Multiplication of polynomials is terrible

[aljce]

Specifically the instance for vectors is awful

polyTimes :: Semiring a => Vector a -> Vector a -> Vector a
polyTimes x y
  = if Vector.null x then x else if Vector.null y then y else
      Vector.cons (times a b) (Vector.map (a *) q + Vector.map (* b) p + (Vector.cons zero (polyTimes p q)))
  where
    a = Vector.unsafeHead x
    b = Vector.unsafeHead y
    p = (\t d -> if Vector.null t then d else t) (Vector.tail x) (Vector.empty)
    q = (\t d -> if Vector.null t then d else t) (Vector.tail y) (Vector.empty)

Technically this is the optimum in terms of asymptotic analysis.
Proof:

C(0) = 0
C(n) = n (Vector.cons) + n (Vector.map) + n (Vector.map) + n (Vector.cons) + C(n - 1) (recursive call)

That recurrence represent the worst case running time of polyTimes. Solving this with the master method C(n) = Θ(n^2). This being said the asymptotics hide some serious constants. You copy the vector into a new vector two times each recursive call because of Vector.cons. Never use Vector.cons it has to copy the entire vector into a new one that is just one element bigger. Instead rethink the algorithm to not use Vector.cons. You could completely redesign the alg or use ST and mutable vectors.

[chessai]

fixed