More than twenty-five years ago, Manickam, Miklós, and Singhi conjectured that for positive integers n, k with n ≥ 4k, every set of n real numbers with nonnegative sum has at least n−1 k−1 k-element subsets whose sum is also nonnegative. We verify this conjecture when n ≥ 8k 2 , which simultaneously improves and simplifies a bound of Alon, Huang, and Sudakov and also a bound of Pokrovskiy when k < 10 45 .Moreover, our arguments resolve the vector space analogue of this conjecture. Let V be an n-dimensional vector space over a finite field. Assign a real-valued weight to each 1-dimensional subspace in V so that the sum of all weights is zero. Define the weight of a subspace S ⊂ V to be the sum of the weights of all the 1-dimensional subspaces it contains. We prove that if n ≥ 3k, then the number of k-dimensional subspaces in V with nonnegative weight is at least the number of k-dimensional subspaces in V that contain a fixed 1-dimensional subspace. This result verifies a conjecture of Manickam and Singhi from 1988.
We construct a lifting of commuting power series from a residue field of positive characteristic to a local ring, in a manner similar to the formal group cohomologies of Lubin-Tate. This lifting is then used to prove that a nontorsion Z p automorphism and a multiplication-by-p endomorphism uniquely define a formal group over fields of a positive characteristic. This suggests that a characteristic 0 conjecture by Lubin could be approached with residue field considerations.
Using results from the theory of the field of norms, we prove that if a height‐one commuting family of power series over ℤp is large enough, then there exists a formal group in the background. We suggest as a consequence an alternate hypothesis for a conjecture by Lubin.
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