Computer Discovery and Analysis of Large Poisson Polynomials

ABSTRACT

In two earlier studies of lattice sums arising from the Poisson equation of mathematical physics, it was established that the lattice sum 1/π · ∑_{m, n odd}cos (mπx)cos (nπy)/(m² + n²) = log A, where A is an algebraic number, and explicit minimal polynomials associated with A were computed for a few specific rational arguments x and y. Based on these results, one of us (Kimberley) conjectured a number-theoretic formula for the degree of A in the case x = y = 1/s for some integer s. These earlier studies were hampered by the enormous cost and complexity of the requisite computations. In this study, we address the Poisson polynomial problem with significantly more capable computational tools. As a result of this improved capability, we have confirmed that Kimberley’s formula holds for all integers s up to 52 (except for s = 41, 43, 47, 49, 51, which are still too costly to test), and also for s = 60 and s = 64. As far as we are aware, these computations, which employed up to 64,000-digit precision, producing polynomials with degrees up to 512 and integer coefficients up to 10²²⁹, constitute the largest successful integer relation computations performed to date. By examining the computed results, we found connections to a sequence of polynomials defined in a 2010 paper by Savin and Quarfoot. These investigations subsequently led to a proof, given in the Appendix, of Kimberley’s formula and the fact that when s is even, the polynomial is palindromic (i.e., coefficient a_k = a_{m − k}, where m is the degree).

KEYWORDS:

• experimental mathematics
• MPFR
• multiprecision computation
• parallel computing
• Poisson equation
• PSLQ algorithm

2000 AMS SUBJECT CLASSIFICATION:

• 11K16
• 11K45

Acknowledgments

The authors wish to acknowledge computer equipment provided for our use by Apple Inc. and by the Lawrence Berkeley National Laboratory.