if it is [$\mathcal{T}_{cliq}$-forcing]{} and [$\mathcal{T}_{bip}$-forcing]{}.

This result allows us to answer the question of whether a fixed graph $H$ is locally forcing or not in terms of whether certain polynomial equations have a common real root in a bounded interval. This can be answered using a combination of Sturm’s Theorem (see [@BPR]) and the Euclidean Algorithm for computing the greatest common divisor of a sequence of polynomials. See the concluding remarks for more details.

**Organisation of the paper:** The short proof of Proposition \[prop: forcing iff transitive\] is given in the next section. In Section \[sec: counting polynomial\] a counting polynomial associated with [$\mathcal{T}_{cliq}$-forcing]{} is introduced. In the same section we show that a locally forcing tournament must be [$\mathcal{T}_{cliq}$-forcing]{} and that this property imposes many restrictions on $H$, which gives us a proof of Theorem \[thm: locally-forcing\]. Despite this in Section \[sec: simple tournaments\] we find many [$\mathcal{T}_{cliq}$-forcing]{} tournaments. In Sections \[sec: reg simple tourns are forcing\] and \[sec: proof of simple and esimple implies local forcing\] we introduce the polynomials related to the property of [$\mathcal{T}_{bip}$-forcing]{}, show that any nearly regular tournament must be [$\mathcal{T}_{cliq}$-forcing]{} and show Theorem \[thm:-simple and esimple iff locally forcing\]. Using this we 