J. Korean Math. Soc.
A -GENERATORS
FOR THE POLYNOMIAL ALGEBRA OF FIVE
VARIABLES IN DEGREE
5(2t − 1) + 6.2t
°ng Vã Phóc Let Ps := F2 [x1 , x2 , . . . , xs ] = Ln>0 (Ps )n be the polynomial algebra viewed as a graded left module over the mod 2 Steenrod algebra, A . The grading is by the degree of the homogeneous terms (Ps )n of degree n in the variables x1 , x2 , . . . , xs of grading 1. We are interested in the hit problem, set up by F.P. Peterson, of finding a minimal system of generators for A -module Ps . Equivalently, we want to find a basis for the F2 -graded vector space F2 ⊗A Ps . In this paper, we explicitly compute the hit problem in the case s = 5 and the degree n = 5(2t − 1) + 6.2t with t an arbitrary positive integer. Abstract.
1. Introduction Denote by
RP ∞
the infinite real projective space and by
F2
the prime field of two ele-
ments. We study the hit problem of determining a minimal set of generators for
F2 -graded
algebra
Ps := F2 [x1 , x2 , . . . , xs ] = H ∗ ((RP ∞ )×s ; F2 ) A . Equivalently, we want F2 ⊗A Ps in each degree n. Such a basis may be represented by a list of monomials of degree n. Here Ps is the polynomial algebra in s generators x1 , x2 , . . . , xs , each of degree 1. The action of A on Ps is determined by the i elementary properties of the Steenrod squares Sq , in grading i > 0, and subject to the
as an unstable left module over the mod 2 Steenrod algebra, to find a basis of the
F2 -vector
space
Cartan formula (see [12]). The hit problem was first studied by Peterson [3, 4], Wood [17], Singer [10], and Priddy [8], who showed its relation to several classical problems in the homotopy theory. Several aspects of the hit problem were then investigated by many authors (see Crabb-Hubbuck [1], Kameko [2], Repka-Selick [9], Singer [11], Wood [18] and others). Let
GLs := GL(s, F2 ) be the general Ps by matrix substitution.
linear group of rank
s
F2 . This group acts A and GLs upon Ps commute with each other, there is an inherited action of GLs on F2 ⊗A Ps . For a non-negative integer n, denote by (F2 ⊗A Ps )n the subspace of F2 ⊗A Ps consisting of GL all the classes represented by the homogeneous polynomials in (Ps )n and by (F2 ⊗A Ps )n s the subspace of (F2 ⊗A Ps )n consisting of all the GLs -invariant classes of degree n. One of
naturally on
over
Since the two actions of
our main tools for studying the hit problem is the so-called Kameko's squaring operation
Sq 0 : F2 ⊗GLs P Hn ((RP ∞ )×s ; F2 ) −→ F2 ⊗GLs P H2n+s ((RP ∞ )×s ; F2 ), P Hn ((RP ∞ )×s ; F2 ) denotes the primitive subspace consisting of all elements in the ∞ ×s homology group Hn ((RP ) ; F2 ) that are annihilated by every positive degree operation GLs 0 0 GL in A . The dual of Sq is the homomorphism Sq∗ : (F2 ⊗A Ps )2n+s −→ (F2 ⊗A Ps )n s . This 0 f ∗ : (F2 ⊗A Ps )2n+s → (F2 ⊗A Ps )n . homomorphism is given by the F2 GLs -homomorphism Sq The latter is given by the F2 -linear map ψ : Ps −→ Ps , determined by ( u if x = x1 x2 . . . xs u2 , ψ(x) = 0 otherwise, where
for any monomial and
ψSq
2i+1
=0
x ∈ Ps . for any
The map
ψ
is not an
A -homomorphism.
However,
ψSq 2i = Sq i ψ
i > 0.
2010 Mathematics Subject Classification. Primary 55S10, 55S05, 55T15. Steenrod squares, Peterson hit problem, Polynomial algebra. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2017.05. Key words and phrases.
1
2
.V. PHÓC From a result of Wood [17], the hit problem is reduced to the case of the degree of the
form
n = r(2t − 1) + 2t m,
(1.1) with
r, t, m
µ(m) < r 6 s (see [15]). Here, µ(m) is P Sum m= (2di − 1), where di > 0.
positive integers such that
smallest number
k
the
for which it is possible to write
16i6k The space case
s>4
F 2 ⊗ A Ps
was explicitly determined for the cases
s 6 4 (see [2, 3, 14, 15]).
The
is an open problem.
In the present paper, we study the hit problem of the degree n of the form (1.1) for r = s = 5 and m = 6. More precisely, we explicitly compute the admissible monomials of t t degree n = 5(2 − 1) + 6.2 in P5 . The main result of the paper is the following.
Main Theorem.
566
for
t = 1,
n = 5(2t −1)+6.2t with t a positive integer. dim(F2 ⊗A P5 )n = 2130 for t > 2.
Let
and
Noting that the theorem has been proved in [6] for
Then,
dim(F2 ⊗A P5 )n =
t = 1.
The paper is divided into four sections and organized as follows. In Section 2, we recall some results related to the admissible monomials in
Ps ,
Singer's criterion on the hit mono-
mials and Kameko's homomorphism. Our main result will be proved in Section 3. Finally, in Section 4, we list all the admissible monomials of degrees
18, 39
in
P4
and
P5 .
2. Some preliminaries on the hit problem In this section, we review some needed information from Kameko [2], Singer [11], PhócSum [5], and Sum [13, 15] which will be used in the next section.
Notation 2.1.
Throught the paper, we use the following notations.
Ns
= {1, 2, . . . , s},
XI
= X{i1 ,i2 ,...,ir } =
Y
xi , I = {i1 , i2 , . . . , ir } ⊆ Ns .
i∈Ns \I In particular,
XNs
=
X∅
= x 1 x 2 . . . xs ,
X{i} Let
1,
= x1 . . . x ˆi . . . xs , i = 1, 2, . . . , s.
αj (n) denote the j -th coefficients in dyadic expansion of a non-negative integer n. n = α0 (n)20 + α1 (n)21 + · · · + αj (n)2j + · · · for αj (n) ∈ {0, 1} with j > 0. x = xa1 1 xa2 2 . . . xas s ∈ Ps . We set
This means Let
Ij (x) = {i ∈ Ns : αj (ai ) = 0}, for
j > 0.
Then,
Definition 2.2
x=
j
Q
j>0
(Kameko [2]).
sequences associated with
P
ωj (x) = sequence ω(x) is
where
XI2j (x) .
16i6s
x
For a monomial
x = xa1 1 xa2 2 . . . xas s ∈ Ps ,
we define two
by
ω(x)
=
(ω1 (x), ω2 (x), . . . , ωj (x), . . .),
σ(x)
=
(a1 , a2 , . . . , as ),
αj−1 (ai ) = degXIj−1 (x) , j > 1.
ωj (x) 6 s for all j. The σ(x) called the exponent
Noting that
x and x. Let ω = (ω1 , ω2 , . . . , ωj , . . .) be a sequence of non-negative integers. The sequence is called the weight vector if ωi = 0 for i 0. The sets of all the weight vectors and the exponent called the weight vector of the monomial
vector of the monomial
vectors are given the left lexicographical order. For a weight vector
ω,
we define
deg ω =
P
2i−1 ωi .
Denote by
Ps (ω)
the subspace of
Ps
i>1 spanned by all monomials subspace of
Ps
x ∈ Ps
deg x = deg ω, ω(x) 6 ω, x ∈ Ps such that ω(x) < ω.
such that
spanned by all monomials
and by
Ps− (ω)
the
THE HIT PROBLEM FOR THE POLYNOMIAL ALGEBRA OF FIVE VARIABLES Definition 2.3
3
(Kameko [2], Sum [15]). Let
ω be a weight vector and f, g two homogeneous Ps . f ≡ g if and only if (f + g) ∈ A + · Ps . If f ≡ 0 then f is called hit. f ≡ω g if and only if (f + g) ∈ A + · Ps + Ps− (ω). If f ≡ω 0 then f is called ω -hit. + Here A denotes the augmentation ideal of A .
polynomials of the same degree in (i) (ii)
≡
Obviously, the relations of
Ps (ω)
≡ω are equivalence ones. ≡ω . Then we have
and
Denote by
QPs (ω)
the quotient
by the equivalence relation
Furthermore,
QPs (ω) = Ps (ω)/((A + · Ps ∩ Ps (ω)) + Ps− (ω)). ∼ L QPs (ω) QPs (ω) is an GLs -module and (F2 ⊗A Ps )n =
(see Sum [15]).
deg ω=n
Definition 2.4
(Kameko [2]). Let
x, y
be monomials in
Ps .
We say that
x 0}
It is easy to see that if opposite is not true.
x
if and only
16j62r −1
and suitable polynomials
hj ∈ Ps .
is strictly inadmissible, then it is inadmissible. In general, the
For instance, the monomial
x = x1 x22 x23 x24 x25 x6 ∈ P6
is inadmissible,
but it is not strictly inadmissible.
Theorem 2.7
0
(Kameko [2], Sum [13]). Let x, y and u be monomials in Ps such that ωi (x) = i > r > 0, ωt (u) 6= 0 and ωi (u) = 0 for i > t > 0. 2r (i) If u is inadmissible, then xu is also inadmissible. 2t (ii) If u is strictly inadmissible, then uy is also strictly inadmissible.
for
For a positive integer
n,
we set
X µ(n) = min k ∈ N : n = (2di − 1), di > 0 . 16i6k
Theorem 2.8
(Kameko [2]). If
0 f Sq ∗
µ(2n + s) = s,
then the homomorphism
g0 )(s,2n+s) : (F2 ⊗A Ps )2n+s −→ (F2 ⊗A Ps )n = (Sq ∗
is an isomorphism of the
F2 GLs -modules.
We now recall a result of Singer [11] on the hit monomials in
Definition 2.9.
dj = 0
for
j > r,
Noting that if
z = xb11 xb22 . . . xbss is called a spike if bi = 2di − 1 for di a i = 1, 2, . . . , s. If z is a spike with d1 > d2 > . . . > dr−1 > dr > 0
A monomial
non-negative integer and and
Ps .
then it is called a minimal spike.
µ(n) 6 s,
then there exists uniquely a minimal spike of degree
n
in
Ps
(see
Singer [11]).
Lemma 2.10
(see [5]).
All the spikes in
Ps
are admissible and their weight vectors are
ω = (ω1 , ω2 , . . .) ω(z) = ω.
weakly decreasing. Furthermore, if a weight vector
ω1 6 s,
then there is a spike
z
in
Ps
such that
The following is a criterion for the hit monomials in
Ps .
is weakly decreasing and
4
.V. PHÓC
Theorem 2.11
s.
z
Let
(Singer [11]). Suppose that
be the minimal spike of degree
n
in
x ∈ Ps is a monomial of degree n, where µ(n) 6 Ps . If ω(x) < ω(z), then x is hit.
We set
Ps0
Then
and
{x = xa1 1 xa2 2 . . . xas s ∈ Ps | a1 a2 . . . as = 0},
Ps0
=
Span
Ps+
=
Span
Ps+
{x = xa1 1 xa2 2 . . . xas s ∈ Ps | a1 a2 . . . as > 0}.
A -submodules F2 -vector spaces
are the
decomposition of the
Ps .
of
F2 ⊗A Ps = (F2 ⊗A Ps0 )
Furthermore, we have a direct summand
M
(F2 ⊗A Ps+ ).
We end this section by recalling some notations and definitions in Sum [15]. We denote
Ns := {(i; I) | I = (i1 , i2 , . . . , ir ), 1 6 i < i1 < i2 < . . . < ir 6 s, 0 6 r < s}. Then,
|Ns | = 2s − 1.
Here, by convention,
I = ∅,
if
r = 0.
Denote by
r = `(I)
the length of
I. (Sum [15]). Let (i; I) ∈ Ns , r = `(I), and let u be an integer with 1 6 u 6 r. as−1 x = xa1 1 xa2 2 . . . xs−1 ∈ Ps−1 is said to be u-compatible with (i; I) if all of the
Definition 2.12 A monomial
following hold: (i) (ii) (iii) (iv)
of
ai1 −1 = ai2 −1 = . . . = ai(u−1) −1 = 2r − 1, aiu −1 > 2r − 1, αr−t (aiu −1 ) = 1, ∀t, 1 6 t 6 u, αr−t (ait −1 ) = 1, ∀t, u < t 6 r. can be u-compatible with a given (i; I) ∈ Ns for at most one value 1-compatible with (i; ∅). the homomorphism τi : Ps−1 → Ps of algebras by substituting xj if 1 6 j < i, τi (xj ) = xj+1 if i 6 j < s.
Clearly, a monomial
x
u.
is
By convention,
For
1 6 i 6 s,
x
define
This map is also a monomorphism of
Definition 2.13
A -module.
(i; I) ∈ Ns , 0 < r < s. Denote Y r−1 r−2 r−u r−t x2it = xi2u +2 +···+2
(Sum [15]). Let
x(I,u)
u 2.
in
P5 .
It is easy to see that
µ(5(2t − 1) + 6.2t ) = 5
for any
So, Theorem 2.8 implies that
g0 )t−2 t (Sq ∗ (5,5(2 −1)+6.2t ) : (F2 ⊗A P5 )5(2t −1)+6.2t −→ (F2 ⊗A P5 )5(22 −1)+6.22 is an isomorphism of
(F2 ⊗A P5 )5(2t −1)+6.2t 3.1. For
The case
GL5 -modules for t = 1, 2.
Consequently,
t > 2.
Therefore, we need only to compute
t=1
t = 1, 5(2t − 1) + 6.2t = 17.
Theorem 3.1.1
for every
The space
(see [6]). There exist exactly
(F2 ⊗A P5 )17 566
has been computed in [6].
admissible monomials of degree
17
in
P5 .
dim(F2 ⊗A P5 )17 = 566.
3.2. The admissible monomials of degree 18 in To compute mials of degree
(F2 ⊗A P5 )5(2t −1)+6.2t 18 in P5 .
Lemma 3.2.1.
If
x
for
t = 2,
P5
we need to determine the admissible mono-
is an admissible monomial of degree
18
in
P5 ,
then
ω(x)
is one of the
following sequences:
(2, 2, 1, 1), (2, 2, 3), (2, 4, 2), (4, 1, 1, 1), (4, 1, 3), (4, 3, 2). 3 z = x15 1 x2
is the minimal spike of degree 18 in P5 and ω(z) = (2, 2, 1, 1). [x] 6= 0, by Theorem 2.11, either ω1 (x) = 2 or ω1 (x) = 4. If ω1 (x) = 2, then x = X{i,j,`} y 2 with y a monomial of degree 8 in P5 and 1 6 i < j < ` 6 5. Since x is admissible, by Theorem 2.7, y is admissible. According to T½n [16], either ω(y) = (2, 1, 1) or ω(y) = (2, 3) 2 or ω(y) = (4, 2). If ω1 (x) = 4, then x = X{i} u with u is an admissible monomial of degree 7 in P5 and 1 6 i 6 5. By T½n [16], ω(u) is one of the sequences (1, 1, 1), (1, 3), and (3, 2). The lemma is proved. L 0 Recall that (F2 ⊗A P5 )18 = (F2 ⊗A P5 )18 (F2 ⊗A P5+ )18 . According to Sum [15], |B4 (18)| = 126 (see Subsection 4.1). Then, by a simple computation, we get
Proof. Observe that Since
dim(F2 ⊗A P50 )18 = |Φ0 (B4 (18))| = 450. Now, from Lemma 3.2.1, we have
(F2 ⊗A P5+ )18
L L =L QP5+ (2, 2, 1, 1) L QP5+ (2, 2, 3) L QP5+ (2, 4, 2) + + QP5 (4, 1, 1, 1) QP5 (4, 1, 3) QP5+ (4, 3, 2).
