2014/03/15 by Michael Farinton Challis, Challis, Michael Farinton
Computer Science · Engineering · #Advanced Numerical Methods in Computational Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Numerical Methods and Algorithms #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1403.3810
openalex publication_date 2014/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Ak = 1, a2, ... ak is an h-basis for X if every positive integer not exceeding X can be expressed as the sum of no more than h values ai; X(h) is called the h-range of the basis. h0 is the smallest value of h for which X(h) is greater than or equal to ak, and h1 is the smallest value for which X(h+1) = X(h) + ak for all h greater than or equal to h1. h2 identifies a further "stabilisation" in the h-range - a definition is included in the body of the paper. It is known that h1 and h2 do not exceed h0 for h-bases A3, but published proofs are complicated (see Ch. VIII of [4], where references [3] and [5] are given). This paper introduces the concept of a "stride generator" A = 1, a2, a3 which, while sharing some of the properties of a basis A3, is simpler to treat mathematically. We establish a relationship between stride generators and h-bases, and show that the result follows immediately if the stride generator underlying a basis has a particular property - here called "canonicality". The proof is lengthy (with a number of special cases to consider), but the underlying principles remain simple.