Dimension Result for the Polynomial Algebra as a Module over the Steenrod Algebra

For n≥1, let P(n)=2[x1,…,xn] be the polynomial algebra in n variables xi, of degree one, over the field 2 of two elements. The mod-2 Steenrod algebra acts on P(n) according to well-known rules. Let +P(n) denote the image of the action of the positively graded part of . A major problem is that of determining a basis for the quotient vector space Q(n)=P(n)/+P(n). Both P(n)=⊕d≥0Pd(n) and Q(n) are graded where Pd(n) denotes the set of homogeneous polynomials of degree d. A spike of degree d is a monomial of the form x12λ1-1⋯xn2λn-1 where λi≥0 for each i. In this paper we show that if n≥2 and d≥1 can be expressed in the form d=d(λ)=(n-1)(2λ-1) with λ≥2, then dim(Qd(λ)(n))≥B(n,d(λ))+{∑q=2λ(nq), if λ

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