On systems of linear inequalities

Abstract

Abstract. -We show in detail that the category of general Roth systems or the category of semi-stable systems of linear inequalities of slope zero is a neutral Tannakian category. On the way, we present a new proof of the semi-stability of the tensor product of semi-stable systems. The proof is based on a numerical criterion for a system of linear inequalities to be semi-stable.Résumé (Sur certains systèmes d'inégalités linéaires). -On démontre en détail que la catégorie des systèmes de Roth généraux ou la catégorie des systèmes semi-stables d'inégalités linéaires de pente zéro est une catégorie tannakienne neutre. En chemin, on présente une nouvelle preuve de la semi-stabilité du produit tensoriel de systèmes semi-stables. La preuve découle d'un critère numérique pour qu'un système d'inégalités linéaires soit semi-stable.

Vertices of the Harder and Narasimhan polygons and the laws of large numbers

“…Now, for all , inside of , define and Then Let us also mention that, with respect to the filtration on that is induced by the given filtration on V , the slope of is (We refer to [9, Section 1], for example, for more details about the behavior of slopes under taking tensor products and quotients. )…”

“…Over the years, this theory has been developed and has produced significant applications. (See, for example, [3, 7, 9, 24] and the references therein. )…”

Connectedness of the Tannakian group attached to semi-stable multiple filtrations

“…In Section 2, a proof of Theorem 1.2 is given for an arbitrary (finite or infinite) GALOIS extension L. It might be useful and natural to bring in a finite e Âtale K-algebra L. But we would like in the present paper to stick to our original setting [5,6] bearing Diophantine approximation in mind. A little deviation is that the index set M is any non-empty set.…”

The algebraic groups leading to the Roth inequalities

“…The definition of the category C ss 0 (Q,Q) is motivated by the strong (or parametric) subspace theorem of Schmidt (e.g. [12,VI §3]) in Diophantine Approximation ( [8], [5]). It originates from the observations of Faltings and Wüstholz ( [7], [6]) that the condition of a linear system being a general Roth system agrees with the semi-stability in Geometric Invariant Theory.…”