# Strange result with homomorphism between Laurent polynomial rings

I am trying to perform a certain computation in group theory. I start with a finitely presented group G, and I compute its Alexander matrix using the built-in method. This gives me a matrix over a multivariate Laurent polynomial ring with generators f4 and f5; I then apply a homomorphism "bar" which sends each of these to the generator t of a univariate Laurent polynomial ring.

```
F.<a,b,c,x,y> = FreeGroup()
G = F/[a*c*(c*a)^-1,a*x*(x*a)^-1,y*c*(c*y)^-1,x*y*x*(y*x*y)^-1,b*a*y*(y*b*a)^-1,(a*b*a*y)*(y*b*a*b)^-1,c*b*x*(x*c*b)^-1,b*c*b*x*(x*c*b*c)^-1]
M = G.abelian_alexander_matrix()[0]
R.<t> = LaurentPolynomialRing(QQ)
bar = M.base_ring().hom([t,t])
Mbar = matrix([[bar(x) for x in m] for m in M])
```

The baffling thing to me is that bar seems to be malfunctioning. For instance, the (0,0) entry of M is 1-f4, but the (0,0) entry of Mbar is t^-1 - 1. Pretty much the entirety of M/Mbar is like this: it seems like the entries in Mbar are correct only up to multiplication by some power of t, which is not uniform across Mbar. If I directly ask bar to evaluate 1-f4, it correctly returns 1-t.

Does anybody know what's going on, and how to fix it? Thanks!

Using

seems to work.

In contrast, the following simpler case works

and the extension works