6
.V. PHÓC
Proposition 3.2.2. (i) (ii)
(iii)
(iv) (v) (vi)
We have the following:
B5+ (2, 2, 1, 1) = Φ+ (B4 (2, 2, 1, 1)) ∪ {x1 x22 x43 x34 x85 , x1 x22 x43 x84 x35 , x1 x22 x43 x94 x25 , x31 x42 x3 x24 x85 }. B5+ (2, 2, 3) is the set of the following monomials: x1 x22 x43 x44 x75 , x1 x22 x43 x74 x45 , x1 x22 x73 x44 x45 , x1 x72 x23 x44 x45 , x71 x2 x23 x44 x45 , x1 x22 x43 x54 x65 , x1 x22 x53 x44 x65 , x1 x22 x53 x64 x45 , x1 x32 x43 x44 x65 , x1 x32 x43 x64 x45 , x1 x32 x63 x44 x45 , x31 x2 x43 x44 x65 , x31 x2 x43 x64 x45 , x31 x2 x63 x44 x45 , x31 x52 x23 x44 x45 . B5+ (2, 4, 2) is the set of the following monomials: x1 x22 x33 x64 x65 , x1 x32 x23 x64 x65 , x1 x32 x63 x24 x65 , x1 x32 x63 x64 x25 , x31 x2 x23 x64 x65 , x31 x2 x63 x24 x65 , x31 x2 x63 x64 x25 , x31 x52 x23 x24 x65 , x31 x52 x23 x64 x25 , x31 x52 x63 x24 x25 . + B5 (4, 1, 1, 1) = Φ+ (B4 (4, 1, 1, 1)) ∪ {x31 x52 x83 x4 x5 }. B5+ (4, 1, 3) = Φ+ (B4 (4, 1, 3)). B5+ (4, 3, 2) = Φ+ (B4 (4, 3, 2)) ∪ D, where D is the set of the following monomials: x31 x42 x3 x34 x75 , x31 x42 x3 x74 x35 , x31 x42 x33 x4 x75 , x31 x42 x33 x74 x5 , x31 x42 x73 x4 x35 , x31 x42 x73 x34 x5 , x31 x72 x43 x4 x35 , x31 x72 x43 x34 x5 , x71 x32 x43 x4 x35 , x71 x32 x43 x34 x5 , x31 x42 x33 x34 x55 , x31 x42 x33 x54 x35 .
By a simple computation, one gets
|B5+ (2, 2, 1, 1)| = 25, |B5+ (4, 1, 1, 1)| = 40, |B5+ (4, 1, 3)| = 10, |B5+ (4, 3, 2)| = 180. These sets are explicitly determined as in Subsection 4.2.
So, the following corollary is
immediate.
Corollary 3.2.3.
dim(F2 ⊗A P5 )18 = 730.
To prove Proposition 3.2.2, we need some lemmas. By a simple computation, we obtain the following.
Lemma 3.2.4.
The following monomials are strictly inadmissible:
x2i xj x3` , x2i xj x7` , x2i x5j x3` , x4i x3j x3` , x6i xj x3` , x2i xj x3` x4t , x2i xj x5` x2t , x2i xj x2` x2t x3m , i < j, 2 2 2 3 3 (ii) xi xj x` xt , xi xj x` xt xm , i < j < `, 6 2 2 6 xi xj x` xt , xi xj x` xt , xi x2j x2` x5t , xi x2j x2` xt x4m , j < ` < t, 2 5 6 2 2 2 4 (iii) xi xj x` xt , xi xj x` xt xm , xi xj x` xt xm , i < j < ` < t, 6 7 2 6 2 2 2 3 3 3 3 2 2 3 3 (iv) xi xj x xt , xi xj x xt xm , x1 x2 x3 x4 x5 , xi xj x xt xm , x1 x2 x3 x4 x5 . ` ` ` Here (i, j, `, t, m) is a permutation of (1, 2, 3, 4, 5). (i)
Lemma 3.2.5.
x1 x6j x3` x6t x2m
If
(j, `, t, m) is a permutation of (2, 3, 4, 5) such that j < ` then the monomial
is strictly inadmissible.
Proof. We have
x1 x6j x3` x6t x2m
= Sq 1 (x21 x3j x3` x5t x4m + x21 x3j x5` x5t x2m + x21 x5j x3` x5t x2m ) +Sq 2 (x1 x5j x3` x5t x2m + x1 x3j x5` x5t x2m + x1 x3j x3` x5t x4m ) +x1 x3j x6` x6t x2m mod(P− 5 (2, 4, 2)).
This relation shows that
Lemma 3.2.6.
x1 x6j x3` x6t x2m
is strictly inadmissible. The lemma is proved.
The following monomials are strictly inadmissible:
x1 x2 x63 x84 x25 , x31 x32 x43 x44 x45 , x31 x42 x3 x4 x95 , x31 x42 x3 x64 x45 , x31 x42 x53 x54 x5 , x31 x42 x3 x84 x25 , x31 x42 x53 x44 x25 ,
x1 x62 x3 x84 x25 , x31 x42 x33 x44 x45 , x31 x42 x3 x94 x5 , x31 x42 x43 x4 x65 , x31 x52 x43 x4 x55 , x31 x42 x83 x4 x25 , x31 x52 x43 x24 x45 ,
x1 x62 x83 x4 x25 , x31 x42 x43 x34 x45 , x31 x42 x93 x4 x5 , x31 x42 x3 x54 x55 , x31 x52 x43 x54 x5 , x31 x42 x43 x54 x25 , x31 x52 x43 x44 x25 ,
x1 x62 x33 x44 x45 , x31 x42 x43 x44 x35 , x31 x42 x3 x44 x65 , x31 x42 x53 x4 x55 , x31 x52 x53 x44 x5 , x31 x42 x53 x24 x45 , x31 x12 2 x3 x4 x5 .
THE HIT PROBLEM FOR THE POLYNOMIAL ALGEBRA OF FIVE VARIABLES Proof. We prove the lemma for the monomials
u = x1 x2 x63 x84 x25
and
v = x31 x32 x43 x44 x45 .
7 The
others can be proven by a similar computation. We have
ω(u) = (2, 2, 1, 1) and ω(v) = (2, 2, 3). By a direct computation using the Cartan
formula, one gets
4 10 2 = x1 x2 x23 x64 x85 + x1 x2 x23 x84 x65 + x1 x2 x43 x24 x10 5 + x1 x2 x3 x4 x5 6 2 8 1 2 4 +x1 x2 x3 x4 x5 + Sq (h1 ) + Sq (h2 ) + Sq (h4 ) mod(P− 5 (2, 2, 1, 1)), where = x1 x42 x33 x44 x55 + x1 x42 x33 x54 x45 + x1 x42 x43 x34 x55 + x1 x42 x43 x54 x35 +x1 x42 x53 x34 x45 + x1 x42 x53 x44 x35 + x21 x2 x3 x54 x85 + x21 x2 x3 x84 x55 +x21 x2 x53 x44 x55 + x21 x2 x53 x54 x45 + x21 x42 x3 x54 x55 + x41 x42 x33 x34 x35 , = x1 x2 x63 x64 x25 + x1 x2 x53 x54 x45 + x1 x2 x3 x54 x85 + x1 x2 x3 x84 x55 +x1 x2 x53 x44 x55 + x1 x2 x63 x24 x65 + x1 x22 x3 x64 x65 + x1 x22 x23 x34 x85 +x1 x22 x23 x84 x35 + x1 x22 x33 x44 x65 + x1 x22 x33 x64 x45 + x1 x22 x63 x34 x45 +x1 x22 x63 x44 x35 + x1 x42 x3 x54 x55 + x1 x42 x23 x34 x65 + x1 x42 x23 x64 x35 +x21 x42 x33 x34 x45 + x21 x42 x33 x44 x35 + x21 x42 x43 x34 x35 , = x1 x2 x43 x24 x65 + x1 x2 x43 x64 x25 + x1 x22 x33 x44 x45 + x1 x22 x43 x34 x45 + x1 x22 x43 x44 x35 .
u h1
h2
h4
By Definition 2.6, we see that
v
=
u
is strictly inadmissible. Similarly, we obtain
1
Sq (x31 x52 x3 x44 x45 + x51 x32 x43 x4 x45 + x51 x52 x3 x24 x45 + x61 x52 x3 x4 x45 ) +Sq 2 (x31 x32 x43 x24 x45 + x31 x52 x23 x24 x45 + x31 x62 x3 x24 x45 + x61 x32 x23 x4 x45
This equality implies that
v
+ x61 x62 x3 x4 x25 ) mod(P− 5 (2, 2, 3)).
is strictly inadmissible. The lemma follows.
Proof of Proposition 3.2.2. We prove the first part of the proposition.
The others can be
proved by a similar computation. Set
B := Φ+ (B4+ (2, 2, 1, 1)) ∪ {x1 x22 x43 x34 x85 , x1 x22 x43 x84 x35 , x1 x22 x43 x94 x25 , x31 x42 x3 x24 x85 }. Suppose that x is an admissible monomial of degree 18 in P5 with ω(x) = (2, 2, 1, 1). Since ω1 (x) = 2, x = X{i,j,`} y 2 with y a monomial of degree 8 in P5 and 1 6 i < j < ` 6 5. On the other hand, x is admissible; hence by Theorem 2.7, y ∈ B5 (8).
u ∈ B5 (8), 1 6 i < j < ` 6 5, such that w which is given in one of Lemmas 3.2.4(i), (ii), (iii), 2 2r and 3.2.5(ii) such that X{i,j,`} u = wu1 with u1 ∈ P5 , and r = max{s ∈ Z : ωs (w) > 0}. 2 2 By Theorem 2.7, X{i,j,`} u is inadmissible. Since x = X{i,j,`} y admissible and y ∈ B5 (8), + one gets x ∈ B. This implies B5 (2, 2, 1, 1) ⊂ B. By a direct computation we see that for all
X{i,j,`} u2 6∈ B,
there is a monomial
[B] is linearly independent in (F2 ⊗A P5 )18 . We denote the classes ak , 1 6 k 6 25 (see Subsection 4.2). Suppose there is a linear relation X S= γk ak ≡ 0,
We now prove the set in
[B]
represented by
16k625
γk ∈ F2 , ∀k, 1 6 k 6 25. We express p(i;I) (S), (i; I) ∈ N5 in terms of the admissible (P4+ )18 . Computing directly from the relations p(1;2) (S) ≡ 0, p(1;3) (S) ≡ 0, p(2;3) (S) ≡ 0, and p(3;4) (S) ≡ 0, we obtain γk = 0, k = 1, 2, . . . , 25. This finishes the proof. where
monomials in
3.3. The case For
t = 2,
t=2
we have
5(2t − 1) + 6.2t = 39.
Since Kameko's homomorphism
g0 )(5,39) : (F2 ⊗A P5 )39 −→ (F2 ⊗A P5 )17 (Sq ∗ L g0 is an epimorphism, (F2 ⊗A P5 )39 = Ker((Sq∗ )(5,39) ) (F2 ⊗A P5 )17 . So, we need to compute g 0 Ker((Sq )(5,39) ). ∗
Set
ω(1) = (3, 2, 2, 1, 1), ω(2) = (3, 2, 2, 3), ω(3) = (3, 2, 4, 2), ω(4) = (3, 4, 1, 1, 1), ω(5) = (3, 4, 1, 3), ω(6) = (3, 4, 3, 2). Then we obtain the following.
Lemma 3.3.1. then
ω(x)
x is an admissible monomial of degree 39 in P5 of the sequences ω(k) , 1 6 k 6 6.
If
is one
and
g0 )(5,39) ), [x] ∈ Ker((Sq ∗
8
.V. PHÓC 7 z = x31 1 x2 x3
is the minimal spike of degree 39 in P5 and ω(z) = ω(1) . Since [x] 6= 0, by Theorem 2.11, either ω1 (x) = 3 or ω1 (x) = 5. If ω1 (x) = 5, 2 then x = X∅ y with y a monomial of degree 17 in P5 . Since x is admissible, by Theorem g0 2.7, y ∈ B5 (17). Hence, (Sq∗ )(5,39) ([x]) = [y] 6= [0]. This contradicts the face that [x] ∈ g0 )(5,39) ), so ω1 (x) = 3. Then, we have x = X{i,j} y 2 with 1 6 i < j 6 5 and y1 an Ker((Sq
Proof. It is easy to see that
∗
1
admissible monomial of degree
18
in
P5 .
Now, the lemma follows from Lemma 3.2.1.
Combining this lemma and a result of Sum [15], we get
g0 )(5,39) ) = (F2 ⊗A P 0 )39 Ker((Sq ∗ 5 g0 )(5,39) ) ∩ (F2 ⊗A P + )39 Ker((Sq ∗ 5
L
g0 )(5,39) ) ∩ (F2 ⊗A P + )39 , Ker((Sq ∗ 5 L = QP5+ (ω(k) ), 16k66
dim(F2 ⊗A P50 )39 = |Φ0 (B4 (39)| = 915. Noting that |B4 (39)| = 225 (see Subsection 4.3).
The set {[dk ] : 1 6 (F2 ⊗A P5+ )39 . Here the monomials dk
Proposition 3.3.2.
g0 )∩ k 6 649} is a basis of the F2 -vector space Ker(Sq ∗ = d39,k , with 1 6 k 6 649, are determined as in
Subsection 4.4.
We prove this proposition by proving some lemmas.
Lemma 3.3.3.
The set
QP5+ (ω(1) )
is spanned by the set
{[dk ]ω(1) : 1 6 k 6 485}.
The following lemma is proved by a direct computation.
Lemma 3.3.4.
If
(i, j, `, t, m) is a permutation of (1, 2, 3, 4, 5), then the following monomials
are strictly inadmissible: (i) (ii)
(iii)
x2i xj x` x3t , xi x2j x6` x7t x7m , xi x6j x2` x7t x7m , x2i xj x2` x3t x3m , x2i x2j x` x3t x3m , i < j < `. 3 4 9 7 3 12 7 3 12 3 5 7 x2i xj x` xt x2m , x1 x22 x23 x4 x5 , x3i x12 j x` xt , xi xj x` xt , xi xj x` xt , xi xj x` xt , x3i x5j x9` x6t , x3i x5j x8` x7t , x71 x82 x33 x54 , xi x6j x3` x6t x7m , xi x6j x6` x3t x7m , i < j < ` < t. x2i x3j x3` x3t .
Lemma 3.3.5.
The following monomials are strictly inadmissible:
x1 x22 x63 x34 x35 , x1 x62 x23 x34 x35 , x31 x42 x3 x84 x75 , x31 x42 x83 x4 x75 , x31 x42 x3 x94 x65 , x31 x42 x93 x4 x65 , 3 12 3 4 x31 x42 x33 x12 4 x 5 , x1 x 2 x 3 x 4 x 5 , 3 5 7 4 4 x1 x2 x3 x4 x5 , x31 x72 x43 x44 x55 , x71 x32 x43 x44 x55 , x71 x32 x43 x54 x45 , 8 6 4 3 5 x31 x42 x11 3 x4 x5 , x1 x2 x3 x4 x5 , 3 5 6 5 4 3 5 6 4 5 x1 x2 x3 x4 x5 , x1 x2 x3 x4 x5 ,
x1 x62 x33 x24 x35 , x31 x42 x83 x74 x5 , x31 x42 x93 x64 x5 , x31 x42 x73 x44 x55 , x31 x72 x43 x54 x45 , x71 x32 x53 x44 x45 , x31 x52 x83 x4 x65 , 6 x31 x12 2 x3 x4 x5 ,
x1 x62 x33 x34 x25 , x71 x82 x33 x44 x5 , x31 x42 x33 x44 x95 , x31 x42 x73 x54 x45 , x31 x72 x53 x44 x45 , x31 x42 x83 x34 x55 , x31 x52 x83 x64 x5 , 6 x31 x12 2 x3 x4 x5 .
The proof of this lemma is straightforward.
Lemma 3.3.6.
