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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.

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  • $\begingroup$ Why the "finite fields" tag? $\endgroup$
    – Gerry Myerson
    Commented Feb 23, 2016 at 21:33
  • $\begingroup$ @Gerry Myerson as a matter of fact I need the solutions over a finite field $\endgroup$
    – Jousha
    Commented Feb 23, 2016 at 22:10
  • $\begingroup$ @jeq yes y_1 is 0. I know I could have modified the other equations accordingly rather than having to add the last equation ... $\endgroup$
    – Jousha
    Commented Feb 23, 2016 at 22:13
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    $\begingroup$ As I understand your question, your system of equations will be inconsistent if it has no solutions. However, it is clear that setting $y_i = 0$ for all $i$ gives a solution, whatever the values of the unfixed coefficients. Hence, since the system always has at least one solution, I don't see how it can ever be inconsistent. Perhaps you need to refine your question to avoid this trivial answer. $\endgroup$
    – Igor Khavkine
    Commented Feb 24, 2016 at 2:08
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    $\begingroup$ @Igor Khavkine: Not quite right. Note that for instance in the first two equations you have constant terms in the equations. So a trivial solution will only be a valid solution if the coefficients involved in those constant terms are also 0. $\endgroup$
    – Jousha
    Commented Feb 24, 2016 at 10:41

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