If we have a multivariate polynomial in SAGE for instance
f=3*x^3*y^2+x*y+3
how can i display the full list of coefficients including the zero ones from missing terms between maximum dregree term and constant.
P.<x,y> = PolynomialRing(ZZ, 2, order='lex')
f=3*x^2*y^2+x*y+3
f.coefficients()
gives me the list
[3, 1, 3]
but i'd like the "full" list to put into a a matrix. In the above example it should be
[3, ,0 , 0, 1, 0, 0, 0, 0, 3]
corresponding to terms:
x^2*y^2, x^2*y, x*y^2, x*y, x^2, y^2, x, y, constant
Am I missing something?
Your desired output isn't quite well defined, because the monomials you listed are not in the lexicographic order (which you used in the first line of your code). Anyway, using a double loop you can arrange coefficients in any specific way you want. Here is a natural way to do this:
coeffs = []
for i in range(f.degree(x), -1, -1):
for j in range(f.degree(y), -1, -1):
coeffs.append(f.coefficient({x:i, y:j}))
Now coeffs is [3, 0, 0, 0, 1, 0, 0, 0, 3], corresponding to
x^2*y^2, x^2*y, x^2, x*y^2, x*y, x, y, constant
The built-in .coefficients() method is only useful if you also use .monomials() which provides a matching list of monomials that have those coefficients.