Sums of Reciprocals of Polynomials over Finite Fields

Consider the following example (typical in college algebra): 1 x2 + 1 x2 + 1 + 1 x2 + x + 1 x2 + x+ 1 = 4x + 6x + 8x + 6x + 3x+ 1 x2(x+ 1) (x2 + 1) (x2 + x+ 1) . Now let’s assume that all the coefficients in the above are from the binary field F2 = Z2 = {0, 1}. The result becomes much cleaner: 1 x2 + 1 x2 + 1 + 1 x2 + x + 1 x2 + x+ 1 = 1 (x2 + x)(x4 + x) . After a brief introduction to finite fields, we consider the sum of the reciprocals of all monic polynomials of a given degree over a finite field Fq each raised to the power of k. When k ≤ q, the sum has a surprisingly simple result due to mysterious cancellations that occur in the sum. We discuss this interesting phenomenon and its connection to a deeper problem. The talk is based on a recent paper in the MAA Monthly: K. Hicks, X. Hou, G. L. Mullen, Sums of reciprocals of polynomials over finite fields, Amer. Math. Monthly, 119 (2012), 313 – 317.