# Evaluate polynomial over extension ring

Given a ring and polynomial like:

_.<I> = RR[]
K.<i> = RR.extension(I^2 + 1)
L.<a, b> = K[]
f = a + i*b


then I would expect that f(i=0) would give me a. But instead it just returns the same value again.

Any ideas how to do this? Thanks

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1

Does not make really sense, but

sage: f.map_coefficients(lambda cf:cf[0])
a

( 2024-01-19 12:12:20 +0200 )edit

thanks a lot!

( 2024-01-20 09:06:08 +0200 )edit

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The issue here is that you want to make substitution not into f but into each of its coefficients. Direct approach using .map_coefficients() is given by @FrédéricC in the comments.

However, with the i being the imaginary unit, there is no need to define it over RR, since one can work in CC where the imaginary unit is already defined as I. So, you can go directly with

L.<a, b> = CC[]
f = a + I*b
f.map_coefficients(real)


Alternative general approach is to define all a, b, i as polynomial variables and work in the quotient ring:

F.<I,A,B> = RR[]
K.<i,a,b> = F.quotient( [I^2 + 1] )
f = a + i*b
f.lift().subs({I:0})


Instead of f.lift().subs({I:0}) giving result in F, you can use f.lift().specialization({I:0}), which gives result in the polynomial ring of A and B only.

more

thanks a lot! Gives me a load of stuff to look into now. Good day

( 2024-01-20 09:05:38 +0200 )edit