Prime ideals of Bhargava domains

Let DD be a domain with quotient field KK. For any non-zero x∈Dx∈D, we consider the ring Bx(D)={f∈K[X]∣∀a∈D,f(xX+a)∈D[X]}Bx(D)={f∈K[X]∣∀a∈D,f(xX+a)∈D[X]}. These domains are subrings of the ring of integer-valued polynomials Int(D). We study here the prime ideals of Bx(D)Bx(D) for DD a Krull domain. Using some localization properties, we focus first to the case of a discrete valuation domain VV, and we determine the prime ideals above mm, the maximal ideal of VV. In contrast to Int(V) where all the primes are of the form Ma={f∈Int(V)∣f(a)∈m̂}, (in one-to-one correspondence with the elements aa of V̂ the mm-adic completion of VV) we prove that there are two classes of prime ideals above mm in Bx(V)Bx(V). We give a complete description of the primes and the maximal ideals of Bx(V)Bx(V). By globalization, we obtain the prime ideals of Bx(D)Bx(D) above a height one prime, for DD a Krull domain.