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R.<z1,z2,z3,z4,p,a1,a2,a3,a4> = PolynomialRing(QQ,9,order='lex(4),lex(1),deglex(4)')
asym1 = -(z1+z2+z3+z4)
asym2 = +(z1*z2+z1*z3+z1*z4+z2*z3+z2*z4+z3*z4)
asym3 = -(z1*z2*z3+z1*z2*z4+z1*z3*z4+z2*z3*z4)
asym4 = +(z1*z2*z3*z4)
Isym = R.ideal([a1-asym1, a2-asym2, a3-asym3, a4-asym4])
pexpr = z1*z3+z2*z4
Ip = R.ideal([p-pexpr])
fexpr = z1^2*z2+z2^2*z3+z3^2*z4+z4^2*z1
fexprConj = fexpr.subs({z1:z3,z3:z1})
coef0 = fexpr*fexprConj
coef1 = -(fexpr+fexprConj)
B = (Isym+Ip).groebner_basis()
coef1.reduce(B)
coef0.reduce(B)