Pete L. Clark

We introduce real induction, a proof technique analogous to Mathematical Induction but applicable to statements indexed by an interval on the real line. We apply these principles to give streamlined, conceptual proofs of basic results in elementary real analysis and topology. Then we pursue inductive principles in arbitrary ordered sets. Applications are given, e.g., to “real analysis in reverse.”

In 1959 Golomb defined a connected topology on ℤ. An analogous Golomb topology on an arbitrary integral domain was defined first by Knopfmacher-Porubský [KP97] and then again in a recent work of Clark [Cl17]. Here we study properties of Golomb topologies.

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.