import os
import sys
sys.path.insert(0, os.path.abspath('../..'))

import sympy as sp
import numpy as np
import itertools as it
import pysymmpol as sy
import pysymmpol.utils as ut
from IPython.display import display, Latex

Class: ElementaryPolynomial

As expected, everything we said about the Complete Homogeneous Polynomials is also true for the Elementary Symmetric Polynomials.

n = 3
elementary = [sy.ElementaryPolynomial(i) for i in range(n+1)]
t_dict = ut.create_miwa(n)

for a in elementary:
    display(a.explicit(t_dict))
    print(10*'-')

1

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$\displaystyle t_{1}$

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$\displaystyle \frac{t_{1}^{2}}{2} - t_{2}$

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$\displaystyle \frac{t_{1}^{3}}{6} - t_{1} t_{2} + t_{3}$

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elementary[2].explicit(t_dict, True)

$\displaystyle \operatorname{Poly}{\left( \frac{1}{2} t_{1}^{2} - t_{2}, t_{1}, t_{2}, domain=\mathbb{Q} \right)}$

m=2
tt = ut.tx_power_sum(elementary[-1].level, m)
for a in elementary:
    display(a.explicit(tt, True))
    print(10*'-')

1

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$\displaystyle \operatorname{Poly}{\left( x_{1} + x_{2}, x_{1}, x_{2}, domain=\mathbb{Q} \right)}$

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$\displaystyle \operatorname{Poly}{\left( x_{1}x_{2}, x_{1}, x_{2}, domain=\mathbb{Q} \right)}$

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$\displaystyle \operatorname{Poly}{\left( 0, x_{1}, x_{2}, domain=\mathbb{Q} \right)}$

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