Sympy matrix determinant of polynomials is not a polynomial

I want to calculate the determinant of some specific matrix. I have simplified my problem and try to demonstrate the problem with the simplest code below.

My problem is :

• Every element in my matrix is a polynomial.
• But the determinant of the matrix is not a polynomial.

I suspect that it should be caused by precision or something else, I would like to know if there is other way to avoid this problem. I hope the code below can successfully demonstrate my issue, thanks everyone.

import sympy as sp

class Matrix:
    def __init__(self, N):
        self.E = sp.symbols('E') # the matrix determinant will be represented as polynomial in E
        self.elements = [] # list of matrix elements
        for i in range(2*N):
            if i >= 2:
                self.elements.append(self.recursion(s=i, X=self.elements[i-1], Y=self.elements[i-2]))
            else:
                self.elements.append(self.recursion(s=i))

        self.m = sp.Matrix([[self.elements[i+j] for j in range(N)] for i in range(N)])

    def recursion(self, s, X=None, Y=None):
        if s == 0:
            return 1
        elif s == 1:
            return self.E
        else:
            return s*self.E*X + 0.5*Y # if change to s*self.E*X + Y will be no  problem

N = 4 # N=1,2,3 works fine
m = Matrix(N).m
det = m.det()

print("Is the determinant polynomial : ", det.is_polynomial()) # will get False
for element in m:
    print("Is this element polynomial : ", element.is_polynomial()) # will get True

print("The determinant looks like : \n", det)