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Positive semigroups in lattices and totally real number fields (Lenny Fukshansky, CMC)
December 6, 2022 @ 12:15 pm - 1:10 pm
Let L be a full-rank lattice in R^n and write L+ for the semigroup of all vectors with nonnegative coordinates in L. We call a basis X for L positive if it is contained in L+. There are infinitely many such bases, and each of them spans a conical semigroup S(X) consisting of all nonnegative integer linear combinations of the vectors of X. Such S(X) is a sub-semigroup of L+, and we investigate the distribution of the gaps of S(X) in L+, i.e. points in L+ outside of S(X). We describe some basic properties and counting estimates for these gaps. Our main focus is on the restrictive successive minima of these sets, for which we produce bounds in the spirit of Minkowski’s successive minima theorem. We apply these results to obtain analogous bounds for the successive minima with respect to Weil height of totally positive sub-semigroups of ideals in totally real number fields. Joint work with Siki Wang (CMC’22).