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In your workflow, can you use the right s (as an element of FF, not SR) from the beginning ?

sage: a,b = var('a b')
sage: FF.<s> = FunctionField(SR)
sage: expr1 = (a^2*s + 2)/(s^3 + s + 3) + s
sage: expr1
(s^4 + s^2 + (a^2 + 3)*s + 2)/(s^3 + s + 3)
sage: expr1.numerator()
s^4 + s^2 + (a^2 + 3)*s + 2
sage: expr1.denominator()
s^3 + s + 3

Or is the fact that s as a symbolic variable in expr1 can not be changed in the beginning ?

In your workflow, can you use the right s (as an element of FF, not SR) from the beginning ?

sage: a,b = var('a b')
sage: FF.<s> = FunctionField(SR)
sage: expr1 = (a^2*s + 2)/(s^3 + s + 3) + s
sage: expr1
(s^4 + s^2 + (a^2 + 3)*s + 2)/(s^3 + s + 3)
sage: expr1.numerator()
s^4 + s^2 + (a^2 + 3)*s + 2
sage: expr1.denominator()
s^3 + s + 3

Or is the fact that s as a symbolic variable in expr1 can not be changed in the beginning absolute ?