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Arithmetic properties of polynomial specializations over finite fields

We present applications of some recent results that establish a partial finite field analogue of Schinzel’s Hypothesis H. For example, we prove that the distribution of gaps between degree n prime polynomials over Fp is close to Poisson for p large compared to n. We also estimate the number of polynomial substitutions without prime factors of large degree (“smooth” polynomial substitutions); this confirms a finite field analogue of a conjecture of Martin in certain ranges of the parameters. Other topics considered include an analogue of Brun’s constant for polynomials and “smooth” values of neighboring polynomials.