  A Boolean network (BN) with $n$ components is a discrete dynamical system
described by the successive iterations of a function $f:\{0,1\}^n \to
\{0,1\}^n$. This model finds applications in biology, where fixed points play a
central role. For example, in genetic regulations, they correspond to cell
phenotypes. In this context, experiments reveal the existence of positive or
negative influences among components: component $i$ has a positive (resp.
negative) influence on component $j$ meaning that $j$ tends to mimic (resp.
negate) $i$. The digraph of influences is called signed interaction digraph
(SID), and one SID may correspond to a large number of BNs (which is, in
average, doubly exponential according to $n$). The present work opens a new
perspective on the well-established study of fixed points in BNs. When
biologists discover the SID of a BN they do not know, they may ask: given that
SID, can it correspond to a BN having at least/at most $k$ fixed points?
Depending on the input, we prove that these problems are in $\textrm{P}$ or
complete for $\textrm{NP}$, $\textrm{NP}^{\textrm{NP}}$,
$\textrm{NP}^{\textrm{#P}}$ or $\textrm{NEXPTIME}$. In particular, we prove
that it is $\textrm{NP}$-complete (resp. $\textrm{NEXPTIME}$-complete) to
decide if a given SID can correspond to a BN having at least two fixed points
(resp. no fixed point).
