We prove the reciprocity law for the twisted second moments of Dirichlet $L$-functions over rational function fields, corresponding to two irreducible polynomials. This formula is the analogue of the formulas for Dirichlet $L$-functions over $\mathbb{Q}$ obtained by Conrey [‘The mean-square of Dirichlet $L$-functions’, arXiv:0708.2699 [math.NT] (2007)] and Young [‘The reciprocity law for the twisted second moment of Dirichlet $L$-functions’, Forum Math. 23(6) (2011), 1323–1337].

A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain ‘Condition (D)’ on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further generalized Alon–Füredi theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon–Füredi. We then discuss the relationship between Alon–Füredi and results of DeMillo–Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon–Füredi theorem and its generalization in terms of Reed–Muller-type affine variety codes is shown, which gives us the minimum Hamming distance of these codes. Then we apply the Alon–Füredi theorem to quickly recover – and sometimes strengthen – old and new results in finite geometry, including the Jamison–Brouwer–Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.

A univariate polynomial f over a field is decomposable if it is the composition f = g ○ h of two polynomials g and h whose degree is at least 2. We determine an approximation to the number of decomposables over a finite field. The tame case, where the field characteristic p does not divide the degree n of f, is reasonably well understood, and we obtain exponentially decreasing relative error bounds. The wild case, where p divides n, is more challenging and our error bounds are weaker.