Prime specialization in higher genus I

Prime Specialization in Higher Genus I

A classical conjecture predicts how often a polynomial in Z[T ] takes prime values. The natural analogous conjecture for prime values of a polynomial f(T ) ∈ κ[u][T ], where κ is a finite field, is false. The conjecture over κ[u] was modified in earlier work by introducing a correction factor that encodes unexpected periodicity of the Möbius function at the values of f on κ[u] when f ∈ κ[u][T ]...

Prime Specialization in Higher Genus Ii

We continue the development of the theory of higher-genus Möbius periodicity that was studied in Part I for odd characteristic, now treating asymptotic questions and the case of characteristic 2. The extra difficulties in characteristic 2 are overcome via rigid geometry in characteristic 0. The results on Möbius periodicity in any positive characteristic are used to incorporate a correction fac...

Prime Specialization in Genus 0

For a prime polynomial f(T ) ∈ Z[T ], a classical conjecture predicts how often f has prime values. For a finite field κ and a prime polynomial f(T ) ∈ κ[u][T ], the natural analogue of this conjecture (a prediction for how often f takes prime values on κ[u]) is not generally true when f(T ) is a polynomial in T p (p the characteristic of κ). The explanation rests on a new global obstruction wh...

Prime Higher Derivations on Algebras

I-prime ideals

In this paper, we introduce a new generalization of weakly prime ideals called $I$-prime. Suppose $R$ is a commutative ring with identity and $I$ a fixed ideal of $R$. A proper ideal $P$ of $R$ is $I$-prime if for $a, b in R$ with $ab in P-IP$ implies either $a in P$ or $b in P$. We give some characterizations of $I$-prime ideals and study some of its properties.  Moreover, we give conditions  ...