The following monomials are strictly inadmissible:
17 x1 x62 x33 x12 4 x5 , 5 16 x x1 x62 x11 x 3 4 5 , 17 4 x1 x72 x10 3 x4 x5 , 16 4 x x x71 x11 2 3 4 x5 , 17 4 x x x31 x42 x11 3 4 5, x31 x72 x83 x44 x17 5 , 5 x x31 x72 x83 x16 4 5, 3 7 12 x1 x2 x3 x4 x16 5 , 4 x x x31 x72 x24 3 4 5, 3 20 x x31 x12 2 3 x4 x5 ,
16 x1 x62 x33 x13 4 x5 , 16 5 x x1 x62 x11 x 3 4 5, 17 4 x71 x2 x10 3 x4 x5 , 4 16 x71 x11 x x 2 3 4 x5 , 4 x31 x52 x93 x18 4 x5 , 4 x31 x72 x83 x17 x 4 5, 7 3 8 5 16 x1 x2 x3 x4 x5 , 16 x31 x72 x12 3 x4 x5 , 7 3 24 x1 x2 x3 x4 x45 , 2 5 x31 x28 2 x3 x4 x5 ,
4 17 6 11 17 4 5 x1 x62 x11 x1 x62 x33 x24 4 x5 , 3 x4 x5 , x1 x2 x3 x4 x5 , 7 10 16 5 20 7 10 16 5 , x x x x , x x x x x x1 x62 x11 1 2 3 4 5 1 2 x3 x4 x5 , 3 4 5 16 4 7 11 16 4 7 11 16 4 x x , x x x x x , x x x x1 x72 x11 1 2 3 x4 x5 , 1 2 3 4 5 3 4 5 14 3 4 17 4 7 11 16 4 16 x , x x x x x x , x x x x x71 x11 x 1 2 3 4 x5 , 1 2 3 4 5 2 3 4 5 3 5 25 2 4 2 5 3 5 24 3 4 x , x x x x x x , x x31 x52 x24 1 x2 x3 x4 x5 , 3 4 5 1 2 3 4 5 7 3 8 17 4 3 7 8 5 16 x71 x32 x83 x44 x17 , x x x x x , x 5 1 2 3 4 5 1 x2 x3 x4 x5 , 7 3 8 16 5 3 7 8 20 x1 x2 x3 x4 x5 , x1 x2 x3 x4 x5 , x71 x32 x83 x20 4 x5 , 16 7 3 12 16 3 7 24 4 x71 x32 x12 x x , x x x x x , x x x x 3 4 5 1 2 3 4 5 1 2 3 4 x5 , 7 3 24 4 3 12 2 21 3 12 3 20 x1 x2 x3 x4 x5 , x1 x2 x3 x4 x5 , x1 x2 x3 x4 x5 , 3 4 3 4 x31 x28 x31 x28 2 x3 x4 x5 , 2 x3 x4 x5 .
THE HIT PROBLEM FOR THE POLYNOMIAL ALGEBRA OF FIVE VARIABLES Proof. We prove the lemma for the monomials
17 4 u = x31 x42 x11 3 x4 x5 ,
and
9
16 4 v = x71 x11 2 x3 x4 x5 .
The others can be proved by a similar computation. By a direct computation, we have
u
17 4 3 13 18 4 3 11 20 4 3 4 9 21 2 = x31 x22 x13 3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 2 9 21 4 3 12 21 2 3 10 21 4 4 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x31 x2 x93 x22 4 x5 2 13 21 2 2 11 21 4 2 7 25 4 3 4 7 17 8 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 8 3 4 11 16 5 3 4 8 21 3 2 4 13 17 3 +x31 x2 x73 x20 4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 2 4 11 17 5 4 18 13 3 4 14 17 3 13 6 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x22 x17 3 x4 x5 2 7 17 12 2 13 17 6 2 7 13 16 3 18 13 4 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 18 5 2 11 20 5 2 17 14 5 3 14 17 4 +x21 x2 x13 3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 2 13 18 5 18 14 5 14 18 5 17 14 6 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 18 6 3 16 14 5 3 13 16 6 3 13 18 4 +x1 x2 x13 3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 17 14 4 1 2 4 8 +x1 x2 x3 x4 x5 + Sq (u1 ) + Sq (u2 ) + Sq (u4 ) + Sq (u8 ) mod(P− 5 (ω(1) )),
u1
=
u2
=
u4
=
u8
=
Hence,
v
v1
u
is strictly inadmissible. By a similar computation, we obtain
8 3 7 4 17 8 3 7 5 16 8 3 7 8 17 4 = x31 x72 x3 x20 4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 7 16 5 8 3 7 17 4 8 3 9 2 21 4 4 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x31 x92 x43 x19 4 x5 3 9 18 5 4 3 9 20 3 4 3 11 20 4 3 11 16 5 4 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 17 4 4 3 12 19 4 3 12 17 3 4 3 14 17 4 +x31 x11 2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 14 17 4 3 17 4 7 8 3 17 6 5 8 3 17 8 7 4 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 9 6 4 3 18 13 4 3 18 5 5 8 3 18 5 9 4 +x31 x17 2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 18 13 4 3 19 12 4 3 19 4 5 8 5 8 4 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x31 x19 2 x3 x4 x5 3 20 7 8 3 20 11 4 3 20 3 5 8 3 20 3 9 4 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 2 9 4 3 21 3 4 8 3 21 3 8 4 3 21 4 3 8 +x31 x21 2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 24 3 5 4 3 24 5 3 4 3 25 3 4 4 4 3 4 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x31 x25 2 x3 x4 x5 4 7 19 8 4 7 17 3 8 4 11 19 4 4 11 17 3 4 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 7 8 4 19 11 4 4 19 3 5 8 4 19 3 9 4 +x41 x19 2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 4 21 3 3 8 4 25 3 3 4 5 3 2 21 8 5 3 2 25 4 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 5 3 5 10 16 5 3 4 11 16 8 5 3 3 24 4 +x51 x32 x33 x20 4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 5 7 2 9 16 5 7 18 8 5 3 8 7 16 5 3 9 6 16 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 18 4 5 11 2 5 16 8 5 11 5 7 18 +x51 x72 x33 x84 x16 5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 5 8 7 8 5 11 18 4 5 18 5 11 3 4 16 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x51 x18 2 x3 x4 x5 7 3 5 8 16 7 3 4 17 8 7 3 4 9 16 5 18 7 8 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 7 5 2 17 8 7 3 9 16 4 8 7 3 8 17 4 +x71 x32 x53 x16 4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 7 7 16 8 16 8 7 5 3 16 8 7 7 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x71 x82 x3 x74 x16 5 7 9 2 5 16 18 4 7 8 7 16 7 9 +x71 x82 x33 x54 x16 5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 7 9 18 4 7 9 3 16 4 4 7 9 3 4 16 +x71 x92 x23 x17 4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 1 2 4 8 7 11 16 4 +x1 x2 x3 x4 x5 + Sq (v1 ) + Sq (v2 ) + Sq (v4 ) + Sq (v8 ) mod(P− 5 (ω(1) )),
=
where
17 4 5 11 13 8 5 7 17 8 5 9 21 2 5 4 13 13 3 x51 x2 x11 3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 13 13 8 5 4 11 13 5 3 4 13 13 5 8 13 13 3 13 5 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x82 x11 3 x4 x5 3 11 18 5 3 11 14 9 13 14 9 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 , 17 4 3 11 18 4 6 7 21 2 3 2 9 21 2 3 10 21 2 x31 x22 x11 3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 9 22 2 2 11 21 2 2 7 25 2 3 11 13 8 3 14 8 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x11 3 x4 x5 3 2 7 17 8 3 7 18 8 3 4 11 13 6 6 4 11 13 3 3 4 11 14 5 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 14 3 3 4 14 13 3 2 8 11 13 3 2 4 11 17 3 8 11 14 3 +x31 x42 x13 3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 2 11 13 10 2 11 18 5 2 11 14 9 5 11 14 6 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 , 7 13 4 4 7 21 2 3 2 11 14 5 4 4 11 13 3 x10 1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 14 13 4 3 2 11 16 3 4 14 13 3 13 6 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x42 x11 3 x4 x5 2 13 13 6 2 7 13 12 2 13 14 5 14 14 5 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 14 6 3 12 14 5 3 11 14 6 3 13 14 4 +x1 x2 x13 3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 , 3 6 7 13 4 x31 x42 x83 x13 4 x5 + x1 x2 x3 x4 x5 .
8 3 7 17 3 8 3 11 19 4 x31 x72 x3 x19 4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + 3 19 7 8 3 19 11 4 3 19 3 5 8 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 9 4 3 21 3 3 8 3 25 3 3 4 +x31 x19 2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 7 13 9 8 7 13 5 5 8 7 13 9 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x85 ,
17 3 4 x31 x11 2 x3 x4 x5
where
10
.V. PHÓC
Hence,
v
v2
8 3 7 9 10 8 3 11 5 10 8 3 11 6 9 8 = x31 x72 x83 x11 4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 11 8 7 8 3 11 10 5 8 3 14 11 8 3 9 8 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x31 x14 2 x3 x4 x5 3 14 11 8 5 7 2 19 4 5 7 18 3 4 5 19 2 7 4 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 6 4 5 22 3 3 4 7 3 2 21 4 7 3 3 20 4 +x51 x19 2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 7 3 4 7 16 7 3 5 6 16 7 3 8 11 8 8 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x71 x32 x93 x10 4 x5 7 7 18 4 7 7 2 5 16 7 7 3 4 16 7 7 8 7 8 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 4 7 9 2 11 8 7 9 3 10 8 +x71 x72 x93 x64 x85 + x71 x72 x18 3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 7 11 6 5 8 7 11 9 2 8 7 11 10 8 7 8 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x71 x14 2 x3 x4 x5 7 14 3 5 8 7 14 7 8 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 ,
v4
4 3 7 5 12 8 3 7 12 9 4 3 7 18 3 4 = x31 x72 x23 x19 4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 11 4 13 4 3 13 4 11 4 3 13 5 6 8 5 10 4 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x31 x13 2 x3 x4 x5 3 13 6 5 8 3 14 13 4 3 14 9 5 4 3 14 13 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x45 2 7 4 3 19 3 6 4 3 22 3 3 4 5 3 2 21 4 +x31 x19 2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 5 3 3 20 4 5 3 4 7 16 5 3 5 6 16 4 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x51 x72 x3 x18 4 x5 5 7 2 5 16 5 7 3 4 16 5 7 8 7 8 5 7 9 6 8 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 4 5 11 5 6 8 5 11 6 5 8 5 14 7 8 +x51 x72 x18 3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 5 14 3 5 8 5 14 7 8 11 3 4 13 4 11 3 5 12 4 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 5 2 13 4 11 5 3 12 4 11 5 4 7 8 11 5 5 6 8 +x11 1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 11 7 12 4 11 7 4 5 8 11 7 5 4 8 11 7 12 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x45 ,
v8
4 3 7 12 5 4 3 13 4 7 4 3 13 5 6 4 = x31 x72 x43 x13 4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 14 5 5 4 7 3 4 13 4 7 3 5 12 4 4 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x71 x52 x23 x13 4 x5 7 5 3 12 4 7 5 4 7 8 7 5 5 6 8 7 7 12 4 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 4 7 8 7 8 +x71 x72 x43 x54 x85 + x71 x72 x53 x44 x85 + x71 x72 x12 3 x4 x5 + x1 x2 x3 x4 x5 7 8 3 5 8 7 8 7 8 +x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 .
is strictly inadmissible. The lemma is proved.
x be an admissible monomial of degree 39 in P5+ such that ω(x) = ω(1) . Then x = X{i,j} u2 with 1 6 i < j 6 5 and u ∈ B5 (2, 2, 1, 1). + 2 Let y ∈ B5 (2, 2, 1, 1) such that X{i,j} y ∈ P5 . By a direct computation using the results 2 in Subsection 3.2, we see that if X{i,j} y 6= d39,k , ∀k, 1 6 k 6 485, then there is a monomial ` w which is given in one of Lemmas 3.3.4, 3.3.5, and 3.3.6 such that X{i,j} y 2 = wy12 with y1 ∈ P5 , and ` = max{r ∈ Z : ωr (w) > 0}. By Theorem 2.7, X{i,j} y 2 is inadmissible. Since x = X{i,j} u2 with u ∈ B5 (2, 2, 1, 1), and x is admissible, one can see that x = dk . for some k, 1 6 k 6 485. This implies B5+ (ω(1) ) ⊂ {dk : 1 6 k 6 485}. The proposition follows.
Proof of Lemma 3.3.3. Let
Lemma 3.3.7.
B5 (ω(2) ) = B5+ (ω(2) ) = ∅.
x be an 1 6 i < j 6 5,
This means
Proof. Let
admissible monomial in
with
and
y ∈ B5 (2, 2, 3).
P5+
QP5+ (ω(2) ) = 0.
such that
ω(x) = ω(2) .
Then
x = X{i,j} y 2
By a direct computation using the results Proposi-
tion 3.2.2, Theorem 2.7, and Lemma 3.3.4(i), (ii), we see that
x
is a permutation of one of
the monomials:
13 3 4 9 10 13 3 4 9 11 12 x31 x42 x83 x11 x31 x42 x93 x94 x14 4 x5 , 5 , x1 x 2 x 3 x 4 x 5 , x1 x 2 x 3 x 4 x 5 , 3 5 9 10 12 3 5 8 11 12 3 5 8 10 13 3 5 8 9 14 x 1 x 2 x 3 x 4 x 5 , x1 x 2 x 3 x 4 x 5 , x1 x 2 x 3 x 4 x 5 , x1 x 2 x 3 x 4 x 5 , 4 9 x71 x72 x83 x84 x95 , x71 x82 x11 x71 x32 x83 x94 x12 x71 x32 x83 x84 x13 3 x4 x5 , 5 , 5 , 15 3 4 8 9 7 9 10 5 8 7 9 10 4 9 7 8 11 5 8 x1 x2 x3 x4 x5 , x1 x2 x3 x4 x5 , x1 x2 x3 x4 x5 , x1 x2 x3 x4 x5 , 3 5 8 8 x15 1 x2 x3 x4 x5 . A simple computation shows that
3 5 2 5 22 3 6 5 22 = Sq 1 x51 x52 x3 x54 x22 + Sq 2 x31 x52 x3 x64 x22 5 5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 5 8 5 14 mod(P− + Sq 8 x31 x52 x43 x54 x14 +Sq 4 x31 x52 x43 x94 x14 5 5 + x1 x2 x3 x4 x5 5 (ω(2) )).
x31 x52 x83 x94 x14 5 Hence
[x31 x52 x83 x94 x14 5 ]ω(2) = [0]ω(2) .
The others can be proved by a similar computation. The
lemma is proved. The follwing lemma is an immediate consequence of Lemmas 3.3.4(i), (ii), and 3.3.5.
Lemma 3.3.8.
We have
B5 (ω(3) ) = B5+ (ω(3) ) = ∅.
Consequently,
QP5+ (ω(3) ) = 0.
THE HIT PROBLEM FOR THE POLYNOMIAL ALGEBRA OF FIVE VARIABLES Lemma 3.3.9.
The space
where the monomials
Lemma 3.3.10.
dk
QP5 (ω(4) )
is spanned by the set
11
{[dk ]ω(4) : 486 6 k 6 604},
are determined as in Subsection 4.4.
The following monomials are strictly inadmissible:
19 x1 x22 x73 x10 4 x5 , 7 2 10 19 x1 x2 x3 x4 x5 , 18 x1 x72 x23 x11 4 x5 , 7 11 2 18 x1 x2 x3 x4 x5 , 3 x1 x22 x73 x26 4 x5 , 7 2 26 3 x1 x2 x3 x4 x5 , 18 3 x1 x72 x10 3 x4 x5 ,
19 x1 x72 x23 x10 4 x5 , 7 10 2 19 x1 x 2 x 3 x 4 x 5 , 2 18 x1 x72 x11 3 x4 x5 , 7 11 18 2 x1 x2 x3 x4 x5 , 3 x1 x72 x23 x26 4 x5 , 7 26 2 3 x1 x2 x3 x4 x5 , 3 18 x71 x2 x10 3 x4 x5 ,
Proof. We prove the lemma for
2 19 x1 x72 x10 3 x4 x5 , 7 10 19 2 x1 x 2 x 3 x 4 x 5 , 18 2 x1 x72 x11 3 x4 x5 , 7 11 2 18 x1 x 2 x 3 x 4 x 5 , 2 3 x1 x72 x26 3 x4 x5 , 7 26 3 2 x1 x2 x3 x4 x5 , 18 3 x71 x2 x10 3 x4 x5 ,
19 x = x1 x22 x73 x10 4 x5 .
19 2 x1 x72 x10 3 x4 x5 , 2 7 11 18 x1 x 2 x 3 x 4 x 5 , 18 x71 x2 x23 x11 4 x5 , 7 11 18 2 x1 x 2 x 3 x 4 x 5 , 3 2 x1 x72 x26 3 x4 x5 , 7 10 3 18 x1 x2 x3 x4 x5 , 17 2 2 x71 x11 2 x3 x4 x5 .
