λ
n
-
1
-
1
∈
P
d
(
n
) for which w
0
(
v
)
=
n
-
1. We will say that a monomial a
=
x
1
m
1
⋯
x
n
m
n
∈
X
d
(
n
) of weight order greater than or equal to that of v isassociatedwith v if, for some pair (
i
,
j
), we have m
j
=
2
λ
i
-
1 and isstrongly associated with v if a is associated with v and if in addition m
j
≤
2
n
-
2
-
1 for some j. Note that a monomial a is associated with a spike v if a
=
h
j
λ
i
(
u
) for some λ
i and some monomial u
∈
X
(
n
-
1
).There are cases, dependent ond, where every monomial in X
d
(
n
) is strongly associated with some spike v of degree d
. In particular we have the following.Lemma 11.
Letd be an integer for which α
(
n
+
d
)
≤
n and let a
∈
X
d
(
n
) be a monomial. Suppose that for each spike v
=
x
1
2
λ
1
-
1
⋯
x
n
-
1
2
λ
n
-
1
-
1 of degree d we have λ
n
-
1
>
0, λ
1
≤
n
-
1, λ
i
-
λ
i
+
1
≤
1, and λ
1
-
λ
n
-
1
≤
2. Then a is strongly associated with at least one of the spikes of degree d
.Proof.
Suppose that every spike of degreed satisfies the hypothesis of the lemma. We first show that if a
=
x
1
m
1
⋯
x
n
m
n
∈
X
d
(
n
) with w
(
a
)
=
w
(
v
) for some spike v then a is strongly associated with v
. To see this represents the spike v in matrix form. Without loss of generality assume that row 1 of the matrix has zero entries so that row i
+
1, column j has entry α
j
(
2
λ
i
-
1
), 1
≤
i
≤
n
-
1. Note that the columns are indexed by j, 0
≤
j
≤
n
-
2. The matrix form of a monomial a of the same weight order as v may be obtained from the matrix form of v by moving ones from the n
-
1 rows of v vertically upwards or downwards to a different row including row 1
. In the worst case scenario we can move at most n
-
2 ones to row 1 from n
-
2 distinct rows of the matrix form of v. Thus, we must be left with at least one row with consecutive ones; that is, for any monomial obtained this way we must have m
j
=
2
λ
i
-
1 for some j
. Since λ
1
≤
n
-
1, we must have m
j
≤
2
n
-
2
-
1 for some j
. Suppose that w
(
a
)
≠
w
(
v
) for any spike v of degree d
. It is easy to see that this is possible only if there is a spike v for which λ
1
-
λ
n
-
1
=
2
. We claim that we must have m
j
=
2
λ
1
-
1
-
1 for some j
. In this case a monomial of higher order than v can only be obtained by applying the splitting 2
λ
+
2
λ
=
2
λ
+
1 to 2
λ
i where λ
i
=
λ
1. The proof is then completed by noting that λ
1
-
1
≤
n
-
2.We are now in a position to show thatd
i
m
Q
d
(
5
)
=
L
B
5
,
d when 1
≤
d
≤
9, d
=
12, and d
=
28. We will make use of the following observation.Suppose thatd is an integer for which the minimal spike v of degree d satisfies the condition w
o
v
=
n
-
1. For each t, 1
≤
t
≤
n, let f
t
:
P
d
(
n
)
→
P
2
d
+
n
-
1
(
n
) be the linear mapping given on monomials by (25)
f
t
x
1
e
1
⋯
x
n
e
n
=
x
1
2
e
1
+
1
⋯
x
t
-
1
2
e
t
-
1
+
1
x
t
2
e
t
x
t
+
1
2
e
t
+
1
+
1
⋯
x
n
2
e
n
+
1
.Then, ignoring monomials of order lower than that of the minimal spike of degree 2
d
+
n
-
1, it is easy to see, by virtue of (10), that(26)
A
+
P
