Is there a way  I can figure out what values of the coefficients of some system of non-linear equations makes the system inconsistent?

Take the following system of equations as an example.  The  unknowns are $(y_0,..., y_8)$.  I do not have fixed values for the coefficients   $a,b,c,d,e,f,g,h,u,v,w$, so I just want to know which  values of  $a,b,c,d,e,f,g,h,u,v,w$ make the system inconsistent. 

Note that in the example I am using the system is underdetermined as I have 9 unknowns and only 8 equations. 

$$\begin{array}\\
u*y_5 +  e*y_7  - e + b*y_2*y_0 + c*y_2 + w*y_0  =0\\
d*y_5 + f*y_7 -  f +  a*y_2 + v*y_0 = 0\\
u*y_6 - w*y_7  + b*y_3*y_0 + b*y_2*y_1 + c*y_3  + w*y_1 =  0\\
u*y_8 - c*y_7 + b*y_4*y_0 + c*y_4 =0\\
d*y_8 - a*y_7  + a*y_4 = 0\\
d*y_6 - v*y_7  + a*y_3    + v*y_1  = 0\\
b*y_7 -  b*y_4*y_1 = 0\\
y_1 = 0\\
\end{array}$$

Any hints or help would be highly appreciated.