The others are proved by a similar
computation. By a direct computation using Cartan formula, we have
2 9 3 23 2 7 3 25 4 7 3 23 4 7 3 23 x = Sq 1 x1 x22 x73 x54 x23 5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 6 15 2 6 3 23 + Sq 8 x1 x22 x73 x64 x15 + Sq 4 x1 x22 x11 +Sq 2 x21 x22 x73 x34 x23 5 3 x4 x5 + x1 x2 x3 x4 x5 5
− 2 7 3 26 +x1 x22 x63 x34 x27 5 + x1 x2 x3 x4 x5 mod(P5 (ω(4) )). This equality implies that
x
is strictly inadmissible. The lemma follows.
x be an admissible monomial such that ω(x) = ω(4) . Then, x = X{i,j} u2 with u ∈ B5 (4, 1, 1). + 2 Let y ∈ B5 (4, 1, 1) such that X{i,j} y ∈ P5 . By a direct computation using Proposition 2 3.2.2, we see that if X{i,j} y 6= dk , 486 6 k 6 604 then there is a monomial w which is ` 2 given in one of Lemmas 3.3.5(i), (iii), 3.3.6, and 3.3.10 such that X{i,j} y = wy12 with 2 suitable monomial y1 ∈ P5 , and ` = max{r ∈ Z : ωr (w) > 0}. By Theorem 2.7, X{i,j} y is 2 inadmissible. Since x = X{i,j} u with u ∈ B5 (4, 1, 1) and x admissible, one gets x = dk , for some k, 486 6 k 6 604. This concludes the proof. Proof of Lemma 3.3.9. Let
Lemma 3.3.11.
The space
where the monomials
dk
QP5 (ω(5) )
is spanned by the set
{[dk ]ω(5) : 605 6 k 6 609},
are determined as in Subsection 4.4.
By a direct computation, one gets the following.
Lemma 3.3.12.
All permutations of the following monomials are strictly inadmissible:
11 11 7 10 10 11 x1 x62 x10 3 x4 x5 , x 1 x2 x3 x4 x5 .
x be an admissible monomial such that ω(x) = ω(5) . Then x = xi xj x` y with 1 6 i < j < ` 6 5 and y ∈ B5 (4, 1, 3). + 2 Let z ∈ B5 (4, 1, 3) such that xi xj x` z ∈ P5 . By a direct computation using the results 2 in Subsection 3.2, we see that if xi xj x` z 6= dk , 605 6 k 6 609, then there is a monomial w 2 2r which is given in one of Lemmas 3.3.5(i), (iii), 3.3.6, and 3.3.12 such that xi xj x` z = wz1 with suitable monomia z1 ∈ P5 , and r = max{s ∈ Z : ωs (w) > 0}. By Theorem 2.7, xi xj x` z 2 is inadmissible. Since x = xi xj x` y 2 and x is admissible, one gets x = dk for some k, 605 6 k 6 609. This implies B5+ (ω(5) ) ⊂ {dk : 605 6 k 6 609}. Proof of Lemma 3.3.11. Let
2
Lemma 3.3.13.
The space
where the monomials
Lemma 3.3.14.
dk
QP5 (ω(6) )
is spanned by the set
{[dk ]ω(6) : 610 6 k 6 649},
are determined as in Subsection 4.4.
All permutations of the following monomials are strictly inadmissible:
15 3 5 2 14 15 3 13 2 6 15 x1 x32 x63 x14 4 x5 , x1 x2 x3 x4 x5 , x1 x2 x3 x4 x5 . Proof. We prove the lemma for
15 x = x1 x32 x63 x14 4 x5 .
The others are proved by a similar com-
putation. A direct computation shows that
15 x1 x32 x63 x14 4 x5
Hence
x
15 3 12 7 15 4 12 3 7 15 2 9 5 7 15 = Sq 1 x1 x32 x63 x13 4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 15 4 3 9 7 15 4 9 3 7 15 +x41 x32 x53 x11 4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 15 2 3 10 7 15 +Sq 2 x21 x32 x63 x11 + Sq 4 x21 x52 x63 x74 x15 mod(P− 4 x5 + x1 x2 x3 x4 x5 5 5 (ω(6) ).
is strictly inadmissible. The lemma follows.
12
.V. PHÓC
Lemma 3.3.15.
The following monomials are strictly inadmissible:
7 14 x1 x32 x14 3 x4 x5 , 3 5 2 15 14 x1 x2 x3 x4 x5 , 5 2 14 x31 x15 2 x3 x4 x5 , 3 5 6 15 10 x1 x2 x3 x4 x5 , 5 6 10 x31 x15 2 x3 x4 x5 , 3 5 6 14 11 x1 x2 x3 x4 x5 , 7 14 x31 x52 x10 3 x4 x5 , 3 5 14 14 3 x1 x2 x3 x4 x5 , 14 3 6 x31 x13 2 x3 x4 x5 , 3 13 6 15 2 x1 x2 x3 x4 x5 , 3 13 2 6 x15 1 x2 x3 x4 x5 , 3 13 7 14 2 x1 x2 x3 x4 x5 , 6 7 10 x31 x13 2 x3 x4 x5 , 7 9 3 14 6 x1 x2 x3 x4 x5 .
14 7 x1 x32 x14 3 x4 x5 , 3 5 14 2 15 x1 x 2 x 3 x 4 x 5 , 5 14 2 x31 x15 2 x3 x4 x5 , 3 5 10 6 15 x1 x 2 x 3 x 4 x 5 , 5 10 6 x31 x15 2 x3 x4 x5 , 3 5 14 6 11 x1 x 2 x 3 x 4 x 5 , 14 7 x31 x52 x10 3 x4 x5 , 3 13 3 6 14 x1 x 2 x 3 x 4 x 5 , 14 6 3 x31 x13 2 x3 x4 x5 , 3 13 15 2 6 x1 x2 x3 x4 x5 , 3 13 6 2 x15 1 x2 x3 x4 x5 , 3 13 14 2 7 x1 x 2 x 3 x 4 x 5 , 6 10 7 x31 x13 2 x3 x4 x5 ,
7 14 x31 x2 x14 3 x4 x5 , 3 5 14 15 2 x1 x 2 x 3 x 4 x 5 , 3 5 2 14 x15 1 x2 x3 x4 x5 , 3 5 10 15 6 x1 x 2 x 3 x 4 x 5 , 3 5 6 10 x15 1 x2 x3 x4 x5 , 3 5 14 11 6 x1 x 2 x 3 x 4 x 5 , 7 10 x31 x52 x14 3 x4 x5 , 3 13 3 14 6 x1 x 2 x 3 x 4 x 5 , 2 6 15 x31 x13 2 x3 x4 x5 , 3 13 15 6 2 x1 x 2 x 3 x 4 x 5 , 2 7 14 x31 x13 2 x3 x4 x5 , 3 13 14 7 2 x1 x 2 x 3 x 4 x 5 , 7 6 10 x31 x13 2 x3 x4 x5 ,
Proof. We prove the lemma for the monomials
14 7 x31 x2 x14 3 x4 x5 , 3 5 15 2 14 x1 x 2 x 3 x 4 x 5 , 3 5 14 2 x15 1 x2 x3 x4 x5 , 3 5 15 6 10 x1 x 2 x 3 x 4 x 5 , 3 5 10 6 x15 1 x2 x3 x4 x5 , 3 5 7 10 14 x1 x2 x3 x4 x5 , 10 7 x31 x52 x14 3 x4 x5 , 3 13 6 3 14 x1 x 2 x 3 x 4 x 5 , 2 15 6 x31 x13 2 x3 x4 x5 , 3 15 13 2 6 x1 x 2 x 3 x 4 x 5 , 2 14 7 x31 x13 2 x3 x4 x5 , 3 13 6 6 11 x1 x2 x3 x4 x5 , 7 10 6 x31 x13 2 x3 x4 x5 ,
14 x = x31 x52 x63 x11 4 x5 ,
15 x31 x52 x23 x14 4 x5 , 3 5 15 14 2 x1 x 2 x 3 x 4 x 5 , 15 x31 x52 x63 x10 4 x5 , 3 5 15 10 6 x1 x 2 x 3 x 4 x 5 , 14 x31 x52 x63 x11 4 x5 , 3 5 7 14 10 x1 x2 x3 x4 x5 , 3 14 x31 x52 x14 3 x4 x5 , 3 13 6 14 3 x1 x 2 x 3 x 4 x 5 , 6 2 15 x31 x13 2 x3 x4 x5 , 3 15 13 6 2 x1 x 2 x 3 x 4 x 5 , 7 2 14 x31 x13 2 x3 x4 x5 , 3 13 6 11 6 x1 x2 x3 x4 x5 , x71 x92 x33 x64 x14 5 ,
and
y = x71 x92 x33 x64 x14 5 .
The others can be proved by a similar computation. By a direct computation, we have
x
13 3 5 3 13 14 3 5 6 11 13 3 6 9 7 13 3 10 5 7 13 = Sq 1 x31 x62 x53 x11 4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 13 6 5 3 11 13 +x61 x32 x53 x11 4 x5 + x1 x2 x3 x4 x5 13 3 6 3 11 14 3 3 5 13 13 3 3 6 11 14 5 3 5 7 17 +Sq 2 x31 x52 x33 x13 4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 5 3 9 9 11 5 5 3 7 17 5 9 3 7 13 5 9 3 9 11 +x51 x32 x93 x74 x13 5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 13 3 9 3 7 13 3 5 3 7 17 3 3 5 11 13 3 5 6 7 14 +Sq 4 x31 x52 x33 x11 4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 3 5 7 17 3 3 9 9 11 3 9 3 9 11 +x31 x32 x93 x74 x13 5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 − 14 3 3 6 13 14 +x31 x52 x33 x14 4 x5 + x1 x2 x3 x4 x5 mod(P5 (ω(6) )).
This relation shows that
y
x
is strictly inadmissible. By a similar computation, we obtain
7 7 5 6 13 7 7 8 5 11 7 7 5 8 11 3 7 9 6 13 = Sq 1 x71 x72 x33 x84 x13 5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 13 7 9 10 5 7 7 9 5 10 7 7 12 9 3 7 7 9 12 3 7 +x31 x72 x53 x10 4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 9 7 7 9 3 12 7 7 3 12 9 7 7 3 9 12 7 7 3 12 5 11 +x71 x12 2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 7 3 5 12 11 7 3 6 9 13 +x71 x32 x93 x64 x13 5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 3 11 3 6 14 3 7 3 10 14 7 7 6 6 11 +Sq 2 x71 x32 x63 x74 x14 5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 6 6 11 7 3 10 10 7 7 3 10 6 11 7 3 6 10 11 +x31 x11 2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 10 3 7 7 7 3 6 14 7 10 3 10 7 +x71 x10 2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 5 7 6 6 11 3 7 6 6 13 11 5 6 6 7 8 7 5 6 6 7 +Sq 4 x51 x72 x33 x64 x14 5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + Sq x1 x2 x3 x4 x5 − 7 3 6 9 14 3 13 3 6 14 3 13 6 6 11 +x71 x32 x93 x64 x14 5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 mod(P5 (ω(6) )).
Hence,
y
is strictly inadmissible. The lemma is proved.
x be an admissible monomial such that ω(x) = ω(6) . Then x = xi xj x` y 2 with 1 6 i < j < ` 6 5 and y ∈ B5 (4, 3, 2). + 2 Let u ∈ B5 (4, 3, 2) such that xi xj x` u ∈ P5 . By a direct computation using Proposition 2 3.2.2, we see that if xi xj x` u 6= dk , 610 6 t 6 649. then there is a monomial w which is r 2 given in one of Lemmas 3.3.5, 3.3.6, 3.3.14, and 3.3.15 such that xi xj x` u = wu21 with 2 suitable monomial u1 ∈ P5 , and r = max{s ∈ Z : ωs (w) > 0}. By Theorem 2.7, xi xj x` u 2 is inadmissible. Since x = xi xj x` y , and x is admissible, one gets x = dk for some k, 610 6 k 6 649. This implies B5+ (ω(6) ) ⊂ {dk : 610 6 t 6 649}. Proof of Lemma 3.3.13. Let
We now prove Proposition 3.3.2.
Proof of Proposition 3.3.2. From Lemmas 3.3.3, 3.3.7, 3.3.8, 3.3.9, 3.3.11, and 3.3.13, we see that the space
g0 )(5,39) ) ∩ (F2 ⊗A P + )39 Ker((Sq ∗ 5
is spanned by the set
{[dk ] : 1 6 k 6 649}.
THE HIT PROBLEM FOR THE POLYNOMIAL ALGEBRA OF FIVE VARIABLES Futhermore, the set
{[dk ] : 1 6 k 6 649}
is linearly independent in
(F2 ⊗A P5 )39
13 . Indeed,
suppose there is a linear relation
X
S=
γk dk ≡ 0,
16k6649
γk ∈ F2 , for 1 6 k 6 649. We explicitly compute p(i;I) (S) + monomials in (P4 )39 . Computing directly from the relations where
in terms of the admissible
p(i;I) (S) ≡ 0, ∀(i; I) ∈ N5 , `(I) > 0, we otain
γk = 0
Recall that we get
for all
k.
The proposition follows.
dim(F2 ⊗A P50 )39
dim(F2 ⊗A P5 )39 = 2130.
= 915.
Hence, from Theorem 3.1.1 and Proposition 3.3.2,
The main result is completely proved.
4. Appendix In this section, we list all admissible monomials of degrees a set of some monomials in
Ps
4.1. The admissible monomials of degree 18 in
B4 (18)
126 monomials: 2 1. x2 x23 x15 2. x2 x15 4 3 x4 3 14 3 15 5. x2 x3 x4 6. x3 x4 3 10. x15 9. x32 x15 3 2 x4 7 10 7 13. x2 x3 x4 14. x2 x3 x10 4 7 11 2 17. x2 x3 18. x32 x13 3 x4 15 2 2 15 21. x2 x3 x4 22. x1 x3 x4 2 25. x1 x22 x15 26. x1 x15 3 2 x4 14 3 3 14 29. x1 x3 x4 30. x1 x2 x4 3 33. x1 x14 x 34. x1 x63 x11 2 3 4 7 10 37. x1 x3 x4 38. x1 x72 x10 4 2 41. x1 x2 x14 42. x1 x22 x3 x14 3 x4 4 6 10 45. x1 x2 x3 x4 46. x1 x62 x3 x10 4 2 49. x1 x32 x23 x12 50. x1 x32 x12 4 3 x4 2 7 8 7 2 8 53. x1 x2 x3 x4 54. x1 x2 x3 x4 57. x1 x2 x33 x13 58. x1 x32 x3 x13 4 4 15 61. x1 x2 x3 x4 62. x1 x15 2 x3 x4 66. x1 x72 x73 x34 65. x1 x72 x33 x74 3 14 69. x1 x2 x3 70. x31 x15 4 3 13 2 2 73. x1 x3 x4 74. x31 x13 2 x4 3 5 10 3 5 10 77. x1 x2 x4 78. x1 x2 x3 81. x31 x32 x12 82. x31 x2 x23 x12 3 4 3 6 8 85. x1 x2 x3 x4 86. x31 x52 x23 x84 89. x31 x2 x3 x13 90. x31 x2 x13 4 3 x4 3 5 9 93. x1 x2 x3 x4 94. x31 x52 x93 x4 97. x31 x72 x73 x4 98. x31 x32 x53 x74 3 5 7 3 101. x1 x2 x3 x4 102. x31 x72 x33 x54 106. x71 x2 x10 105. x71 x3 x10 4 4 7 11 110. x71 x11 109. x1 x3 2 113. x71 x2 x73 x34 114. x71 x32 x3 x74 117. x71 x72 x33 x4 118. x71 x32 x33 x54 2 2 121. x15 122. x15 1 x2 x4 1 x2 x3 15 3 15 125. x1 x2 126. x1 x2 x3 x4 .