n
∩
P
d
′
n
=
S
q
1
P
n
∩
P
d
′
n
⊕
⨁
t
=
1
n
f
t
A
+
P
n
∩
P
d
n
,where d
′
=
2
d
+
n
-
1 and S
q
1
(
P
(
n
)
) is the image of the action of S
q
1 on P
(
n
). In other words the image of S
q
k, k
>
1, is given by ⨁
t
=
1
n
f
t
A
+
P
(
n
)
∩
P
d
(
n
).Whenn
=
5 then the spikes x
1
7
x
2
7
x
3
7
x
4
7, x
1
15
x
2
7
x
3
3
x
4
3, of degree 28, satisfy the hypothesis of Lemma 11 and each monomial of this degree is of the weight order of one of these spikes. Let a
=
x
1
m
1
x
2
m
2
x
3
m
3
x
4
m
4
x
5
m
5
∈
X
28
(
5
). Then m
j
=
3,7
,
15 for some j or equivalently we can find σ
∈
S
(
5
) and a monomial u
∈
X
(
4
) such that a
=
h
σ
λ
(
u
) where λ
=
2,3, or 4. To prove that d
i
m
(
Q
28
(
5
)
)
=
L
B
(
5,28
), it is sufficient to show that whenever a is admissible then we can find an admissible monomial u
∈
X
(
4
) and a permutation σ
∈
S
(
5
) such that h
σ
λ
(
u
)
=
a (λ
=
2,3, or 4). Since a is of the form h
σ
λ
(
u
) for some u
∈
X
(
4
) it is sufficient to show that if f is a hit polynomial that has a as a term, then the error terms are either hit or of lower order than a
. It is easy to see that if a is of the weight order of the spike x
1
7
x
2
7
x
3
7
x
4
7 and S
q
k
(
w
) is a hit polynomial that has a as a term, then all the error terms are of lower weight order than that of a
. Clearly this suffices to show that every admissible monomial of the weight order of the spike x
1
7
x
2
7
x
3
7
x
4
7 is of the form h
σ
3
(
u
) for some admissible monomial u
∈
X
(
4
).That leaves us with the case wherea has weight order that of the minimal spike v
=
x
1
15
x
2
7
x
3
3
x
4
3
. In this case we can find j, 1
≤
j
≤
5, such that m
j is equal to at least one 3,7
,
15
. If m
j
=
15 for some j, then the hypothesis of Corollary 5 is satisfied if we take d
′
=
13 and λ
=
4; hence, if a is admissible then a
=
h
σ
4
(
u
) for some admissible monomial u
∈
X
13
(
4
)
. We may therefore suppose that m
j
≠
15 for any j
. We show that in all cases each element of A
+
P
(
5
)
∩
P
28
(
5
) that has a as a term has error terms that consist of hit monomials. When n
=
5, then 2
d
+
n
-
1
=
28 implies d
=
12 and the minimal spike when d
=
12 is v
=
x
1
7
x
2
3
x
3
x
4
. Since w
0
(
v
)
=
4 we have (27)
A
+
P
5
∩
P
28
5
=
S
q
1
P
5
∩
P
28
n
⊕
⨁
t
=
1
5
f
t
A
+
P
5
∩
P
12
5
.Since f
t preserves the order of monomials for each t we may, by induction, assume that whenever a
∈
X
12
(
5
) is admissible then we can find an admissible monomial u
∈
X
(
4
) and a permutation σ
∈
S
(
5
) such that h
σ
λ
(
u
)
=
a (λ
=
1,2, or 3). Thus, we need only to consider the case when a is a term in S
q
1
(
P
(
5
)
)
. Since w
0
(
a
)
=
w
1
(
a
)
=
4, we see that the error part of a hit polynomial that has a as a term consists of monomials which are hit unless if in the matrix form of a we have α
o
(
m
i
)
=
α
1
(
m
i
)
=
0 for some i
. In this case a is a permutation representative of one of x
1
7
x
2
3
x
3
3
x
4
3
x
5
12, x
1
7
x
2
3
x
3
3
x
4
4
x
5
11, or x
1
7
x
2
7
x
3
3
x
4
3
x
5
8
