Effective results for polynomial values of (alternating) power sums of arithmetic progressions

We prove effective finiteness results concerning polynomial values of the sums bᵏ + (a+b) ᵏ + + (a (x-1) + b) ᵏ and bᵏ - (a+b) ᵏ + (2a+b) ᵏ - + (-1) ^x-1 (a (x-1) + b) ᵏ, where a 0, b, k are given integers with (a, b) =1 and k 2.