We generalise the Elekes-Szab 'o theorem to arbitrary arity and dimension and characterise the complex algebraic varieties without power saving. The characterisation involves certain algebraic subgroups of commutative algebraic groups endowed with an extra structure arising from a skew field of endomorphisms. We also extend the Erd o s-Szemer 'edi sum-product phenomenon to elliptic curves. Our approach is based on Hrushovski's framework of pseudo-finite dimensions and the abelian group configuration theorem.
Meanwhile, Chernikov and Starchenko @cite_0 recently proved a version of Theorem in strongly minimal structures which are reducts of distal structures. This direction of generalisation is orthogonal to the one we consider here, where we restrict to the case of @math (this restriction is used in Lemma and in Proposition ).
@cite_0
2787605512 2147668104 2136727539 2168328960
Author(s): Chernikov, A; Starchenko, S | Abstract: We prove a generalizations of the Elekes-Szab #x27;o theorem for relations definable in strongly minimal structures that are interpretable in distal structures. We show that the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap @math . This is achieved by exhibiting @math -point metric spaces of negative type whose @math distortion is @math . Our result is based on quantitative bounds on the rate of degeneration of Lipschitz maps from the Heisenberg group to @math when restricted to cosets of the center. Recently an orthogonal basis ofWN -algebra (AFLT basis) labeled by N-tuple Young diagrams was found in the context of 4D 2D duality. Recursion relations among the basis are summarized in the form of an algebra SH c which is universal for any N. We show that it has an S3 automorphism which is referred to as triality. We study the level- rank duality between minimal models, which is a special example of the automorphism. It is shown that the nonvanishing states in both systems are described by N or M Young diagrams with the rows of boxes appropriately shued. The reshuing of rows implies there exists partial ordering of the set which labels them. For the simplest example, one can compute the partition functions for the partially ordered set (poset) explicitly, which reproduces the Rogers-Ramanujan identities. We also study the description of minimal models by SH c . Simple analysis reproduces some known properties of minimal models, the structure of singular vectors and the N-Burge condition in the Hilbert space. We prove a normalized version of the restricted invertibility principle obtained by Spielman-Srivastava. Applying this result, we get a new proof of the proportional Dvoretzky-Rogers factorization theorem recovering the best current estimate. As a consequence, we also recover the best known estimate for the Banach-Mazur distance to the cube: the distance of every n-dimensional normed space from ^n is at most (2n)^(5 6). Finally, using tools from the work of Batson-Spielman-Srivastava, we give a new proof for a theorem of Kashin-Tzafriri on the norm of restricted matrices.