Canonical expansion of integers for families of roofline polynomials

Abstract. We investigate the existence and length of canonical expansion of integers in polynomial bases where the bases are taken from some infinite families of polynomials which we call roofline polynomials. A roofline polynomial's coefficients form a sequence that is first increasing, then constant, then decreasing, with an additional symmetry condition. We prove necessary conditions for the CNS property, and in special cases we also prove that the condition is sufficient. We provide results on the length of the expansion of \(-1\) as a function of the degree of the base. We also formulate some open problems about the canonical number system property of these families.