I want to test linearly dependence of a polynomial **f** on a list of polynomials **F** by **additional condition on parametric coefficients** of linear parametric polynomial (linear for variables not parameters). Please note that:

- The polynomial
** f **and the members of **F **are always homogenous in the variables.
- The coefficients of
**f**, the coefficients of the members of **F** are all always polynomials in the parameters or contant and the members of **N** and** W** are all always polynomials in the parameters.

For example let

and

(a,b,c,d,e,h are parameters and A1,A2,A3 are variables).

If I use **PolyLinearCombo(F,f,{A1,A2,A3}) (**see *http://www.mapleprimes.com/questions/204469-How-Can-I-Find-The-Coefficients-Of-Linear#comment217621***)**then its output is** false,[]**.

Now we let to condition sets for parameters as the following:

**N:=[ebc+ahd]**

**W:=[a,c]**

The elements of N must be zero means that ebc+ahd=0

and the elements of W are non-zero that is a<>0 and c <>0.

Let a=b=c=d=h=1 and e=-1. This specialization satisfy in the above condition sets **N **and** W**. By this specialization we have:

and

Now if I use **PolyLinearCombo(F,f,{A1,A2,A3}) **then its output is** true,[-1,1].**

By this additional two condition sets I have to check that whether f is linearly independent of F or not. How can I do this without specialization? In fact I want an algorithm that its input is (null condition **N**, not-null condition **W**, list of polynomials **F**, a polynomial **f**, the set of variables) and its output is true and coefficients if **f** is linearly dependent of **F** w.r.t. null and not-null conditions **N **and** W**, else its output is false.

If the name of new procedure is **ExtPolyLinearCombo **and

**N:=[ebc+ahd]**

**W:=[a,c]**

I want the output of

**ExtPolyLinearCombo**(N,W,F,f,{A1,A2,A3}) be** true,[coefficients]**

Thank you very much in advance.