Let I  be a polynomial in K[A][X] s.t. A is a sequence of parameters (coefficients of f in F) and X is a sequence of variables. I want to extract the variables from ideal I.

For example if I=[(a-1)x*y^2-b+x, x-y+x^2-c] s.t. a,b,c are parameters and x,y are variables. I want {x,y} as the output of algorithm.

How can I decide that a polynomial is univariate? I want an algorithm that gives a polynomial and its output be true if f is univariate, and be false otherwise.

Let I=<3x^2+2xy+x, y-xy+3, y^2-2x+4> be a polynomial ideal in K[x,y]. I want to form a matrix M corresponding to this ideal as the following:      

                                 x^2     xy     x      y^2      y      constant

                               -----     ----   ----    ----     ----     ------

                                  [3       2       1       0       0           0]

                             M= [0      -1      0        0       1           3]

                                  [0       0     -2        1       0           4]

Please note that in the first, the all monomials appeared in generators of I,  sorted by lexicographic ordering x>y. How can I from matrix M from polynomial I?

 Let M be a matrix with polynomial array f_i's such that any array is in K[a_1,..,a_m][x_1,..,x_n] where a=a_1,,,a_m are sequence of parameters and x=x_1,..,x_n are sequense of variables. Now, I want to extract the coefficients of  f_i that are in K[a_1,a_2,..,a_m]. For example if how can I extract the matrix coefficint ?

Please note that a,b,c,d are parameters and x,y are variables.