. Consider the cases where a is a permutation representative of one of x
1
7
x
2
3
x
3
3
x
4
3
x
5
12 or x
1
7
x
2
3
x
3
3
x
4
4
x
5
11
. If we take m
j
=
7, then the error terms in a hit polynomial, S
q
1
(
w
), that has a
=
h
j
3
(
u
) as a term are hit monomials. Finally it is easy to check that the case where S
q
1
(
P
(
5
)
) has an element which has a permutation representative of x
1
7
x
2
7
x
3
3
x
4
3
x
5
8 as a term is when such an element is a sum of hit polynomials in P
28
(
5
)
. Such action makes no contribution (in terms of adding or deducting) to the basis of Q
28
(
5
)
. A similar argument applies to monomials of weight order of the spike x
1
7
x
2
3
x
3
x
4 of degree 12 which is also the base of our induction in the case d
=
28
.We now show thatd
i
m
(
Q
d
(
5
)
)
=
L
B
(
5
,
d
) when 1
≤
d
≤
9
. There is nothing to show in the cases 1
≤
d
≤
4 as d
i
m
(
Q
d
(
5
)
)
=
L
B
(
5
,
d
) whenever X
d
(
5
)
=
0
. In the case d
=
5 the spike v
=
x
1
x
2
x
3
x
4
x
5 is the only element in W
5
(
5
)
. In the cases d
=
6,7 each monomial in X
d
(
5
) is associated with one of the respective spikes x
1
3
x
2
x
3
x
4 and x
1
3
x
2
3
x
3, x
1
3
x
2
x
3
x
4
x
5 and it is easy to verify that those of the form h
σ
λ
(
u
) for some u
∈
W
(
4
) form basis for W
6
(
5
) and W
7
(
5
), respectively. Similarly in the cases d
=
8,9
.Ifd
=
10, then one can verify that in addition to the elements obtained from W
(
4
) via the mappings h
σ
λ the five monomials x
1
x
2
x
3
2
x
4
2
x
5
4, x
1
x
2
x
3
2
x
4
4
x
5
2, x
1
x
2
2
x
3
x
4
2
x
5
4, x
1
x
2
2
x
3
x
4
4
x
5
2, and x
1
x
2
2
x
3
4
x
4
x
5
2 are also admissible so that d
i
m
(
Q
d
(
5
)
)
=
280
. If d
=
11, then it is easy to show that in addition to the monomials generated from W
(
4
) the two monomials x
1
x
2
x
3
2
x
4
2
x
5
5
,
x
1
x
2
2
x
3
x
4
2
x
5
5 also belong to W
11
(
5
) or are also admissible so that d
i
m
(
Q
11
(
5
)
)
=
315
.Finally we note that ifv
∈
P
d
(
n
) is a spike that satisfies the hypothesis of Lemma 11 and is one for which λ
1
-
λ
n
-
1
≤
1, then v is the maximal spike and w
(
v
)
≥
w
(
a
) for every monomial a
∈
P
d
(
n
)
. In these cases monomials a
∈
X
d
(
n
) for which w
(
a
)
=
w
(
v
) are strongly associated with v
. Even though there might be other spikes of degree d which do not satisfy the hypothesis of the lemma, one may all the same obtain partial results for d
i
m
(
Q
d
(
n
)
) by applying an inductive procedure on n as above to compute the dimension of the subspace of Q
d
(
n
) generated by monomials in the weight class of v
.In [19] one such case is considered. The spike v
=
x
1
2
λ
-
1
⋯
x
n
-
1
2
λ
-
1 of degree d
λ
=
n
-
1
2
λ
-
1 satisfies the hypothesis of Lemma 11 and is one for which λ
1
-
λ
n
-
1
=
0
. It is shown that if λ
≤
n
-
1, then the dimension of the subspace of Q
d
λ
n generated by monomials in the weight class of v is equal to n
+
∑
q
=
2
λ
(
n
q
) and that if λ
≥
n, then the dimension of the respective subspace is 2
n
-
1
.



---

*Source: 235806-2015-08-13.xml*