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We determine the largest integer i such that 0 < i ≤ n and the coefficient of ti is odd in the polynomial (1 + t + t2 + · · · + tn)n+1. We apply this to prove that the co-index of the tangent bundle over FP n is stable if 2r ≤ n < 2r + 1 3 (2r − 2) for some integer r.

A CERTAIN PROPERTY OF POLYNOMIALS AND THE CI-STABILITY OF TANGENT BUNDLE OVER PROJECTIVE SPACES