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These are questions asked by MDD

Hi every one

Let F= [x^2-z^2, x*y+y*z, x*z-y*z, y^2+y*z,w^2] be a special list of monomials and binomials. How to convert F into the following list:

FF=[x^2=z^2, x*y=-y*z, x*z=y*z, y^2=-y*z,w^2=0]. Please note that the left side of each equation of FF is greater than the right side w.r.t. lex ordering plex(x,y,z,w). 

Thank you in advance.

Hi every one

How do I automatically change the polynomial ring R=K[x1,...,xn] into the quotient ring R/I when I is a homogeneous ideal of degree d in R?

For example, let I be a polynomial ideal generated by [x-y, x^2+y^2+z^2] in R=K[x,y,z]. In the middle of the computations, the ideal B=[x-y] is computed and now we have to continue the calculation at the quotient ring R/B. How does change R into R/B automatically in Maple and then [x^2+y^2+z^2] changes into [2y^2+z^2] and K[x,y,z]/[x-y] ----> K[y,z]?

Thank you in advance.

Hi every one

Let F=[f1,...,f10] be a  list of homogeneous polynomials with different degrees. I want to create a list of lists s.t. any list satisfies the following conditions:

1. all elements of a list have the same degree.

2. the lists contained in the main list sort increasingly.

For example if F=[x-y, y+z, x^3-xyz+z^3, y^3-xz^2, y^6-x^4y^2, x^5y-z^6+x^2y^2z^2] then 

L=[[x-y, y+z],[x^3-xyz+z^3, y^3-xz^2],[y^6-x^4y^2, x^5y-z^6+x^2y^2z^2]] is the output.

Thanks for your answers.


I have a question concerning the matrix. Is there any Maple command or function for counting the nonzero elements in any row of a matrix?

Thanks for your help.

Let us consider the following assumptions:

Any set of binomials $B \in R=K[x_1, \cdots, x_n]$ induces an equivalence relation on the set of monomials in $R$ under which $m_1∼m_2$ if and only if $m_1−tm_2\in \langle B \rangle$ for some non-zero $t\in K$. As a k-vector space, the quotient ring $R/B$ s spanned by the equivalence classes of monomials. Now let $f =x^2−y^2$ be a binomial in $K[x, y]$. Among monomials of total degree three, $x^3$ and $xy^2$, as well as $x^2y$ and $y^3$ become equal in $K[x, y] / \langle f\rangle$.

Why the degree three part in the quotient is two-dimensional with one basis vector per equivalence class?

Also, why does the polynomial $f=x^3+xy^2+y^3$ map to a binomial with a coefficient matrix [2, 1]? I think this matrix arises from the matrix [1, 1, 1, 0] by summing the columns corresponding to $x^3$ and $xy^2$ and those for $x^2y$ and $y^3$. 

How can I implement a simple code to obtain these results in {\tt Maple}?

I am looking forward to hearing any help and guidance.

Thank you in advance

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