Question: How to select monomials based on a rule?


Unfortunately I got stuck again when trying to work with monomials.  Consider the following set of set of monomials:

f := [theta[1]*y+theta[2]*z,theta[3]+theta[4]*x+theta[5]*y+theta[6]*z+theta[7]*x*y+theta[8]*x*z+theta[9]*y*z,theta[10]*x+theta[11]*y+theta[12]*z+theta[14]*x*y+theta[15]*x*z+theta[16]*y*z+theta[17]*x^2+theta[18]*y*y+theta[19]*z*z+theta[20]];


x, y and z are the variables and thetas are the coefficients.  The coefficients theta can be zero and I need to classify the resulting set as valid or not.   Here are some examples of not valid sets



fffff :=[theta[1]*y+theta[2]*z,theta[3]+theta[4]*x+theta[5]*y+theta[6]*z+theta[7]*x*y+theta[8]*x*z+theta[9]*y*z,theta[12]*z+theta[19]*z*z+theta[20]];

that is, the first coordinate of the set cannot be a function of x alone, the second coordinate cannot be a function of y alone and the third coordinate cannot be a function of z only.  


I could not figure out how to do that automatically, can you help me, please?


Many thanks.




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