In short my question is how do I show that a superpotential of the form,

\begin{equation}

W = \sum _{i = 1 } ^{ N _Y } Y _i f _i ( X _1 , ... , X _{ N _X } )

\end{equation}

(generic O'Raifertaigh models) lead to $ N _Y - N _X $ flat directions after spontaneous SUSY breaking? I describe the context and the details below.

In Weinberg Vol III (pg. 84) he introduces two fields,

\begin{equation} Y = \left( Y _1, ... , Y _{N _Y } \right) , \quad X = \left( X _1 , ... , X _{ N _X } \right) \end{equation}

where $ Y $ has an $R$ charge of $2 $ and $ X $ has an $R$ charge of $0$. Then superpotential has to take the form,

\begin{equation}

W = \sum _i Y _i f _i ( X _1 , ... , X _{ N _X } )

\end{equation}

which gives the SUSY conserving conditions,

\begin{align}

f _i ( X ) & = 0 \mbox{ for $ i = 1 , ... , N _Y $}\\

\sum _i Y _i \frac{ \partial f ( X ) }{ \partial X _n } & = 0 \mbox{ for $n = 1 , ... , N _X $}

\end{align}

The first set of equations is made up of $ N _X $ unknowns and $ N _Y $ equations, thus if $ N _Y > N _X $ it can't be generically solved. He goes on to define

\begin{equation}

v _{n,i} \equiv \frac{ \partial f _i }{ \partial x _n } \bigg|_{ x = x _0 }

\end{equation}

which gives the potential,

\begin{equation}

V ( x, y ) = \sum _i \left| f _i \right| ^2 + \sum _n \left| \sum _i y _i v _{n,i} \right| ^2

\end{equation}

Up to this point I have no problems. However, then he goes on to say that the second term vanishes if $ y $ is orthogonal to $ v _{ n} $. Since ``$ v _n $ cannot span the space of $ y _i $s and there will be at least $ N _Y - N _X $ flat directions''. Where does this conclusion come from? How can I see these ``flat directions''?