full_simplify produces unexpected result

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The problem is that you use floating point numbers. For them the operation are unexact and lcm(p^2, p) is likely to be p^3 rather than being p^2 even with simple expressions

sage: p = 2.1*z^2 + 0.1
sage: f = (1/p + 1/p**2).full_simplify()
sage: f
(4.41*z^4 + 2.52*z^2 + 0.11)/(9.261000000000001*z^6 + 1.323*z^4 + 0.06300000000000001*z^2 + 0.001)
sage: (p^3).expand()
9.26100000000000*z^6 + 1.32300000000000*z^4 + 0.0630000000000000*z^2 + 0.00100000000000000

If you exact numbers like integers or rationals this is fine

sage: g1=12108*s^4 + 3016*s^3 + 10600*s^2 + 1937*s + 409/s + 472/s^2 + 28/s^3 + 24/s^4 + 3397
sage: g2=g1.subs(s=((z-1)/(z+1)))  
sage: g2.full_simplify()
(31991*z^8 - 158168*z^7 + 408320*z^6 - 668856*z^5 + 758902*z^4 - 603528*z^3 + 332472*z^2 - 116200*z + 21211)/(z^8 - 4*z^6 + 6*z^4 - 4*z^2 + 1)

Here is a simpler example showing the problem (which comes from using floating-point entries).

sage: s, z = var('s z')
sage: f = s^2 + s + 1/s + 1/s^2 + 1.
sage: g = f.subs(s=(z-1)/(z+1))
sage: g.full_simplify()
(5.0*z^6 + 5.0*z^4 - 9.0*z^2 - 1.0)/(z^6 - 3.0*z^4 + 3.0*z^2 - 1.0)

By contrast, with integer entries:

sage: s, z = var('s z')
sage: f = s^2 + s + 1/s + 1/s^2 + 1 
sage: g = f.subs(s=(z-1)/(z+1))
sage: g = f.subs(s=(z-1)/(z+1))
sage: g.full_simplify()
(5*z^4 + 10*z^2 + 1)/(z^4 - 2*z^2 + 1)

So applying simplify_full in the first seems to introduce an extra factor (z^2-1) in the numerator and denominator.

Exercise: find out what simplify_full does, follow each step and figure out where things go wrong.