Betting strategies are often expressed formally as martingales. A martingale is called integer-valued if each bet must be an integer value. Integer-valued strategies correspond to the fact that in most betting situations, there is a minimum amount that a player can bet. According to a well known paradigm, algorithmic randomness can be founded on the notion of betting strategies. A real X is cal...

Let a be an integer with |a| > 1. Let f(T ) ∈ Q[T ] be a nonconstant, integer-valued polynomial with positive leading term, and suppose that there are infinitely many primes p for which f does not possess a root modulo p. Under these hypotheses, Ballot and Luca showed that almost all primes p do not divide any number of the form a−1. More precisely, assuming the Generalized Riemann Hypothesis (...

Motivated by Wilmshurst’s conjecture, we investigate the zeros of harmonic polynomials. We utilize a certified counting approach which is a combination of two methods from numerical algebraic geometry: numerical polynomial homotopy continuation to compute a numerical approximation of each zero and Smale’s alpha-theory to certify the results. Using this approach, we provide new examples of harmo...

This note investigates two long-standing conjectures on the Krull dimension of integer-valued polynomial rings and of polynomial rings, respectively, in the context of (locally) essential domains.

This paper describes a probabilistic method for verifying the equivalence of two multiple-valued functions. Each function is hashed to an integer code by transforming it to a integer-valued polynomial and the equivalence of two polynomials is checked probabilistically. The hash codes for two equivalent functions are always the same. Thus, the equivalence of two functions can be verified with a ...