Test for inconsistency of system of nonlinear equations

[e2m2a]

Is there a quick test to determine if a system of nonlinear equations is inconsistent. For example, suppose there is a system of equations such as:

3x cubed + 2y cubed = z cubed
2x cubed + 5y cubed = z cubed

Since these two equations are clearly not dependent, could we say that since they both are equal to the same term (z cubed), that they are therefore, inconsistent?

 

[mfb]

x=y=z=0 is a solution, for example.

Both left hand sides are equal to the same thing, which means $3x^2+2y^3=2x^3+5y^3$, that is an equation with solutions. An infinite set of solutions, actually. And for every solution of that equation there is a z that fits.

If you can derive something impossible, (like $x^2=-1$ if you work with real numbers), then there is no solution, but it is not always directly obvious if you can do that.

 

[fresh_42]

Is there a quick test to determine if a system of nonlinear equations is inconsistent. For example, suppose there is a system of equations such as:

3x cubed + 2y cubed = z cubed
2x cubed + 5y cubed = z cubed

Since these two equations are clearly not dependent, could we say that since they both are equal to the same term (z cubed), that they are therefore, inconsistent?

They are not "inconsistent" which presumable means "without solution". Those kind of questions define an algebraic variety and are subject to commutative algebra and algebraic geometry. As far as I know, there is no test in "P" that decides the shape of the zeros, but this is more of a guess.