We prove that a nonlinear version of the Grothendieck-Katz conjecture (essentially in the form given by Ekhedahl, Shepherd-Barron and Taylor) is equivalent to the original Grothendieck-Katz conjecture together with a certain differential algebraic geometric/model-theoretic statement: a type over C(t) with “p-curvature 0 for almost all p” is nonorthogonal to the constants.

In [Tama], a proof of the Grothendieck Conjecture (reviewed below) was given for smooth affine hyperbolic curves over finite fields (and over number fields). The purpose of this paper is to show how one can derive the Grothendieck Conjecture for arbitrary (i.e., not necessarily affine) smooth hyperbolic curves over number fields from the results of [Tama] for affine hyperbolic curves over finit...

The little Grothendieck problem (a special case of Boolean quadratic optimization) consists of maximizing ∑ ij Cijxixj over binary variables xi ∈ {±1}, where C is a positive semidefinite matrix. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C ∈ Rdn×dn a positive semidefinite matrix, the objective is to maximi...

We prove an explicit combinatorial formula for the structure constants of the Grothendieck ring of a Grassmann variety with respect to its basis of Schubert structure sheaves. We furthermore relate K-theory of Grassmannians to a bialgebra of stable Grothendieck polynomials, which is a K-theory parallel of the ring of symmetric functions.

0. Introduction. This paper continues the study of the noncommutative infinitesimal cohomology we introduced in [3]. This is the cohomology of sheaves on a noncommutative version of the commutative infinitesimal site of Grothendieck ([8]). Grothendieck showed that, for a smooth scheme X of characteristic zero, the cohomology of the structure sheaf on the infinitesimal site gives de Rham cohomol...

We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco–Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial properties of the reduced rectangular plactic algebra and associated Cauchy kernels.

We establish the conjecture of Reiner and Yong for an explicit combinatorial formula expansion a Grothendieck polynomial into basis Lascoux polynomials. This is subtle refinement its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, Yong, which gives stable polynomials indexed by permutations Grassmannian Our <mml:math xmlns:mml="ht...