We have
=
in
P4
and
P5 . We
order
P4
is the set of
3. x2 x33 x14 4 3 7. x15 3 x4 3 11. x15 2 x3 7 11 15. x3 x4 19. x32 x53 x10 4 2 23. x1 x15 3 x4 15 2 27. x1 x2 x3 31. x1 x32 x14 3 35. x1 x62 x11 4 39. x1 x72 x10 3 2 43. x1 x14 2 x3 x4 2 3 12 47. x1 x2 x3 x4 51. x1 x22 x43 x11 4 55. x1 x32 x43 x10 4 59. x1 x32 x13 3 x4 63. x1 x32 x53 x94 67. x31 x3 x14 4 71. x31 x15 3 2 75. x31 x13 2 x3 3 3 12 79. x1 x3 x4 2 83. x31 x2 x12 3 x4 3 3 4 8 87. x1 x2 x3 x4 91. x31 x13 2 x3 x4 95. x31 x2 x73 x74 99. x31 x32 x73 x54 103. x31 x72 x53 x34 107. x71 x2 x10 3 111. x71 x2 x23 x84 115. x71 x32 x73 x4 119. x71 x32 x53 x34 3 123. x15 1 x4
4.2. The admissible monomials of degree 18 in
B5+ (18)
18, 39
by using the order as in Definition 2.4.
B5 (18) = B50 (18) ∪ B5+ (18),
B5+ (2, 2, 1, 1)
∪
B5+ (2, 2, 3)
∪
where
3 4. x2 x14 3 x4 3 15 8. x2 x4 12. x2 x63 x11 4 16. x72 x11 4 20. x32 x33 x12 4 24. x1 x22 x15 4 28. x1 x33 x14 4 3 32. x1 x14 2 x4 6 11 36. x1 x2 x3 40. x1 x2 x23 x14 4 2 44. x1 x22 x13 3 x4 2 12 3 48. x1 x2 x3 x4 52. x1 x22 x53 x10 4 56. x1 x32 x63 x84 60. x1 x2 x3 x15 4 64. x1 x32 x73 x74 68. x31 x2 x14 4 72. x31 x15 2 76. x31 x53 x10 4 80. x31 x32 x12 4 84. x31 x2 x43 x10 4 88. x31 x52 x83 x24 92. x31 x2 x53 x94 96. x31 x72 x3 x74 100. x31 x52 x33 x74 104. x31 x52 x53 x54 108. x71 x11 4 112. x71 x2 x33 x74 116. x71 x72 x3 x34 2 120. x15 1 x3 x4 15 3 124. x1 x3
P5
B50 (18) = Φ0 (B4 (18)), |B50 (18)| = 450,
B5+ (2, 4, 2)
∪
B5+ (4, 1, 1)
∪
B5+ (4, 1, 3)
∪
and
B5+ (4, 3, 2).
14
.V. PHÓC B5+ (2, 2, 1, 1) is the set of 1. x1 x2 x23 x24 x12 5 2 5. x1 x22 x12 3 x4 x5 2 6 8 9. x1 x2 x3 x4 x5 13. x1 x22 x53 x24 x85 17. x1 x32 x43 x24 x85 21. x31 x2 x43 x84 x25 25. x1 x22 x43 x94 x25 .
25 monomials: 2 2. x1 x2 x23 x12 4 x5 2 4 10 6. x1 x2 x3 x4 x5 10. x1 x2 x63 x24 x85 14. x1 x22 x53 x84 x25 18. x1 x32 x43 x84 x25 22. x1 x22 x43 x34 x85
B5+ (2, 2, 3) is the set of 15 monomials: 1. x1 x22 x43 x44 x75 2. x1 x22 x43 x74 x45 7 2 4 4 5. x1 x2 x3 x4 x5 6. x1 x22 x43 x54 x65 3 4 4 6 10. x1 x32 x43 x64 x45 9. x1 x2 x3 x4 x5 3 4 6 4 13. x1 x2 x3 x4 x5 14. x31 x2 x63 x44 x45
3. x1 x22 x3 x24 x12 5 7. x1 x22 x3 x44 x10 5 11. x1 x22 x3 x64 x85 15. x1 x22 x33 x44 x85 19. x31 x2 x23 x44 x85 23. x1 x22 x43 x84 x35
3. x1 x22 x73 x44 x45 4. x1 x72 x23 x44 x45 2 5 4 6 7. x1 x2 x3 x4 x5 8. x1 x22 x53 x64 x45 3 6 4 4 11. x1 x2 x3 x4 x5 12. x31 x2 x43 x44 x65 15. x31 x52 x23 x44 x45 .
B5+ (2, 4, 2) is the set of 10 monomials: 1. x1 x22 x33 x64 x65 2. x1 x32 x23 x64 x65 3. x1 x32 x63 x24 x65 3 2 6 6 3 6 2 6 5. x1 x2 x3 x4 x5 6. x1 x2 x3 x4 x5 7. x31 x2 x63 x64 x25 3 5 2 6 2 3 5 6 2 2 9. x1 x2 x3 x4 x5 10. x1 x2 x3 x4 x5 . B5+ (4, 1, 1, 1) is the set of 40 monomials: 1. x1 x2 x3 x4 x14 2. x1 x2 x3 x14 5 4 x5 2 13 5. x1 x2 x3 x4 x5 6. x1 x2 x23 x4 x13 5 9. x1 x22 x3 x13 10. x1 x22 x13 4 x5 3 x4 x5 3 12 13. x1 x2 x33 x12 4 x5 14. x1 x2 x3 x4 x5 3 12 3 12 17. x1 x2 x3 x4 x5 18. x1 x2 x3 x4 x5 21. x1 x22 x3 x54 x95 22. x1 x22 x53 x4 x95 25. x1 x32 x3 x44 x95 26. x1 x32 x43 x4 x95 29. x31 x2 x43 x4 x95 30. x31 x2 x43 x94 x5 33. x1 x32 x53 x4 x85 34. x1 x32 x53 x84 x5 37. x31 x2 x53 x84 x5 38. x31 x52 x3 x4 x85
3. x1 x2 x14 3 x4 x5 7. x1 x2 x23 x13 4 x5 11. x1 x2 x3 x34 x12 5 15. x1 x32 x3 x12 4 x5 19. x31 x2 x12 3 x4 x5 23. x1 x22 x53 x94 x5 27. x1 x32 x43 x94 x5 31. x1 x2 x33 x54 x85 35. x31 x2 x3 x54 x85 39. x31 x52 x3 x84 x5
B5+ (4, 1, 3) is the set of 10 monomials: 1. x1 x22 x53 x54 x55 2. x1 x32 x43 x54 x55 3. x1 x32 x53 x44 x55 3 4 5 5 3 5 4 5 5. x1 x2 x3 x4 x5 6. x1 x2 x3 x4 x5 7. x31 x2 x53 x54 x45 3 5 5 4 3 5 5 4 9. x1 x2 x3 x4 x5 10. x1 x2 x3 x4 x5 . B5+ (4, 3, 2) is the set of 180 monomials: 1. x1 x2 x23 x74 x75 2. x1 x2 x73 x24 x75 2 7 7 5. x1 x2 x3 x4 x5 6. x1 x22 x73 x74 x5 7 2 7 9. x1 x2 x3 x4 x5 10. x1 x72 x23 x74 x5 7 2 7 13. x1 x2 x3 x4 x5 14. x71 x2 x3 x74 x25 17. x71 x2 x73 x4 x25 18. x71 x2 x73 x24 x5 21. x1 x2 x33 x64 x75 22. x1 x2 x33 x74 x65 25. x1 x2 x73 x34 x65 26. x1 x2 x73 x64 x35 29. x1 x32 x63 x4 x75 30. x1 x32 x63 x74 x5 33. x1 x62 x3 x34 x75 34. x1 x62 x3 x74 x35 37. x1 x62 x73 x4 x35 38. x1 x62 x73 x34 x5 41. x1 x72 x33 x4 x65 42. x1 x72 x33 x64 x5 45. x31 x2 x3 x64 x75 46. x31 x2 x3 x74 x65 49. x31 x2 x73 x4 x65 50. x31 x2 x73 x64 x5 53. x71 x2 x3 x34 x65 54. x71 x2 x3 x64 x35 57. x71 x2 x63 x4 x35 58. x71 x2 x63 x34 x5 61. x1 x22 x33 x54 x75 62. x1 x22 x33 x74 x55 65. x1 x22 x73 x34 x55 66. x1 x22 x73 x54 x35 69. x1 x32 x53 x24 x75 70. x1 x32 x53 x74 x25 73. x1 x72 x23 x34 x55 74. x1 x72 x23 x54 x35 77. x31 x2 x23 x54 x75 78. x31 x2 x23 x74 x55 81. x31 x2 x73 x24 x55 82. x31 x2 x73 x54 x25
2 4. x1 x22 x3 x12 4 x5 2 4 10 8. x1 x2 x3 x4 x5 12. x1 x62 x3 x24 x85 16. x1 x32 x23 x44 x85 20. x31 x2 x43 x24 x85 24. x31 x42 x3 x24 x85
3. x1 x2 x73 x74 x25 7. x1 x72 x3 x24 x75 11. x1 x72 x73 x4 x25 15. x71 x2 x23 x4 x75 19. x71 x72 x3 x4 x25 23. x1 x2 x63 x34 x75 276. x1 x32 x3 x64 x75 31. x1 x32 x73 x4 x65 35. x1 x62 x33 x4 x75 39. x1 x72 x3 x34 x65 43. x1 x72 x63 x4 x35 47. x31 x2 x63 x4 x75 51. x31 x72 x3 x4 x65 55. x71 x2 x33 x4 x65 59. x71 x32 x3 x4 x65 63. x1 x22 x53 x34 x75 67. x1 x32 x23 x54 x75 71. x1 x32 x73 x24 x55 75. x1 x72 x33 x24 x55 79. x31 x2 x53 x24 x75 83. x31 x52 x3 x24 x75
4. x1 x32 x63 x64 x25 8. x31 x52 x23 x24 x65
4. x1 x14 2 x3 x4 x5 8. x1 x22 x3 x4 x13 5 12. x1 x2 x33 x4 x12 5 16. x1 x32 x12 3 x4 x5 20. x1 x2 x23 x54 x95 24. x1 x2 x33 x44 x95 28. x31 x2 x3 x44 x95 32. x1 x32 x3 x54 x85 36. x31 x2 x53 x4 x85 40. x31 x52 x83 x4 x5 . 4. x1 x32 x53 x54 x45 8. x31 x52 x3 x44 x55
4. x1 x22 x3 x74 x75 8. x1 x72 x3 x74 x25 12. x1 x72 x73 x24 x5 16. x71 x2 x23 x74 x5 20. x71 x72 x3 x24 x5 24. x1 x2 x63 x74 x35 28. x1 x32 x3 x74 x65 32. x1 x32 x73 x64 x5 36. x1 x62 x33 x74 x5 40. x1 x72 x3 x64 x35 44. x1 x72 x63 x34 x5 48. x31 x2 x63 x74 x5 52. x31 x72 x3 x64 x5 56. x71 x2 x33 x64 x5 60. x71 x32 x3 x64 x5 64. x1 x22 x53 x74 x35 68. x1 x32 x23 x74 x55 72. x1 x32 x73 x54 x25 76. x1 x72 x33 x54 x25 80. x31 x2 x53 x74 x25 84. x31 x52 x3 x74 x25
THE HIT PROBLEM FOR THE POLYNOMIAL ALGEBRA OF FIVE VARIABLES 85. x31 x52 x23 x4 x75 89. x31 x72 x3 x24 x55 93. x71 x2 x23 x34 x55 97. x71 x32 x3 x24 x55 101. x1 x32 x33 x44 x75 105. x1 x32 x73 x34 x45 109. x31 x2 x33 x44 x75 113. x31 x2 x73 x34 x45 117. x31 x32 x43 x4 x75 121. x31 x72 x3 x34 x45 125. x71 x2 x33 x34 x45 129. x71 x32 x33 x4 x45 133. x1 x32 x53 x34 x65 137. x1 x62 x33 x34 x55 141. x31 x2 x53 x34 x65 145. x31 x32 x3 x54 x65 149. x31 x52 x3 x34 x65 153. x31 x52 x63 x4 x35 157. x31 x52 x23 x34 x55 161. x31 x32 x33 x44 x55 165. x31 x32 x53 x34 x45 169. x31 x42 x3 x34 x75 173. x31 x42 x73 x4 x35 177. x71 x32 x43 x4 x35
86. x31 x52 x23 x74 x5 90. x31 x72 x3 x54 x25 94. x71 x2 x23 x54 x35 98. x71 x32 x3 x54 x25 102. x1 x32 x33 x74 x45 106. x1 x32 x73 x44 x35 110. x31 x2 x33 x74 x45 114. x31 x2 x73 x44 x35 118. x31 x32 x43 x74 x5 122. x31 x72 x3 x44 x35 126. x71 x2 x33 x44 x35 130. x71 x32 x33 x44 x5 134. x1 x32 x53 x64 x35 138. x1 x62 x33 x54 x35 142. x31 x2 x53 x64 x35 146. x31 x32 x3 x64 x55 150. x31 x52 x3 x64 x35 154. x31 x52 x63 x34 x5 158. x31 x52 x23 x54 x35 162. x31 x32 x33 x54 x45 165. x31 x32 x53 x44 x35 170. x31 x42 x3 x74 x35 174. x31 x42 x73 x34 x5 178. x71 x32 x43 x34 x5
87. x31 x52 x73 x4 x25 91. x31 x72 x53 x4 x25 95. x71 x2 x33 x24 x55 99. x71 x32 x53 x4 x25 103. x1 x32 x43 x34 x75 107. x1 x72 x33 x34 x45 111. x31 x2 x43 x34 x75 115. x31 x32 x3 x44 x75 119. x31 x32 x73 x4 x45 123. x31 x72 x33 x4 x45 127. x71 x32 x3 x34 x45 131. x1 x32 x33 x54 x65 135. x1 x32 x63 x34 x55 139. x31 x2 x33 x54 x65 143. x31 x2 x63 x34 x55 147. x31 x32 x53 x4 x65 151. x31 x52 x33 x4 x65 155. x31 x32 x53 x24 x55 159. x31 x52 x33 x24 x55 163. x31 x32 x43 x34 x55 167. x31 x52 x33 x34 x45 171. x31 x42 x33 x4 x75 175. x31 x72 x43 x4 x35 179. x31 x42 x33 x34 x55
4.3. The admissible monomials of degree 39 in
B4 (39)
is the set of
225
1. x2 x73 x31 4 5. x32 x73 x29 4 5 9. x32 x31 3 x4 7 11 21 13. x2 x3 x4 3 21 17. x15 2 x3 x4 31 7 21. x2 x3 x4 25. x1 x2 x63 x31 4 7 29. x1 x2 x30 3 x4 2 13 23 33. x1 x2 x3 x4 37. x1 x32 x43 x31 4 23 41. x1 x32 x12 3 x4 3 28 7 45. x1 x2 x3 x4 49. x1 x62 x3 x31 4 5 53. x1 x62 x27 3 x4 7 2 29 57. x1 x2 x3 x4 21 61. x1 x72 x10 3 x4 7 30 65. x1 x2 x3 x4 23 69. x1 x14 2 x3 x4 15 3 20 73. x1 x2 x3 x4 3 5 77. x1 x30 2 x3 x4 31 2 5 81. x1 x2 x3 x4 85. x31 x53 x31 4 7 89. x31 x29 3 x4 3 6 29 93. x1 x2 x3 x4 21 97. x31 x2 x14 3 x4 3 30 5 101. x1 x2 x3 x4 21 105. x31 x32 x12 3 x4 3 4 31 109. x1 x2 x3 x4 5 113. x31 x42 x27 3 x4 3 5 2 29 117. x1 x2 x3 x4
88. x31 x52 x73 x24 x5 92. x31 x72 x53 x24 x5 96. x71 x2 x33 x54 x25 100. x71 x32 x53 x24 x5 104. x1 x32 x43 x74 x35 108. x1 x72 x33 x44 x35 112. x31 x2 x43 x74 x35 116. x31 x32 x3 x74 x45 120. x31 x32 x73 x44 x5 124. x31 x72 x33 x44 x5 128. x71 x32 x3 x44 x35 132. x1 x32 x33 x64 x55 136. x1 x32 x63 x54 x35 140. x31 x2 x33 x64 x55 144. x31 x2 x63 x54 x35 148. x31 x32 x53 x64 x5 152. x31 x52 x33 x64 x5 156. x31 x32 x53 x54 x25 160. x31 x52 x33 x54 x25 164. x31 x32 x43 x54 x35 168. x31 x52 x33 x44 x35 172. x31 x42 x33 x74 x5 176 x31 x72 x43 x34 x5 180. x31 x42 x33 x54 x35 .
