Let
H
\mathcal {H}
be a space of analytic functions on the unit ball
B
d
{\mathbb {B}_d}
in
C
d
\mathbb {C}^d
with multiplier algebra
M
u
l
t
(
H
)
\mathrm {Mult}(\mathcal {H})
. A function
f
∈
H
f\in \mathcal {H}
is called cyclic if the set
[
f
]
[f]
, the closure of
{
φ
f
:
φ
∈
M
u
l
t
(
H
)
}
\{\varphi f: \varphi \in \mathrm {Mult}(\mathcal {H})\}
, equals
H
\mathcal {H}
. For multipliers we also consider a weakened form of the cyclicity concept. Namely for
n
∈
N
0
n\in \mathbb {N}_0
we consider the classes
C
n
(
H
)
=
{
φ
∈
M
u
l
t
(
H
)
:
φ
≠
0
,
[
φ
n
]
=
[
φ
n
+
1
]
}
.
\begin{equation*} \mathcal {C}_n(\mathcal {H})=\{\varphi \in \mathrm {Mult}(\mathcal {H}):\varphi \ne 0, [\varphi ^n]=[\varphi ^{n+1}]\}. \end{equation*}
Many of our results hold for
N
N
:th order radially weighted Besov spaces on
B
d
{\mathbb {B}_d}
,
H
=
B
ω
N
\mathcal {H}= B^N_\omega
, but we describe our results only for the Drury-Arveson space
H
d
2
H^2_d
here.
Letting
C
s
t
a
b
l
e
[
z
]
\mathbb {C}_{stable}[z]
denote the stable polynomials for
B
d
{\mathbb {B}_d}
, i.e. the
d
d
-variable complex polynomials without zeros in
B
d
{\mathbb {B}_d}
, we show that
a
m
p
;
if
d
is odd, then
C
s
t
a
b
l
e
[
z
]
⊆
C
d
−
1
2
(
H
d
2
)
,
and
a
m
p
;
if
d
is even, then
C
s
t
a
b
l
e
[
z
]
⊆
C
d
2
−
1
(
H
d
2
)
.
\begin{align*} &\text { if } d \text { is odd, then } \mathbb {C}_{stable}[z]\subseteq \mathcal {C}_{\frac {d-1}{2}}(H^2_d), \text { and }\\ &\text { if } d \text { is even, then } \mathbb {C}_{stable}[z]\subseteq \mathcal {C}_{\frac {d}{2}-1}(H^2_d). \end{align*}
For
d
=
2
d=2
and
d
=
4
d=4
these inclusions are the best possible, but in general we can only show that if
0
≤
n
≤
d
4
−
1
0\le n\le \frac {d}{4}-1
, then
C
s
t
a
b
l
e
[
z
]
⊈
C
n
(
H
d
2
)
\mathbb {C}_{stable}[z]\nsubseteq \mathcal {C}_n(H^2_d)
.
For functions other than polynomials we show that if
f
,
g
∈
H
d
2
f,g\in H^2_d
such that
f
/
g
∈
H
∞
f/g\in H^\infty
and
f
f
is cyclic, then
g
g
is cyclic. We use this to prove that if
f
,
g
f,g
extend to be analytic in a neighborhood of
B
d
¯
\overline {{\mathbb {B}_d}}
, have no zeros in
B
d
{\mathbb {B}_d}
, and the same zero sets on the boundary, then
f
f
is cyclic in
∈
H
d
2
\in H^2_d
if and only if
g
g
is. Furthermore, if the boundary zero set of
f
∈
H
d
2
∩
C
(
B
d
¯
)
f\in H^2_d\cap C(\overline {{\mathbb {B}_d}})
embeds a cube of real dimension
≥
3
\ge 3
, then
f
f
is not cyclic in the Drury-Arveson space.