@Thomas Richard I did type i mod 2 in maple but after copying here I saw it is written `mod` (i,2) and I thought `` are extra. Now I corrected it in question's face.

@Carl Love 

Thanks.

@acer I tried to use your idea but it was not successful, maybe there is something that I didn't pay attention to it, but may you take a look at my example?

with(Optimization):
Minimize(1, [0.1e-2 <= x[1], 0.1e-2 <= x[2], 0.1e-2 <= x[3], 0.1e-2 <= x[4], 0.1e-2 <= y[1], 0.1e-2 <= y[2], 0.1e-2 <= y[3], 0.1e-2 <= y[4], x[1]+x[3] = 0, -x[1]+x[4] = 0, y[1]+y[2]+y[4] = 0, y[3]-y[4] = 0]);

The error note is:

Error, (in Optimization:-LPSolve) no feasible solution found

outo := 1;
try
    Minimize(1, [0.1e-2 <= x[1], 0.1e-2 <= x[2], 0.1e-2 <= x[3], 0.1e-2 <= x[4], 0.1e-2 <= y[1], 0.1e-2 <= y[2], 0.1e-2 <= y[3], 0.1e-2 <= y[4], x[1]+x[3] = 0, -x[1]+x[4] = 0, y[1]+y[2]+y[4] = 0, y[3]-y[4] = 0])
catch "Error, (in Optimization:-LPSolve) no feasible solution found":
    outo := 0; 
end try;

But it doesn't work as outo remains still 1.

I typed ?homalg in my Maple (2015) and searched the Maplehelp but "No Matches Found". I really like to know if it is possible to compute SyZyGies and kernel of module homomorphisms in Maple using a package.

About the question, I always think having computational things inside of abstract things can have more benefit than only going with abstract without thinking how to do or see them @Axel Vogt.

@Joe Riel  I'm not asking about a specific example, what I'm looking for is how to compute kernel of these maps between polynomial rings using Maple. I changed the example to one with non-trivial one but it's not important if again the kernel is same for other charactersitic, you can add another variable z to the first ring and map it to 2 then when characteristic is 2 the kernel contains z also but otherwise it doesn't have z.

@Markiyan Hirnyk I mentioned an example of my question, there by C I meant complex numbers and C[X[1],X[2]] is the polynomial ring with coefficietns in C, and it has two variables X[1] and X[2]. An example of how to compute that kernel in Singular https://www.singular.uni-kl.de/Manual/latest/sing_324.htm#SEC363 There the field is a field of characteristic 32003 and they mapped x, y, z to themselves but w to 0, trivially the kernel is the ideal generated by w. But in my example characteristic is zero. It's easy to work with Singular but doing some computations in one software and other parts in another software is not very interesting so it will be very nice if I can do this computation in Maple too.

@Markiyan Hirnyk Thank you, primitive element was a good choice, why didn't I think about it yesterday.

For example I sometimes have big matrices or vectors, then when I encounter those Forton blob blob instead of seeing entries of my result, I use "for".

Specially when I want to see generators of an ideal in a column way which is easier for eyes to follow, then if my column is G, I write 

for i from 1 by 1 to nops(G) do G[i]; end do;

For matrices you can use double for and depending to what is important for you and you want to compare what you can use an order. So far I did this way. But sometimes the result is not really that big and commands that can force Maple to show the result or changing the bound which Maple detect bigness of the matrix are useful ^_^

@vv 

Yes, you're right! H[j] is the quotient of division of f on x^{deg(f)-j}. If you put it as the answer I will sign it as the accepted answer. Thanks.

Thank you vv, but I didn't mean Horener of a part of my polynomial. Let me explain ith Horner by an example instead of typing symbols for the general definition,

Let f=x^2+3x-1 then 

H[0]=1

H[1]=x+3

H[2]=x^2+3x-1

Pay attention that f=(((1)x+3)x-1) if you come out from inside step by step and compute you will encounter H[0] then H[1] then H[2]. In fact if f=\sum_{i=0}^n a_i x^i then H[j]=\sum_{i=0}^j a_{n-i}x^{j-i}.

@tomleslie 

Thanks.

@tomleslie 

It treats the notes in labels as math symbols not a text like 

labels = [Total Ligand Concentration(L[T])arbitrary units, Steady-state units (arbitrary concentration)]

It doesn't show paranthesis and doesn't observe the order of words in the second part. How can I tell Maple to treat these completely as a text in your command for lebals?