P4
monomials:
23 2. x2 x15 3 x4 3 13 23 6. x2 x3 x4 10. x72 x3 x31 4 5 14. x72 x27 3 x4 15 23 18. x2 x3 x4 22. x1 x73 x31 4 26. x1 x2 x73 x30 4 6 30. x1 x2 x31 3 x4 2 15 21 34. x1 x2 x3 x4 38. x1 x32 x53 x30 4 22 42. x1 x32 x13 3 x4 3 29 6 46. x1 x2 x3 x4 50. x1 x62 x33 x29 4 54. x1 x62 x31 3 x4 58. x1 x72 x33 x28 4 20 62. x1 x72 x11 3 x4 7 31 66. x1 x2 x3 23 70. x1 x15 2 x4 15 22 74. x1 x2 x3 x4 7 78. x1 x30 2 x3 x4 31 3 4 82. x1 x2 x3 x4 86. x31 x73 x29 4 5 90. x31 x31 3 x4 3 7 28 94. x1 x2 x3 x4 20 98. x31 x2 x15 3 x4 3 31 4 102. x1 x2 x3 x4 20 106. x31 x32 x13 3 x4 3 4 3 29 110. x1 x2 x3 x4 114. x31 x42 x31 3 x4 118. x31 x52 x33 x28 4
7 3. x2 x31 3 x4 3 15 21 7. x2 x3 x4 11. x72 x33 x29 4 15. x72 x31 3 x4 7 19. x31 2 x3 x4 15 23 23. x1 x3 x4 23 27. x1 x2 x14 3 x4 2 5 31 31. x1 x2 x3 x4 7 35. x1 x22 x29 3 x4 3 6 29 39. x1 x2 x3 x4 21 43. x1 x32 x14 3 x4 3 30 5 47. x1 x2 x3 x4 51. x1 x62 x73 x25 4 55. x1 x72 x31 4 59. x1 x72 x63 x25 4 5 63. x1 x72 x26 3 x4 14 67. x1 x2 x3 x23 4 22 71. x1 x15 2 x3 x4 15 23 75. x1 x2 x3 7 79. x1 x31 2 x4 31 6 83. x1 x2 x3 x4 23 87. x31 x13 3 x4 3 4 31 91. x1 x2 x3 x4 23 95. x31 x2 x12 3 x4 3 28 7 99. x1 x2 x3 x4 103. x31 x32 x43 x29 4 5 107. x31 x32 x28 3 x4 3 4 7 25 111. x1 x2 x3 x4 115. x31 x52 x31 4 119. x31 x52 x63 x25 4
4. x32 x53 x31 4 7 8. x32 x29 3 x4 7 7 25 12. x2 x3 x4 23 16. x15 2 x3 x4 31 3 5 20. x2 x3 x4 7 24. x1 x31 3 x4 15 22 28. x1 x2 x3 x4 32. x1 x22 x73 x29 4 5 36. x1 x22 x31 3 x4 3 7 28 40. x1 x2 x3 x4 20 44. x1 x32 x15 3 x4 3 31 4 48. x1 x2 x3 x4 21 52. x1 x62 x11 3 x4 7 30 56. x1 x2 x3 x4 60. x1 x72 x73 x24 4 4 64. x1 x72 x27 3 x4 14 3 21 68. x1 x2 x3 x4 2 21 72. x1 x15 2 x3 x4 30 76. x1 x2 x3 x74 6 80. x1 x31 2 x3 x4 31 7 84. x1 x2 x3 21 88. x31 x15 3 x4 5 30 3 92. x1 x2 x3 x4 22 96. x31 x2 x13 3 x4 3 29 6 100. x1 x2 x3 x4 104. x31 x32 x53 x28 4 4 108. x31 x32 x29 3 x4 3 4 11 21 112. x1 x2 x3 x4 116. x31 x52 x3 x30 4 120. x31 x52 x73 x24 4
15
16
.V. PHÓC 121. 125. 129. 133. 137. 141. 145. 149. 153. 157. 161. 165. 169. 173. 177. 181. 185. 189. 193. 197. 201. 205. 209. 213. 217. 221. 225.
21 x31 x52 x10 3 x4 3 5 30 x 1 x2 x3 x4 x31 x72 x43 x25 4 4 x31 x72 x25 3 x4 3 13 x1 x2 x3 x22 4 23 x31 x13 2 x3 7 x31 x29 2 x4 3 29 6 x1 x2 x3 x4 4 x31 x31 2 x3 x4 7 7 25 x1 x3 x4 x71 x2 x31 4 x71 x2 x63 x25 4 5 x71 x2 x26 3 x4 7 3 29 x 1 x2 x4 x71 x32 x83 x21 4 x71 x32 x29 3 x71 x72 x93 x16 4 20 x71 x11 2 x3 x4 7 27 4 x 1 x2 x3 x4 23 x15 1 x3 x4 15 x1 x2 x3 x22 4 23 x15 1 x2 x3 23 x15 1 x2 x4 31 7 x 1 x3 x4 3 4 x31 1 x2 x3 x4 31 3 x1 x2 x3 x44 7 x31 1 x2 x3 .
122. 126. 130. 134. 138. 142. 146. 150. 154. 158. 162. 166. 170. 174. 178. 182. 186. 190. 194. 198. 202. 206. 210. 214. 218. 222.
20 x31 x52 x11 3 x4 3 5 31 x 1 x2 x3 x31 x72 x53 x24 4 x31 x72 x28 3 x4 2 21 x31 x13 2 x3 x4 3 15 21 x 1 x2 x4 6 x31 x29 2 x3 x4 3 29 7 x 1 x2 x3 5 x31 x31 2 x3 7 11 21 x 1 x3 x4 x71 x2 x3 x30 4 x71 x2 x73 x24 4 4 x71 x2 x27 3 x4 7 3 28 x 1 x2 x3 x4 x71 x32 x93 x20 4 x71 x72 x25 4 x71 x72 x24 3 x4 21 x71 x11 2 x3 7 27 5 x 1 x2 x3 3 21 x15 1 x3 x4 15 x1 x2 x23 x21 4 3 21 x15 1 x2 x4 23 x15 1 x2 x3 31 x1 x2 x74 6 x31 1 x2 x3 x4 31 3 4 x 1 x2 x3 x4
123. 127. 131. 135. 139. 143. 147. 151. 155. 159. 163. 167. 171. 175. 179. 183. 187. 191. 195. 199. 203. 207. 211. 215. 219. 223.
5 x31 x52 x26 3 x4 3 7 29 x 1 x2 x4 x31 x72 x83 x21 4 x31 x72 x29 3 3 20 x31 x13 2 x3 x4 3 15 x1 x2 x3 x20 4 2 5 x31 x29 2 x3 x4 5 x31 x31 2 x4 7 31 x 1 x3 x4 5 x71 x27 3 x4 7 2 29 x 1 x2 x3 x4 21 x71 x2 x10 3 x4 7 30 x 1 x2 x3 x4 x71 x32 x43 x25 4 4 x71 x32 x25 3 x4 7 7 24 x 1 x2 x3 x4 x71 x72 x25 3 5 x71 x27 2 x4 7 31 x 1 x2 x4 23 x15 1 x3 x4 15 x1 x2 x33 x20 4 3 20 x15 1 x2 x3 x4 31 7 x 1 x3 x4 6 x31 1 x2 x3 x4 31 7 x 1 x2 x3 3 5 x31 1 x2 x3
4.4. The admissible monomials of degree 39 in
124. 128. 132. 136. 140. 144. 148. 152. 156. 160. 164. 168. 172. 176. 180. 184. 188. 192. 196. 200. 204. 208. 212. 216. 220. 224.
4 x31 x52 x27 3 x4 3 7 28 x 1 x2 x3 x4 x31 x72 x93 x20 4 23 x31 x13 2 x4 22 x31 x13 2 x3 x4 3 15 21 x 1 x2 x3 3 4 x31 x29 2 x3 x4 3 31 x1 x2 x3 x44 x71 x33 x29 4 x71 x31 3 x4 x71 x2 x33 x28 4 20 x71 x2 x11 3 x4 7 31 x 1 x2 x3 x71 x32 x53 x24 4 x71 x32 x28 3 x4 x71 x72 x83 x17 4 21 x71 x11 2 x4 4 x71 x27 2 x3 x4 7 31 x 1 x2 x3 23 x15 1 x2 x4 15 x1 x2 x22 3 x4 3 21 x15 1 x2 x3 3 5 x31 1 x3 x4 31 x1 x2 x23 x54 3 5 x31 1 x2 x4 31 7 x 1 x2 x4
P5
0 f ∗ )(5,39) ) , B5 (39) = B50 (39) ∪ ϕ(B5 (17)) ∪ B5+ (39) ∩ Ker((Sq B50 (39) = Φ0 (B4 (39)), |B50 (39)| = 915, |ϕ(B5 (17))| = 566 with
We have where
ϕ : P5 → P5 , ϕ(u) = x1 x2 x3 x4 x5 u2 , ∀u ∈ P5 , 0
and
f ∗ )(5,39) ) = B + (ω(1) ) ∪ B + (ω(4) ) ∪ B + (ω(5) ) ∪ B + (ω(6) ). B5+ (39) ∩ Ker((Sq 5 5 5 5
Here
ω(1) = (3, 2, 2, 1, 1), ω(4) = (3, 4, 1, 1, 1), ω(5) = (3, 4, 1, 3), ω(6) = (3, 4, 3, 2).
B5+ (ω(1) )
is the set of
1. x1 x2 x3 x64 x30 5 6 5. x1 x2 x30 3 x4 x5 6 9. x1 x30 x x x 2 3 4 5 22 13. x1 x2 x14 x 3 4 x5 4 17. x1 x2 x23 x31 4 x5 31 2 4 21. x1 x2 x3 x4 x5 2 4 25. x1 x31 2 x3 x4 x5 31 2 29. x1 x2 x3 x4 x45 2 5 33. x1 x2 x30 3 x4 x5 2 5 30 37. x1 x2 x3 x4 x5 41. x1 x2 x63 x24 x29 5 6 45. x1 x22 x29 3 x4 x5 49. x1 x2 x73 x24 x28 5 53. x1 x22 x73 x28 4 x5 57. x1 x72 x23 x4 x28 5 61. x71 x2 x23 x28 4 x5 23 65. x1 x22 x12 3 x4 x5 2 13 22 69. x1 x2 x3 x4 x5 2 21 73. x1 x14 2 x3 x4 x5
485
monomials
dk = d39,k , 1 6 k 6 485 :
6 2. x1 x2 x3 x30 4 x5 6 6. x1 x2 x30 x x 3 4 5 6 10. x1 x30 2 x3 x4 x5 22 14. x1 x14 x x 2 3 4 x5 31 2 4 18. x1 x2 x3 x4 x5 22. x1 x22 x43 x31 4 x5 2 4 26. x1 x31 x x 2 3 4 x5 31 2 4 30. x1 x2 x3 x4 x5 34. x1 x22 x3 x54 x30 5 2 5 38. x1 x30 2 x3 x4 x5 42. x1 x22 x3 x64 x29 5 46. x1 x62 x3 x24 x29 5 50. x1 x22 x3 x74 x28 5 7 54. x1 x22 x28 3 x4 x5 7 2 28 58. x1 x2 x3 x4 x5 23 62. x1 x2 x23 x12 4 x5 2 13 22 66. x1 x2 x3 x4 x5 21 70. x1 x2 x23 x14 4 x5 2 15 20 74. x1 x2 x3 x4 x5
3. x1 x2 x63 x4 x30 5 7. x1 x62 x3 x4 x30 5 22 11. x1 x2 x3 x14 4 x5 22 15. x1 x14 x x x 2 3 4 5 19. x1 x22 x3 x44 x31 5 4 23. x1 x22 x31 3 x4 x5 31 2 4 27. x1 x2 x3 x4 x5 31. x1 x2 x23 x54 x30 5 5 35. x1 x22 x3 x30 4 x5 2 6 29 39. x1 x2 x3 x4 x5 6 43. x1 x22 x3 x29 4 x5 2 7 28 47. x1 x2 x3 x4 x5 7 51. x1 x22 x3 x28 4 x5 2 28 7 55. x1 x2 x3 x4 x5 59. x71 x2 x3 x24 x28 5 23 63. x1 x22 x3 x12 4 x5 2 13 22 67. x1 x2 x3 x4 x5 2 21 71. x1 x2 x14 3 x4 x5 15 2 20 75. x1 x2 x3 x4 x5
4. x1 x2 x63 x30 4 x5 8. x1 x62 x3 x30 4 x5 22 12. x1 x2 x14 3 x4 x5 16. x1 x2 x23 x44 x31 5 4 20. x1 x22 x3 x31 4 x5 2 31 4 24. x1 x2 x3 x4 x5 2 4 28. x31 1 x2 x3 x4 x5 2 30 5 32. x1 x2 x3 x4 x5 36. x1 x22 x53 x4 x30 5 6 40. x1 x2 x23 x29 4 x5 2 29 44. x1 x2 x3 x4 x65 7 48. x1 x2 x23 x28 4 x5 2 7 28 52. x1 x2 x3 x4 x5 56. x1 x72 x3 x24 x28 5 60. x71 x2 x23 x4 x28 5 23 64. x1 x22 x12 3 x4 x5 2 13 68. x1 x2 x3 x4 x22 5 21 72. x1 x22 x3 x14 4 x5 2 15 20 76. x1 x2 x3 x4 x5
THE HIT PROBLEM FOR THE POLYNOMIAL ALGEBRA OF FIVE VARIABLES 20 77. x1 x22 x15 3 x4 x5 15 2 20 81. x1 x2 x3 x4 x5 85. x1 x2 x33 x44 x30 5 4 89. x1 x32 x3 x30 4 x5 3 30 4 93. x1 x2 x3 x4 x5 97. x31 x2 x3 x44 x30 5 4 101. x31 x2 x30 3 x4 x5 3 6 28 105. x1 x2 x3 x4 x5 6 109. x1 x32 x3 x28 4 x5 3 28 6 113. x1 x2 x3 x4 x5 117. x31 x2 x3 x64 x28 5 6 121. x31 x2 x28 3 x4 x5 3 12 125. x1 x2 x3 x4 x22 5 22 129. x31 x2 x12 3 x4 x5 20 133. x1 x32 x14 3 x4 x5 3 14 20 137. x1 x2 x3 x4 x5 4 141. x1 x62 x3 x27 4 x5 7 26 4 145. x1 x2 x3 x4 x5 4 149. x71 x2 x26 3 x4 x5 6 6 25 153. x1 x2 x3 x4 x5 157. x1 x62 x3 x74 x24 5 161. x1 x72 x63 x4 x24 5 165. x71 x2 x63 x24 4 x5 20 169. x1 x62 x3 x11 4 x5 7 10 20 173. x1 x2 x3 x4 x5 20 177. x71 x2 x10 3 x4 x5 2 3 4 29 181. x1 x2 x3 x4 x5 185. x1 x32 x23 x44 x29 5 189. x31 x2 x23 x44 x29 5 193. x31 x42 x3 x24 x29 5 197. x1 x22 x33 x54 x28 5 201. x1 x32 x23 x54 x28 5 205. x31 x2 x23 x54 x28 5 209. x31 x52 x3 x24 x28 5 3 21 213. x1 x22 x12 3 x4 x5 2 21 217. x31 x2 x12 3 x4 x5 2 20 221. x1 x32 x13 x 3 4 x5 2 20 x x 225. x31 x13 2 3 4 x5 229. x1 x22 x43 x74 x25 5 233. x1 x72 x23 x44 x25 5 237. x1 x22 x43 x94 x23 5 2 4 17 241. x1 x15 2 x3 x4 x5 6 245. x1 x22 x53 x25 4 x5 2 7 24 5 249. x1 x2 x3 x4 x5 5 253. x71 x2 x23 x24 4 x5 2 5 11 20 257. x1 x2 x3 x4 x5 2 5 16 261. x1 x15 2 x3 x4 x5 7 2 8 21 265. x1 x2 x3 x4 x5 17 269. x1 x22 x73 x12 4 x5 7 2 13 16 273. x1 x2 x3 x4 x5 277. x1 x32 x43 x34 x28 5 281. x31 x2 x43 x34 x28 5 285. x31 x32 x43 x4 x28 5 289. x31 x42 x3 x34 x28 5 20 293. x31 x2 x33 x12 4 x5 3 3 12 20 297. x1 x2 x3 x4 x5
