We prove that if G is an Abelian group and A1, . . . , Ak ⊆ G satisfy mAi = G (the m-fold sumset), then A1 + · · · + Ak = G provided that k > cm log log |G|. This generalizes a result of Alon, Linial, and Meshulam [Additive bases of vector spaces over prime fields. J. Combin. Theory Ser. A, 57(2):203–210, 1991] regarding so-called additive bases.

Geometric complexity theory (GCT) is an approach to P vs. NP and related problems in complexity theory using algebraic geometry and representation theory. A fundamental problem in representation theory, believed to be important for this approach, is the Kronecker problem, which asks for a positive combinatorial formula for the multiplicity gλμν of an irreducible representation Mν of the symmetr...

Any configuration of lattice vectors gives rise to a hierarchy of higher-dimensional configurations which generalize the Lawrence construction in geometric combinatorics. We prove finiteness results for the Markov bases, Graver bases and facet posets of these configurations, and we discuss applications to the statistical theory of log-linear models.

We investigate, for quotients of the non-commutative polynomial ring, a property that implies finiteness of Gröbner bases computation, and examine its connection with Noetherianity. We propose a Gröbner bases theory for our factor algebras, of particular interest for one-sided ideals, and show a few applications, e.g. how to compute (one-sided) syzygy modules.