The Integer Linear Recurrence Zero Set Under the Microscope

The zero set of a linear recurrence is {m | xm = 0}, where xm is the m-th term. For a linear recurrence over the complex numbers C it is classical that xm = P1(m) · α1 + · · ·+ Pd(m) ·αd where P1(m), . . . , Pd(m) are polynomials in m, and α1, . . . , αd are the distinct zeros of the recurrence’s characteristic polynomial. We show that if the zero set of a linear recurrence over C is infinite, then Pi = 0, identically for every aperiodic αi, or there are no aperiodic zeros. A complex number α is aperiodic if α/|α| is not a root of unity. In the case where all zeros are simple, we also show that deciding whether Pi = 0 for all aperiodic αi is in P. We conjecture this remains true in general. The key result used in our analysis is the Weyl-von Neumann Theorem. 1 Linear recurrence zero set problems Let K be any commutative ring with identity. Chiefly we will be interested in K = Z, the ring of integers, and K = C, the field of complex numbers. A K-linear recurrence has the form xr+m = a1 · xr+m−1 + · · ·+ ar · xm , (1) where x0, . . . , xr−1, a1, . . . , a)r ∈ K. When referring to a linear recurrence, we will always have the notation of Eq. 1 in mind. The zero set of a linear recurrence is defined to be {m | xm = 0}. The characteristic polynomial of a K-linear recurrence is defined to be y − a1 · yr−1 − · · · − ar . This notion is especially important when K is an algebraically closed field, and in that case, one has xm = d ∑ i=1 Pi(m) · αi , (2) where α1, . . . , αd are the distinct zeros of the characteristic polynomial, and the Pi are polynomials. See [4] for details. §School of Information Technology, James Cook University, Townsville, Qld. 4811, Australia [email protected]