1. Introduction

Let be the polynomial algebra over the prime field of two elements,, in variables , each of degree 1. The mod-2 Steenrod algebra acts on by the formula

and subject to the Cartan formula

for (see Steenrod – Epstein ^[8]).

Many authors study the hit problem of determination of the minimal set of generators for as a module over the Steenrod algebra, or equivalently, a basis of . A polynomial in is called hit if it can be written as a finite sum for some polynomials . That means belongs to whehe denotes the augmentation ideal in . This problem has first been studied by Peterson ^[4], Wood ^[12], Singer ^[7], Priddy ^[5], who show its relationship to several classical problems in homotopy theory. Peterson conjectured in ^[4] that as a module over the Steenrod algebra , is generated by monomials in degrees that satisfy , where denotes the number of ones in dyadic expansion of , and proved it for . The conjecture was established in general by Wood ^[12]. This is a useful tool for determining -general for . The tensor product has explicitly been calculated by Peterson^[4] for , Kameko for in his thesis ^[2] and Sum ^[10] for . The hit prolem were then investigated by many authors. (See Nam ^[3], Singer ^[7], Silverman ^[6], Hung ^[1], Walker-Wood ^[12] and others)

One of important tools in Kameko’s computation of -general for is the squaring operation which is determined for all positive intergers . Kameko showed in ^[2] that if then is an isomorphism of -vector spaces, where . From this result and Wood’s theorem, the hit problem is reduced to the cases of degree with .

In this paper, we explicitly determine for and degree . We have

Theorem 1.1. is an -vector space of dimension 174 with a basis consisting of all the classes represented by the following admissible monomials:

i₁) The 20 monomials are all permutations of the spikes

i₂) The 10 monomials of the form

i₃) The 10 monomials of the form

i₄) The 10 monomials of the form

i₅) The 10 monomials of the form

i₆) The 20 monomials of the form

i₇) The 15 monomials of the form

i₈) The 20 monomials of the form

i₉) The 30 monomials are all permutations of the spikes

i₁₀) 15 monomials are some permutations of the monomial , namely:

i₁₁) 4 monomials are some permutations of the monomial , namely:

In Section 2, we recall some results on the admissible monomials and hit monomials in . Theorem 1.1 will be proved in Section 3.

2. Preliminaries

In this section, we recall some results in Sum^[10], Singer ^[7] on the admissible monomials and the hit monomials in . Let denote the -th coefficient in dyadic expansion of a nonnegativeinteger . That means for or 1 and . Let . Following Kameko ^[2], we define two sequences associated with by

where .

Definition 2.1. Let be the monomials in . We say that if and only if one of the following holds

i₁)

i₂) and

Here, the order on the set of sequences of non-negative intergers is the lexicographical one.

Letbe homogeneous polynomials of the same degree in . We denote if and only if . If then is called hit.

Definition 2.2. A monomial is said to be inadmissible if there exists the monomials such that and .

A monomial is said to be admissible if it is not inadmissible.

Obviously, the set of all admissible monomials in is a minimal set of -generators of.

Definition 2.3. A monomial is said to be strictly inadmissible if and only if there exists the monomials such that and , with and .

The following propositon is one of our main tools.

Proposition 2.4 (Sum ^[10]). Let be an admissible monomial in . Then, we have:

i₁) If there is such that then , for all .

i₂) If there is such that then , for all .

Now, we recall a result of Singer ^[7] on the hit monomials in .

Definition 2.5. A monomial is called the spike if for a non-negative interger and . If is the spike with and for then it is called the minimal spike.

The following is a criterion for the hit monomials in .

Theorem 2.6 (Singer ^[7]). Suppose is a monomial of degree, where . Let be the minimal spike of degree . If then is hit.

For later use, we set

It is easy to see that and are the -submodules of . Furthermore, we have . For a polynomial in , we denote by the class in represented by .

3. Proof of Theorem 1.1

For , define the homomorphism of algebras by substituting

Notation is also the admissible monomials set of degree in . It is clear that, for , we have .Hence, .

We recall the following result.

Theorem 3.1 (Sum^[10]) is an -vector space of dimension 55 with a basis consisting of all the classes represented by the following admissible monomials:

i₁)

i₂)

i₃)

i₄)

i₅) ,

i₆)

i₇)

i₈)

From this Theorem, by a direct computation, we easily obtain

Proposition 3.2. is an -vector space of dimension 155 with a basis consisting of all the classes represented by the admissible monomials.

Now Theorem 1.1 is proved by computing . From now on, let us write the monomial by for abbreviation.

Proposition 3.3. is generated by 19 elements .

Proposition 3.4. The elements , for are linearly independent in .

We prepare some lemma for the proof of this propositions.

By a direct computation, we have

Lemma 3.5. We have the relations in

Lemma 3.6. The following monomials are inadmissible:

Proof. We have

Since and

We get is admissible. The other monomials are proved similar.

Proof of Proposition 3.3. There are 35 monomials of degree eight in ; they are and their permutations.

There are 5 monomials in with -sequence associated (4;0;1); they are and their permutations. Using Proposition 2.4, we see that the following monomials are inadmissible:

There are 20 monomials in with - sequence associated (4;2); they are and their permutations. Using Lemma 3.6, we see that the following monomials are inadmissible:

There are 10 monomials in with -sequence associated (2;3); they are and their permutations. Using Lemma 3.6, we see that the following monomials are inadmissible:

So, is generated by 19 elements .

Proof of Proposition 3.4. Suppose there is a linear relation

(1)

with

Consider the homomorphisms , induced by the ring homomorphisms , which are determined by the following tables:

By the direct computation, we see that the homomorphism send the relation (1), we obtain

Then, we get

Similarly, the homomorphisms send the relation (1), we obtain

Hence, we get for The proposition is proved.

Acknowledgements

It is a pleasure for me to express my hearty gratitude to Professor Nguyen Sum for his valuable suggestions and constant encouragement. My thanks also go to all colleagues at the Foundation Sciences Faculty, University of Technical Education of Ho Chi Minh city.

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