We study for each n a one-parameter family of complex-valued measures on the symmetric group Sn, which interpolate the probability of a monic, degree n, square-free polynomial in Fq[x] having a given factorization type. For a fixed factorization type, indexed by a partition λ of n, the measure is known to be a Laurent polynomial. We express the coefficients of this polynomial in terms of characters associated to Sn-subrepresentations of the cohomology of the pure braid group H • (Pn, Q). We deduce that the splitting measures for all parameter values z = − 1 m resp. z = 1 m , after rescaling, are characters of Sn-representations (resp. virtual Sn-representations.)

We use generating functions to relate the expected values of polynomial factorization statistics over Fq to the cohomology of ordered configurations in R 3 as a representation of the symmetric group. Our methods lead to a new proof of the twisted Grothendieck-Lefschetz formula for squarefree polynomial factorization statistics of Church, Ellenberg, and Farb.

Let φ ∈ Q(z) be a polynomial or rational function of degree 2. A special case of Morton and Silverman's Dynamical Uniform Boundedness Conjecture states that the number of rational preperiodic points of φ is bounded above by an absolute constant. A related conjecture of Silverman states that the canonical heightĥ φ (x) of a non-preperiodic rational point x is bounded below by a uniform multiple of the height of φ itself. We provide support for these conjectures by computing the set of preperiodic and small height rational points for a set of degree 2 maps far beyond the range of previous searches.

Let M d,n (q) denote the number of monic irreducible polynomials in Fq[x1, x2, . . . , xn] of degree d. We show that for a fixed degree d, the sequence M d,n (q) converges coefficientwise to an explicitly determined rational function M d,∞ (q). The limit M d,∞ (q) is related to the classic necklace polynomial M d,1 (q) by an involutive functional equation we call liminal reciprocity. The limiting first moments of factorization statistics for squarefree polynomials are expressed in terms of symmetric group characters as a consequence of liminal reciprocity, giving a liminal analog of the twisted Grothendieck-Lefschetz formula of Church, Ellenberg, and Farb.

For the class of functions fc(x)=x^2+c, we prove a conditional bound on the number of rational solutions to fc^N(x) =−1 and make computational conjectures for a bound on the number of rational solutions to fc^N(x)=a fora in a specific subset of the rationals.