20 78. x1 x22 x15 3 x4 x5 15 82. x1 x2 x3 x24 x20 5 4 86. x1 x2 x33 x30 4 x5 3 4 30 90. x1 x2 x3 x4 x5 3 4 94. x1 x30 2 x3 x4 x5 3 30 4 98. x1 x2 x3 x4 x5 4 102. x31 x2 x30 3 x4 x5 3 28 6 106. x1 x2 x3 x4 x5 110. x1 x32 x63 x4 x28 5 114. x1 x62 x3 x34 x28 5 6 118. x31 x2 x3 x28 4 x5 3 28 6 122. x1 x2 x3 x4 x5 22 126. x1 x32 x12 3 x4 x5 3 14 20 130. x1 x2 x3 x4 x5 20 134. x1 x32 x14 3 x4 x5 3 14 138. x1 x2 x3 x4 x20 5 4 142. x1 x62 x27 3 x4 x5 4 146. x1 x72 x26 3 x4 x5 7 26 4 150. x1 x2 x3 x4 x5 154. x1 x62 x3 x64 x25 5 158. x1 x62 x73 x4 x24 5 162. x1 x72 x63 x24 4 x5 21 166. x1 x2 x63 x10 4 x5 6 11 170. x1 x2 x3 x4 x20 5 20 174. x1 x72 x10 3 x4 x5 178. x1 x22 x53 x24 x29 5 4 182. x1 x22 x33 x29 4 x5 3 2 29 4 186. x1 x2 x3 x4 x5 4 190. x31 x2 x23 x29 4 x5 3 29 2 4 194. x1 x2 x3 x4 x5 5 198. x1 x22 x33 x28 4 x5 3 2 28 5 202. x1 x2 x3 x4 x5 5 206. x31 x2 x23 x28 4 x5 210. x31 x52 x23 x4 x28 5 21 214. x1 x32 x23 x12 4 x5 20 218. x1 x22 x33 x13 x 4 5 20 222. x31 x2 x23 x13 4 x5 2 20 226. x31 x13 x x x 2 3 4 5 7 230. x1 x22 x43 x25 4 x5 4 234. x1 x72 x23 x25 x 4 5 21 238. x1 x22 x43 x11 x 4 5 15 2 4 17 242. x1 x2 x3 x4 x5 246. x1 x22 x53 x74 x24 5 250. x1 x72 x23 x54 x24 5 254. x1 x22 x53 x84 x23 5 17 258. x1 x22 x53 x14 4 x5 15 2 5 16 262. x1 x2 x3 x4 x5 266. x1 x22 x73 x94 x20 5 17 270. x1 x72 x23 x12 4 x5 7 2 13 16 274. x1 x2 x3 x4 x5 3 4 278. x1 x32 x28 3 x4 x5 3 28 3 4 282. x1 x2 x3 x4 x5 286. x31 x32 x43 x28 4 x5 290. x31 x42 x33 x4 x28 5 3 20 294. x31 x2 x12 3 x4 x5 4 298. x1 x32 x43 x27 4 x5
2 20 79. x1 x15 2 x3 x4 x5 15 2 83. x1 x2 x3 x4 x20 5 3 4 87. x1 x2 x30 3 x4 x5 91. x1 x32 x43 x30 4 x5 3 4 95. x1 x30 2 x3 x4 x5 3 4 30 99. x1 x2 x3 x4 x5 103. x31 x42 x3 x4 x30 5 107. x1 x2 x63 x34 x28 5 111. x1 x32 x63 x28 4 x5 115. x1 x62 x33 x4 x28 5 119. x31 x2 x63 x4 x28 5 22 123. x1 x2 x33 x12 4 x5 3 12 22 127. x1 x2 x3 x4 x5 3 20 131. x1 x2 x14 3 x4 x5 14 135. x1 x2 x3 x34 x20 5 20 139. x31 x2 x14 3 x4 x5 4 143. x1 x62 x27 3 x4 x5 7 26 4 147. x1 x2 x3 x4 x5 5 151. x1 x2 x63 x26 4 x5 6 7 24 155. x1 x2 x3 x4 x5 159. x1 x62 x73 x24 4 x5 163. x71 x2 x3 x64 x24 5 21 167. x1 x62 x3 x10 4 x5 7 10 20 171. x1 x2 x3 x4 x5 20 175. x71 x2 x3 x10 4 x5 2 29 2 5 179. x1 x2 x3 x4 x5 183. x1 x22 x43 x34 x29 5 187. x1 x32 x43 x24 x29 5 191. x31 x2 x43 x24 x29 5 2 4 195. x31 x29 2 x3 x4 x5 2 5 3 28 199. x1 x2 x3 x4 x5 203. x1 x32 x53 x24 x28 5 207. x31 x2 x53 x24 x28 5 211. x31 x52 x23 x28 4 x5 2 21 215. x1 x32 x12 3 x4 x5 3 20 219. x1 x22 x13 x 3 4 x5 2 20 223. x31 x2 x13 3 x4 x5 5 227. x1 x22 x43 x27 4 x5 231. x1 x22 x73 x44 x25 5 235. x71 x2 x23 x44 x25 5 17 239. x1 x22 x43 x15 4 x5 2 5 26 5 243. x1 x2 x3 x4 x5 7 247. x1 x22 x53 x24 4 x5 7 2 24 5 251. x1 x2 x3 x4 x5 255. x1 x22 x53 x94 x22 5 16 259. x1 x22 x53 x15 4 x5 2 7 8 21 263. x1 x2 x3 x4 x5 267. x1 x72 x23 x94 x20 5 17 271. x71 x2 x23 x12 4 x5 3 3 4 28 275. x1 x2 x3 x4 x5 279. x31 x2 x33 x44 x28 5 283. x31 x32 x3 x44 x28 5 4 287. x31 x32 x28 3 x4 x5 3 3 12 20 291. x1 x2 x3 x4 x5 20 295. x31 x32 x3 x12 4 x5 3 4 27 4 299. x1 x2 x3 x4 x5
2 20 80. x1 x15 2 x3 x4 x5 15 2 20 84. x1 x2 x3 x4 x5 88. x1 x32 x3 x44 x30 5 4 92. x1 x32 x30 3 x4 x5 30 3 4 96. x1 x2 x3 x4 x5 100. x31 x2 x43 x30 4 x5 104. x31 x42 x3 x30 4 x5 108. x1 x32 x3 x64 x28 5 6 112. x1 x32 x28 3 x4 x5 6 3 28 116. x1 x2 x3 x4 x5 120. x31 x2 x63 x28 4 x5 22 124. x1 x32 x3 x12 4 x5 3 12 22 128. x1 x2 x3 x4 x5 20 132. x1 x32 x3 x14 4 x5 14 3 136. x1 x2 x3 x4 x20 5 4 140. x1 x2 x63 x27 4 x5 4 144. x1 x2 x73 x26 4 x5 7 26 4 148. x1 x2 x3 x4 x5 5 152. x1 x62 x3 x26 4 x5 7 6 24 156. x1 x2 x3 x4 x5 160. x1 x72 x3 x64 x24 5 164. x71 x2 x63 x4 x24 5 20 168. x1 x2 x63 x11 4 x5 7 10 20 172. x1 x2 x3 x4 x5 20 176. x71 x2 x10 3 x4 x5 2 13 2 21 180. x1 x2 x3 x4 x5 3 4 184. x1 x22 x29 3 x4 x5 3 29 2 4 188. x1 x2 x3 x4 x5 2 4 192. x31 x2 x29 3 x4 x5 3 29 2 4 196. x1 x2 x3 x4 x5 3 5 200. x1 x22 x28 3 x4 x5 3 28 2 5 204. x1 x2 x3 x4 x5 2 5 208. x31 x2 x28 3 x4 x5 21 212. x1 x22 x33 x12 4 x5 21 216. x31 x2 x23 x12 x 4 5 20 220. x1 x32 x23 x13 4 x5 2 20 224. x31 x13 x x x 2 3 4 5 4 228. x1 x22 x53 x27 4 x5 4 232. x1 x22 x73 x25 x 4 5 4 236. x71 x2 x23 x25 x 4 5 4 17 240. x1 x22 x15 x x 3 4 5 244. x1 x22 x53 x64 x25 5 248. x1 x22 x73 x54 x24 5 252. x71 x2 x23 x54 x24 5 21 256. x1 x22 x53 x10 4 x5 2 15 5 16 260. x1 x2 x3 x4 x5 264. x1 x72 x23 x84 x21 5 268. x71 x2 x23 x94 x20 5 16 272. x1 x22 x73 x13 4 x5 3 3 28 4 276. x1 x2 x3 x4 x5 4 280. x31 x2 x33 x28 4 x5 3 3 28 4 284. x1 x2 x3 x4 x5 4 288. x31 x32 x28 3 x4 x5 3 12 3 20 292. x1 x2 x3 x4 x5 20 296. x31 x32 x12 3 x4 x5 3 4 27 4 300. x1 x2 x3 x4 x5
17
18
.V. PHÓC 301. 305. 309. 313. 317. 321. 325. 329. 333. 337. 341. 345. 349. 353. 357. 361. 365. 369. 373. 377. 381. 385. 389. 393. 397. 401. 405. 409. 413. 417. 421. 425. 429. 433. 437. 441. 445. 449. 453. 457. 461. 465. 469. 473. 477. 481. 485.
4 x31 x42 x27 3 x4 x5 3 5 26 4 x 1 x2 x3 x4 x5 4 x31 x52 x26 3 x4 x5 3 6 25 4 x 1 x2 x3 x4 x5 6 x31 x2 x43 x25 4 x5 3 4 7 24 x 1 x2 x3 x4 x5 x1 x72 x33 x44 x24 5 x31 x2 x73 x44 x24 5 x31 x72 x3 x44 x24 5 x71 x2 x33 x44 x24 5 x71 x32 x43 x4 x24 5 x1 x32 x43 x94 x22 5 21 x31 x42 x3 x10 4 x5 3 4 11 x1 x2 x3 x4 x20 5 4 17 x31 x2 x14 3 x4 x5 16 x31 x2 x43 x15 4 x5 15 3 4 16 x 1 x2 x3 x4 x5 5 x1 x32 x63 x24 4 x5 3 6 5 24 x 1 x2 x3 x4 x5 x31 x52 x63 x24 4 x5 20 x31 x2 x53 x10 4 x5 3 5 14 16 x 1 x2 x3 x4 x5 x1 x32 x63 x84 x21 5 x1 x62 x33 x94 x20 5 16 x1 x32 x63 x13 4 x5 3 7 8 20 x 1 x2 x3 x4 x5 x71 x32 x3 x84 x20 5 16 x31 x2 x73 x12 4 x5 7 10 4 17 x1 x2 x3 x4 x5 4 16 x71 x11 2 x3 x4 x5 7 6 8 17 x 1 x2 x3 x4 x5 x71 x2 x63 x94 x16 5 x71 x72 x83 x4 x16 5 2 4 17 x31 x13 2 x3 x4 x5 x31 x52 x23 x94 x20 5 x71 x92 x23 x44 x17 5 4 x31 x42 x33 x25 4 x5 4 x x31 x32 x53 x24 4 5 x31 x32 x43 x84 x21 5 4 17 x31 x32 x12 3 x4 x5 x31 x32 x53 x84 x20 5 16 x31 x52 x33 x12 4 x5 3 5 11 16 4 x 1 x2 x3 x4 x5 x31 x72 x93 x44 x16 5 4 x71 x32 x93 x16 4 x5 3 5 6 8 17 x 1 x2 x3 x4 x5 x71 x32 x53 x84 x16 5 .
B5+ (ω(4) ) 486. 490. 494. 498. 502. 506. 510. 514. 518.
is the set of
x1 x22 x23 x34 x31 5 2 3 x1 x22 x31 3 x4 x5 2 2 x1 x32 x31 3 x4 x5 3 2 2 31 x 1 x2 x3 x4 x5 2 2 3 x31 1 x2 x3 x4 x5 2 2 7 27 x 1 x2 x3 x4 x5 2 x1 x72 x23 x27 4 x5 7 27 2 2 x 1 x2 x3 x4 x5 x1 x32 x23 x34 x30 5
302. 306. 310. 314. 318. 322. 326. 330. 334. 338. 342. 346. 350. 354. 358. 362. 366. 370. 374. 378. 382. 386. 390. 394. 398. 402. 406. 410. 414. 418. 422. 426. 430. 434. 438. 442. 446. 450. 454. 458. 462. 466. 470. 474. 478. 482. 119
5 x1 x32 x43 x26 4 x5 3 4 26 5 x 1 x2 x3 x4 x5 x1 x32 x43 x64 x25 5 x1 x62 x33 x44 x25 5 x31 x2 x63 x44 x25 5 7 x1 x32 x43 x24 4 x5 7 3 24 4 x 1 x2 x3 x4 x5 4 x31 x2 x73 x24 4 x5 3 7 24 4 x 1 x2 x3 x4 x5 4 x71 x2 x33 x24 4 x5 7 3 4 24 x 1 x2 x3 x4 x5 x31 x2 x43 x94 x22 5 20 x1 x32 x43 x11 4 x5 3 4 14 17 x 1 x2 x3 x4 x5 16 x1 x32 x43 x15 4 x5 3 15 4 16 x 1 x2 x3 x4 x5 x1 x32 x53 x64 x24 5 x1 x62 x33 x54 x24 5 5 x31 x2 x63 x24 4 x5 3 5 8 22 x 1 x2 x3 x4 x5 20 x31 x52 x3 x10 4 x5 3 14 5 16 x 1 x2 x3 x4 x5 x1 x62 x33 x84 x21 5 x31 x2 x63 x94 x20 5 16 x31 x2 x63 x13 4 x5 3 7 8 20 x 1 x2 x3 x4 x5 x71 x32 x83 x4 x20 5 16 x31 x72 x3 x12 4 x5 7 10 4 17 x 1 x2 x3 x4 x5 5 16 x1 x72 x10 3 x4 x5 7 6 8 17 x 1 x2 x3 x4 x5 x1 x72 x73 x84 x16 5 x71 x72 x83 x16 4 x5 x31 x52 x23 x54 x24 5 17 x31 x52 x23 x12 4 x5 x71 x92 x23 x54 x16 5 x31 x32 x43 x54 x24 5 x31 x42 x33 x54 x24 5 x31 x32 x43 x94 x20 5 16 x31 x32 x43 x13 4 x5 x31 x52 x33 x84 x20 5 4 17 x31 x52 x10 3 x4 x5 x31 x72 x43 x84 x17 5 4 x31 x72 x93 x16 4 x5 7 9 3 4 16 x 1 x2 x3 x4 x5 x31 x52 x63 x94 x16 5
dk = 2 2 31 3 x 1 x2 x3 x4 x5 3 2 x1 x22 x31 3 x4 x5 31 2 2 3 x 1 x2 x3 x4 x5 2 x31 x2 x23 x31 4 x5 31 2 3 2 x 1 x2 x3 x4 x5 x1 x22 x73 x24 x27 5 2 2 x1 x72 x27 3 x4 x5 2 2 x71 x27 2 x3 x4 x5 3 2 30 3 x 1 x2 x3 x4 x5
monomials
487. 491. 495. 499. 503. 507. 511. 515. 519.
303. 307. 311. 315. 319. 323. 327. 331. 335. 339. 343. 347. 351. 355. 359. 363. 367. 371. 375. 379. 383. 387. 391. 395. 399. 403. 407. 411. 415. 419. 423. 427. 431. 435. 439. 443. 447. 451. 455. 459. 463. 467. 471. 475. 479. 483.
