# Functions for Polynomials

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# for a symmetric function 'f' in 'n' variabels 'DV' of the polynomial ring 'PR'
# generate the representation for the 'n' elementary symmetric polynomials

def symmetric_reduction(f, SV, DV, SymR, PR, n):
    e = SymmetricFunctions(PR).elementary()
    SE = [PR(e([i+1]).expand(n)) for i in range(n)]
    s = SymR(0)
    fs = recursive_symmetric_reduction([f, s], SE, SV, DV, SymR, PR, n)
    return(fs[1])

def recursive_symmetric_reduction(fs, SE, SV, DV, SymR, PR, n):
    if fs[0] == 0:
       return(fs)
    else:
       f = PR(fs[0])
       s = fs[1]
       lm = f.monomials()[0]
       lc = f.coefficients()[0]
       lt = lm.exponents()[0]
       f = f - lc*mul([(SE[i]**(lt[i] - lt[i+1]) if i != n - 1 else SE[i]**(lt[i])) for i in range(n)])
       s = s + SymR(lc*mul([SV[i]**(lt[i] - lt[i+1]) if i != n - 1 else SV[i]**(lt[i]) for i in range(n)]))
       return(recursive_symmetric_reduction([f, s], SE, SV, DV, SymR, PR, n))

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