Singer’s algebraic transfer for all ranks $s$ in some degrees and the
negation of the dimension results for the graded spaces
$\mathbb
F_2\otimes_{\mathscr A}
\mathbb F_2[x_1, \ldots, x_s]$ in
degrees $s+5$

Let $P_s:= \mathbb
F_2[x_1,x_2,\ldots ,x_s]$ be the graded
polynomial algebra over the prime field of two elements,
$\mathbb F_2$, in $s$ variables $x_1, x_2,
\ldots , x_s$, each of degree one. This algebra is
considered as a graded module over the mod-2 Steenrod algebra,
$\mathscr {A}$. The classical “hit problem”,
initiated by Frank Peterson [Abstracts Amer. Math. Soc. 833 (1987),
55-89], concerned with seeking a minimal set of
$\mathscr A$-module $P_s.$ Equivalently, when
$\mathbb F_2$ is an $\mathscr
A$-module concentrated in degree 0, one can write down explicitly a
monomial basis for the $\mathbb Z$-graded vector space
over $\mathbb F_2$:

$$ QP_s:= \mathbb F_2
\otimes_{\mathscr A} P_s =
P_s/\mathscr A^+\cdot P_s,$$

where $\mathscr A^{+}$ denotes the augmentation
ideal of $\mathscr A.$ The problem is unresolved in
general. In this paper, we study the hit problem for $P_s$ with
$s\geq 5.$ More explicitly, we first compute explicitly
the dimension of $QP_s$ for $s = 5$ in the generic degree
$21\cdot 2^{t}-5$ with $t = 1.$ Note that the
problem when $t = 0$ was solved by Nguyễn Sum [Vietnam J. Math. 49
(2021), 1079-1096]. Next, we shall reject a result of Meshack Moetele
and Mbakiso Fix Mothebe [East-West Journal of Mathematics 18 (2016),
151-170] on the dimension of $QP_s$ in degrees $s+5$ for
$8\leq s\leq 9.$ We also give an
explicit formula for the dimension of $QP_s$ in degree $14$ for all
$s > 0$ and in degree $15$ for all $s >
0,\, s\neq 10.$ In addition, a conjecture
on the dimension of $QP_s$ in degree $2^{s-1}-s$ has been
given. As applications, we investigate William Singer’s conjecture
[Math. Z. 202 (1989), 493-523] on the algebraic transfer of rank
$5$ in degrees $21\cdot 2^{t}-5$ for all
$t\geq 0$ and of ranks $s > 0$ in
internal degrees $d,\, 13\leq
d\leq 15.$ This is a completely new result of proving
the Singer conjecture for all ranks $s$ in certain internal degrees.
In particular, our results have shown that any element in the
$Sq^{0}$-families
$\{\chi_t=(Sq^{0})^{t}(\chi_0)\in
{\rm Ext}_{\mathscr A}^{5,
21\cdot 2^{t+1}}(\mathbb F_2,
\mathbb F_2)|\,
t\geq 0\}$ and
$\{D_1(t)=(Sq^{0})^{t}(D_1(0))\in
{\rm Ext}_{\mathscr A}^{5,
57\cdot 2^{t}}(\mathbb F_2,
\mathbb F_2)|\,
t\geq 0\}$ belongs to the image of the
algebraic transfer of rank $5.$ We also study the behavior of the
algebraic transfer of rank 7 in the generic degrees
$23\cdot 2^{t}-7$ for $t = 0$ and
$\ell\cdot 2^{t}-7$ for
$\ell\in
\{9,\,
16\},\, t\leq 3.$ Our
results then claim that the non-zero elements $Pc_0\in
{\rm Ext}_{\mathscr A}^{7,
23\cdot 2^{0}}(\mathbb F_2,
\mathbb F_2),$\ $k_0 =
k\in {\rm
Ext}_{\mathscr A}^{7, 9\cdot
2^{2}}(\mathbb F_2, \mathbb F_2)$
and $h_6D_2\in {\rm
Ext}_{\mathscr A}^{7,
2^{7}}(\mathbb F_2, \mathbb F_2)$
are not in the image of the transfer, and that every indecomposable
element in the $Sq^{0}$-family $\{Q_2(t) =
(Sq^{0})^{t}(Q_2(0))\in {\rm
Ext}^{7, 2^{t+6}}_{\mathscr
A}(\mathbb F_2, \mathbb
F_2):\, t\geq 0\}$
belongs to the image of the transfer.