4 x1 x32 x53 x26 4 x5 3 5 26 4 x 1 x2 x3 x4 x5 6 x1 x32 x43 x25 4 x5 6 3 25 4 x1 x2 x3 x4 x5 4 x31 x2 x63 x25 4 x5 3 7 4 24 x 1 x2 x3 x4 x5 x31 x2 x43 x74 x24 5 x31 x42 x3 x74 x24 5 x31 x72 x43 x4 x24 5 x71 x32 x3 x44 x24 5 x1 x32 x43 x84 x23 5 21 x1 x32 x43 x10 4 x5 3 4 11 20 x 1 x2 x3 x4 x5 4 17 x1 x32 x14 3 x4 x5 3 15 4 16 x 1 x2 x3 x4 x5 4 16 x31 x15 2 x3 x4 x5 3 5 24 6 x1 x2 x3 x4 x5 x31 x2 x53 x64 x24 5 x31 x52 x3 x64 x24 5 x31 x2 x53 x84 x22 5 20 x31 x52 x10 3 x4 x5 3 5 14 16 x 1 x2 x3 x4 x5 x31 x2 x63 x84 x21 5 17 x1 x32 x63 x12 4 x5 3 7 8 20 x 1 x2 x3 x4 x5 x31 x72 x83 x4 x20 5 16 x1 x32 x73 x12 4 x5 7 3 12 16 x 1 x2 x3 x4 x5 4 16 x1 x72 x11 3 x4 x5 7 10 5 16 x 1 x2 x3 x4 x5 x1 x62 x73 x94 x16 5 x71 x2 x73 x84 x16 5 x31 x52 x23 x44 x25 5 5 x31 x52 x23 x24 4 x5 16 x31 x52 x23 x13 x 4 5 x31 x32 x43 x44 x25 5 5 x31 x32 x43 x24 4 x5 x31 x52 x33 x44 x24 5 x31 x42 x33 x94 x20 5 4 16 x31 x32 x13 3 x4 x5 16 x31 x32 x53 x12 4 x5 3 5 10 17 4 x 1 x2 x3 x4 x5 x71 x32 x43 x84 x17 5 x71 x32 x43 x94 x16 5 5 16 x31 x52 x10 3 x4 x5 x31 x52 x73 x84 x16 5
304. 308. 312. 316. 320. 324. 328. 332. 336. 340. 344. 348. 352. 356. 360. 364. 368. 372. 376. 380. 384. 388. 392. 396. 400. 404. 408. 412. 416. 420. 424. 428. 432. 436. 440. 444. 448. 452. 456. 460. 464. 468. 472. 476. 480. 484.
5 x31 x2 x43 x26 4 x5 3 5 26 4 x 1 x2 x3 x4 x5 x1 x32 x63 x44 x25 5 x31 x2 x43 x64 x25 5 x31 x42 x3 x64 x25 5 4 x1 x32 x73 x24 4 x5 3 4 24 7 x1 x2 x3 x4 x5 x31 x42 x73 x4 x24 5 x31 x72 x43 x24 4 x5 4 x71 x32 x3 x24 4 x5 3 4 8 23 x 1 x2 x3 x4 x5 21 x31 x2 x43 x10 4 x5 3 4 11 20 x 1 x2 x3 x4 x5 17 x31 x2 x43 x14 4 x5 15 3 4 16 x1 x2 x3 x4 x5 3 4 16 x15 1 x2 x3 x4 x5 3 6 5 24 x 1 x2 x3 x4 x5 6 x31 x2 x53 x24 4 x5 3 5 6 24 x1 x2 x3 x4 x5 20 x1 x32 x53 x10 4 x5 3 5 10 20 x 1 x2 x3 x4 x5 5 16 x31 x2 x14 3 x4 x5 3 6 9 20 x 1 x2 x3 x4 x5 17 x31 x2 x63 x12 4 x5 7 3 8 20 x 1 x2 x3 x4 x5 x71 x2 x33 x84 x20 5 16 x1 x72 x33 x12 4 x5 7 3 12 16 x 1 x2 x3 x4 x5 4 16 x71 x2 x11 3 x4 x5 6 7 8 17 x 1 x2 x3 x4 x5 x1 x72 x63 x94 x16 5 x71 x72 x3 x84 x16 5 4 x31 x52 x23 x25 4 x5 x31 x52 x23 x84 x21 5 2 5 16 x31 x13 2 x3 x4 x5 4 x31 x32 x43 x25 4 x5 x31 x32 x53 x44 x24 5 4 x31 x52 x33 x24 4 x5 17 x31 x32 x43 x12 x 4 5 3 4 16 x31 x13 x x x 2 3 4 5 5 16 x31 x32 x12 3 x4 x5 3 5 11 4 16 x 1 x2 x3 x4 x5 x31 x72 x43 x94 x16 5 x71 x32 x93 x44 x16 5 16 5 x31 x52 x10 3 x4 x5 x31 x72 x53 x84 x16 5
d39,k , 486 6 k 6 604 : 488. 492. 496. 500. 504. 508. 512. 516. 520.
x1 x22 x33 x24 x31 5 x1 x32 x23 x24 x31 5 2 3 2 x1 x31 2 x3 x4 x5 2 2 x31 x2 x31 3 x4 x5 31 3 2 2 x 1 x2 x3 x4 x5 2 x1 x22 x73 x27 4 x5 7 2 2 27 x1 x2 x3 x4 x5 x1 x22 x33 x34 x30 5 x1 x32 x33 x24 x30 5
489. 493. 497. 501. 505. 509. 513. 517. 521.
2 x1 x22 x33 x31 4 x5 3 2 31 2 x 1 x2 x3 x4 x5 3 2 2 x1 x31 2 x3 x4 x5 3 31 x1 x2 x3 x24 x25 3 2 2 x31 1 x2 x3 x4 x5 7 2 2 27 x 1 x2 x3 x4 x5 2 x71 x2 x23 x27 4 x5 2 3 30 3 x 1 x2 x3 x4 x5 2 x1 x32 x33 x30 4 x5
THE HIT PROBLEM FOR THE POLYNOMIAL ALGEBRA OF FIVE VARIABLES 522. 526. 530. 534. 538. 542. 546. 550. 554. 558. 562. 566. 570. 574. 578. 582. 586. 590. 594. 598. 602.
2 3 x1 x32 x30 3 x4 x5 3 3 2 30 x 1 x2 x3 x4 x5 x31 x32 x3 x24 x30 5 x1 x32 x63 x24 x27 5 2 x31 x2 x63 x27 4 x5 3 7 2 26 x1 x2 x3 x4 x5 2 x1 x72 x33 x26 4 x5 3 7 2 26 x1 x2 x3 x4 x5 2 x71 x2 x33 x26 4 x5 3 6 3 26 x1 x2 x3 x4 x5 3 x31 x2 x63 x26 4 x5 3 6 11 18 x1 x2 x3 x4 x5 18 x31 x2 x73 x10 4 x5 3 3 29 2 2 x 1 x2 x3 x4 x5 x31 x52 x23 x24 x27 5 2 2 x71 x32 x25 3 x4 x5 3 x31 x52 x23 x26 4 x5 3 5 26 3 2 x 1 x2 x3 x4 x5 18 x31 x52 x23 x11 4 x5 3 7 9 18 2 x 1 x2 x3 x4 x5 18 x31 x52 x33 x10 4 x5
B5+ (ω(5) )
is the set of
605. 608. B5+ (ω(6) ) 610. 614. 618. 622. 626. 630. 634. 638. 642. 646.
523. 527. 531. 535. 539. 543. 547. 551. 555. 559. 563. 567. 571. 575. 579. 583. 587. 591. 595. 599. 603. 5
3 2 x1 x32 x30 3 x4 x5 3 3 30 2 x 1 x2 x3 x4 x5 2 x31 x32 x3 x30 4 x5 3 6 27 2 x 1 x2 x3 x4 x5 x1 x22 x33 x74 x26 5 2 x1 x32 x73 x26 4 x5 3 2 7 26 x 1 x2 x3 x4 x5 2 x31 x72 x3 x26 4 x5 7 3 2 26 x 1 x2 x3 x4 x5 3 x1 x32 x63 x26 4 x5 3 3 6 26 x1 x2 x3 x4 x5 18 x31 x2 x63 x11 4 x5 3 7 10 18 x1 x2 x3 x4 x5 2 2 3 x31 x29 2 x3 x4 x5 3 5 2 27 2 x 1 x2 x3 x4 x5 x31 x32 x53 x24 x26 5 x31 x52 x33 x24 x26 5 19 x31 x52 x23 x10 4 x5 3 5 11 2 18 x 1 x2 x3 x4 x5 x71 x32 x93 x24 x18 5 3 18 x31 x52 x10 3 x4 x5
monomials
10 11 x31 x52 x10 3 x4 x5 3 7 9 10 10 x 1 x2 x3 x4 x5
is the set of
14 x1 x32 x73 x14 4 x5 7 3 14 14 x 1 x2 x3 x4 x5 6 14 x71 x2 x11 3 x4 x5 3 7 13 2 14 x 1 x2 x3 x4 x5 5 2 14 x71 x11 2 x3 x4 x5 3 3 5 14 14 x 1 x2 x3 x4 x5 6 14 x31 x52 x11 3 x4 x5 7 3 5 10 14 x 1 x2 x3 x4 x5 x71 x32 x93 x64 x14 5 6 10 x71 x32 x13 3 x4 x5
Acknowledgment.
40
x31 x2 x23 x34 x30 5 2 3 x31 x2 x30 3 x4 x5 x1 x22 x33 x64 x27 5 x31 x2 x23 x64 x27 5 x1 x22 x73 x34 x26 5 x1 x72 x23 x34 x26 5 x31 x2 x73 x24 x26 5 x71 x2 x23 x34 x26 5 2 x71 x32 x3 x26 4 x5 3 3 6 26 x 1 x2 x3 x4 x5 19 x1 x32 x63 x10 4 x5 3 7 10 18 x 1 x2 x3 x4 x5 18 x71 x2 x33 x10 4 x5 3 29 2 3 2 x 1 x2 x3 x4 x5 2 2 x31 x52 x27 3 x4 x5 3 3 5 26 2 x 1 x2 x3 x4 x5 2 x31 x52 x33 x26 4 x5 3 5 10 2 19 x 1 x2 x3 x4 x5 18 2 x31 x52 x11 3 x4 x5 7 3 9 18 2 x 1 x2 x3 x4 x5 18 3 x31 x52 x10 3 x4 x5 .
525. 529. 533. 537. 541. 545. 549. 553. 557. 561. 565. 569. 573. 577. 581. 585. 589. 593. 597. 601.
3 x31 x2 x23 x30 4 x5 3 30 3 2 x 1 x2 x3 x4 x5 x1 x32 x23 x64 x27 5 x31 x2 x63 x24 x27 5 x1 x32 x23 x74 x26 5 x1 x72 x33 x24 x26 5 2 x31 x2 x73 x26 4 x5 7 3 2 26 x1 x2 x3 x4 x5 x1 x32 x33 x64 x26 5 x31 x2 x63 x34 x26 5 19 x31 x2 x63 x10 4 x5 7 3 10 18 x 1 x2 x3 x4 x5 18 x71 x32 x3 x10 4 x5 3 29 3 2 2 x1 x2 x3 x4 x5 2 2 x31 x72 x25 3 x4 x5 3 5 2 3 26 x1 x2 x3 x4 x5 2 3 x31 x52 x26 3 x4 x5 3 5 10 19 2 x1 x2 x3 x4 x5 x31 x72 x93 x24 x18 5 18 x31 x32 x53 x10 4 x5
dk = d39,k , 605 6 k 6 609 :
11 10 606. x31 x52 x10 3 x4 x5 7 3 9 10 10 609. x1 x2 x3 x4 x5 .
monomials
611. 615. 619. 623. 627. 631. 635. 639. 643. 647.
524. 528. 532. 536. 540. 544. 548. 552. 556. 560. 564. 568. 572. 576. 580. 584. 588. 592. 596. 600. 604.
19
10 10 607. x31 x52 x11 3 x4 x5
dk = d39,k , 610 6 k 6 649 :
14 x1 x72 x33 x14 4 x5 7 3 14 14 x1 x2 x3 x4 x5 14 6 x71 x2 x11 3 x4 x5 3 7 13 14 2 x 1 x2 x3 x4 x5 5 14 2 x71 x11 2 x3 x4 x5 3 5 3 14 14 x 1 x2 x3 x4 x5 14 6 x31 x52 x11 3 x4 x5 7 3 5 14 10 x 1 x2 x3 x4 x5 6 x71 x32 x93 x14 4 x5 7 3 13 10 6 x 1 x2 x3 x4 x5
612. 616. 620. 624. 628. 632. 636. 640. 644. 648.
14 x31 x2 x73 x14 4 x5 7 11 6 14 x 1 x2 x3 x4 x5 6 14 x71 x11 2 x3 x4 x5 7 3 13 2 14 x 1 x2 x3 x4 x5 13 2 6 x71 x11 2 x3 x4 x5 3 3 13 6 14 x 1 x2 x3 x4 x5 14 x31 x72 x53 x10 4 x5 3 7 9 6 14 x 1 x2 x3 x4 x5 6 10 x31 x72 x13 3 x4 x5 7 11 5 6 10 x 1 x2 x3 x4 x5
613. 617. 621. 625. 629. 633. 637. 641. 645. 649.
14 x31 x72 x3 x14 4 x5 7 11 14 6 x 1 x2 x3 x4 x5 14 6 x71 x11 2 x3 x4 x5 7 3 13 14 2 x 1 x2 x3 x4 x5 13 6 2 x71 x11 2 x3 x4 x5 3 3 13 14 6 x 1 x2 x3 x4 x5 10 x31 x72 x53 x14 4 x5 3 7 9 14 6 x1 x2 x3 x4 x5 10 6 x31 x72 x13 3 x4 x5 7 11 5 10 6 x1 x2 x3 x4 x5 .
I would like to express my warmest thanks to Assoc.
Prof.
Nguyen
Sum for every helpful piece of advice and his inspiring guidance during my work on my PhD degree. My thanks also go to all colleagues at the Faculty of Education Studies, University of Khanh Hoa. The results in Section 3 is based on the author's PhD. thesis [7] written under the supervision of Assoc. Prof. Nguyen Sum.
References
[1] M.C. Crabb and J.R. Hubbuck, Representations of the homology of BV and the Steenrod algebra II, Algebra Topology: new trend in localization and periodicity, Progr. Math. 136 (1996), 143-154. [2] M. Kameko, Products of projective spaces as Steenrod modules, PhD. thesis, The Johns Hopkins University, ProQuest LLC, Ann Arbor, MI, 1990, 29 pages. [3] F.P. Peterson, Generators of H ∗ (RP ∞ × RP ∞ ) as a module over the Steenrod algebra, Abstracts Amer. Math. Soc. No. 833, April 1987. [4] , A -generators for certain polynomial algebras, Math. Proc. Cambridge Philos. Soc. 105 (1989), 311-312. [5] .V. Phóc and N. Sum, On the generators of the polynomial algebra as a module over the Steenrod algebra, C.R.Math. Acad. Sci. Paris, Ser. I 353 (2015), 1035-1040. [6] .V. Phóc, The hit problem for the polynomial algebra of five variables in degree seventeen and its application, East-West J. Math. 18 (2016), 27-46. [7] , The Peterson hit problem in some types of degrees and applications, PhD. thesis, Quy Nhon University, 2018, 117 pages.
20
.V. PHÓC
[8] S. Priddy, On characterizing summands in the classifying space of a group, I, Amer. Jour. Math. 112 (1990), 737-748. [9] J. Repka and P. Selick, On the subalgebra of H∗ ((RP ∞ )n ; F2 ) annihilated by Steenrod operations, J. Pure Appl. Algebra 127 (1998), 273-288. [10] W.M. Singer, The transfer in homological algebra, Math. Zeit. 202 (1989), 493-523. On the action of the Steenrod squares on polynomial algebras, Proc. Amer. Math. Soc. 111 [11] , (1991), 577-583. [12] N.E. Steenrod and D.B.A. Epstein, Cohomology operations, Annals of Mathematics Studies 50, Princeton University Press, Princeton N.J (1962). [13] N. Sum, The negative answer to Kameko's conjecture on the hit problem, Adv. Math. 225 (2010), 2365-2390. [14] , On the hit problem for the polynomial algebra, C. R. Math. Acad. Sci. Paris, Ser. I 351 (2013), 565-568. [15] , On the Peterson hit problem , Adv. Math. 274 (2015), 432-489. [16] N.K. T½n, The admissible monomial basis for the polynomial algebra of five variables in degree 2s+1 + 2s − 5, East-West J. Math. 16 (2014), 34-46. [17] R.M.W. Wood, Steenrod squares of polynomials and the Peterson conjecture, Math. Proc. Cambriges Phil. Soc. 105 (1989), 307-309. , Problems in the Steenrod algebra, Bull. London Math. Soc. 30 (1998) 449-517. [18] Faculty of Education Studies University of Khanh Hoa Nha Trang, Vietnam
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