Extract coefficients and corresponding monomials from a given polynomial in SymPy

Question

Given a symbolic multivariate polynomial P, I need to extract both its coefficients and corresponding monomials as lists:

def poly_decomp(P):
    ....
    return coeffs, monoms

such that P is the dot product of coefficients and monomials, e.g., if P(x,y) = ax**2 + bxy + cy**2 then we should get coeffs = [a, b, c] and monoms = [x**2, x*y, y**2].

Getting the coefficients is easy since the function is built in coeffs = P.coeffs(). However, I'm having trouble getting the monomials. Here the build in function returns a list of exponents, e.g., in the example above we would get P.monoms() = [(2,0),(1,1),(0,2)].

Obviously the idea would be, provided a list of the variables var=[x,y], to do something like

powers = P.monoms() 
monoms = [sympy.prod(x**k for x,k in zip(var, mon)) for mon in powers ]

However the polynomial class doesn't seem to offer a function that returns a list of variables. All I could find were the methods free_symbols and free_symbols_in_domain which return the sets {a, b, c, x, y} and {a, b, c}. So by taking their difference one could get the set {x, y}.

However then we are faced with the issue that the sets are unordered, hence converting it into a list might mess up the order in different ways depending on the number of variables.

I am kind of at a loss here. Any tips?

Answer 1

The property gens (short for generators) holds a tuple of symbols or other suitable objects that the polynomial is defined over.

from sympy import symbols, Poly

x, y = symbols('x y')
p = Poly(x**3 + 2*x**2 + 3*x*y + 4*y**2 + 5*y**3, x, y)
q = Poly(x**3 + 2*x**2 + 3*x*y + 4*y**2 + 5*y**3, y, x)
print(p.gens)  # (x, y)
print(q.gens)  # (y, x)

So,

[prod(x**k for x, k in zip(p.gens, mon)) for mon in p.monoms()]

returns [x**3, x**2, x*y, y**3, y**2].

Note also that the generators can be types other than symbols, for example:

import sympy

x = sympy.symbols('x')
poly = sympy.poly(sympy.sqrt(2) * x**2)
print('generators: {g}'.format(g=poly.gens))
print('monomials: {m}'.format(m=poly.monoms()))
print('coefficients: {c}'.format(c=poly.coeffs()))

which prints:

generators: (x, sqrt(2))
monomials: [(2, 1)]
coefficients: [1]

where:

• type(poly.gens[0]) is <class 'sympy.core.symbol.Symbol'>, and
• type(poly.gens[1]) is <class 'sympy.core.power.Pow'>.

A relevant method is sympy.polys.polytools.Poly.as_dict, which returns a dict with keys that are monomials, and values that are the corresponding coefficients.

Answer 2

Maybe late to answer, but just to supplement the great answer above:

In Sympy 1.8, a Poly object x has the attribute .args, i.e.:

import sympy as sp  

x = sp.Poly(x**3 + 2*x**2 + 3*x*y + 4*y**2 + 5*y**3)
x.args[1:] #returns (x,y)

and x.args[1:] will be sorted based on how you defined your symbols. For instance, if your polynomial has n variables in terms of the symbols:

a = sp.symbols('a:n')

then x.args[1:] would be in the order (a0, ..., an). Knowing this and x.monoms(), I'm sure it could be put in the format that you wanted.