MDD

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@Dmitry Lyakhov 

You can use double sequences for your purpose. This is faster from loop for. See the following example:

> A := [1, 2, 3]; B := [a, b, c];
                          
> seq(seq({A[i], B[j]}, j = 1 .. nops(B)), i = 1 .. nops(A));


{1, a}, {1, b}, {1, c}, {2, a}, {2, b}, {2, c}, {3, a}, {3, b}, {3, c}

@Carl Love Yes it is good for some example so far.

@Carl Love I think it is better that in Homogenize procedure you remove V from its input and set V=[op(T)] at inside the procedure. What is your idea?

@Carl Love I need both f and F in my main procedure. It is possible that the monomials appears in F are different by monomial in f. In fact the input of this algorithm are (f , F , a monomial ordering) and its output is homogeneous ff and FF.

@MDD Consider the following example:

its output is:

Also we know that its output can be "true,[a,0,0]". This example shows that in your implementation the coefficients are not uniqe. I think this happened since the polynomials

are linearly dependent. I deduce that when the list of polynomials be linearly independent then the coefficients are uniqe. What is your idea about this example and my statements?

@Carl Love This is complete now. Thank you so much for your implementation.

@Carl Love Consider the following examples, too:

In the above examples $a$ is a nonzero parameter and the outputs must be true but both outputs are false.

 

@Carl Love I think that there is a minor error in your procedure, Please run the following example:

Could you please remove this error? If the polynomial is zero then it is not a problem. Please set the output true with zero coefficients.

@Carl Love This is very complete and I have to say thank you so much. I know implementation of some computer algebra algorithms but your implementation is very professional.

@Carl Love Thank you so much. I know these command! I want to know in general the way of your implementation. Thank you so much.

@Carl Love This is very good. Could you please explain your way? I dont know some command that you use in the above your implementation such as "C*" and "~" .

@Carl Love Could you please combine this way with your LinearCombo for checking whether a polynomial f is linear independent of some other polynomials F. Please note that f is a polynomial with parametric coefficients and F is a list of polynomials with parametric coefficients. I want the output is

1- true(f is linear dependent of F) and the coefficients vectors

or

2- false(f is linear independent of F)

@Carl Love Thanks for this good way. I have a suggestion: When its output is false (the polynomials are linearly independent) then the finding coefficients is meaningless. For saving memory and time it is better do not compute the coefficient, when it is false. What is your idea? 

@Markiyan Hirnyk I want a general procedeure to use it in my main algorithm some times, not for special example.

@Markiyan Hirnyk I want to the parameters specialize automatically means that the input of algorithm is a list of parametric polynomials only.

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