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Maps of monomials to monomials

Let us assume I have a function F that takes say degree 1 monomials to degree 2 monomials and degree 3 monomials in variable x,y (example: multiplication by $x+y^{2}$). I have a basis for degree 1 monomials given by x,y; a basis for degree 2 given by $x^{2},xy,y^{2}$, and a basis for degree 3 monomials given by $x^{3},x^{2}y,xy^{2},y^{3}$. So if I want to express F as a matrix, I can say g1= F(x), g2=F(y).

Then the first row of my matrix will be coefficient of $x^{2}$ in g1, coefficient of $xy$ in g1, coefficient of $y^{2}$ in g1, coefficient of $x^{3}$ in g1, ...., coefficient of $y^{3}$ in g1. The second row will just be the same with g1 replaced with g2.

The issue is if I want degree 10 monomials, the writing out each basis element is too much, especially if there were more variables. I was wondering if there was a simpler way in Sage to write this in coding language without writing out each basis element. Suppose I want degree k monomials to degree m and n monomials. Basically, I feel I need some command like the following. I feel there should be a package for this maybe?

  1. Define F. Let B be a k x (m+n) matrix. Take the space of all degree k monomials (maybe ordered lexigraphically?). Apply F to that space. For each element F is applied to, in the image, extract degree m part. Write it out as a vector in the space of degree m monomials. Same for degree n part. Put vector into part of matrix B.