ИСТИНА

We propose to study algebraic and geometric properties of polynomial maps on simple associative and Lie algebras and word maps on simple algebraic groups, being mainly interested in their images, i.e., aiming to solve equations in such algebras and groups. A seminal theorem by Armand Borel establishes the dominance of any non- indentity word map on any connected semisimple algebraic group, and the subsequent fascinating results of the past decade provide various results of similar flavour for finite simple groups. Furthermore, some important parallels between the group and algebra cases have been discovered, and substantial progress towards a long-standing conjecture of Irving Kaplansky has been achieved. We focus on the case where the relevant groups and algebras are defined over valued rings and fields, such as the ring of formal power series C[[t]], its field of fractions C((t)) (the field of Laurent series), the algebraic closure of the latter field Cfftgg (the field ^ of Puiseux series), the completion Cfftgg of the latter one, as well as close rings and fields such as the ring of Laurent polynomials C[t; t−1], the ring of polynomials C[t] and the field of rational functions C(t). On the one hand, better understanding of the behaviour of these maps over such special fields may give a way to the study of more general fields and rings, including those of arithmetic nature, such as Q , Z , F ((t)), etc. On the other hand, such p p q fields and rings appear in a natural way in various versions of deformation methods applied to relevant algebraic varieties over C, eventually giving way to important appli- cations. In this context, arising relationships with local analytic geometry and tropical geometry could be interesting in their own right. Finally, this set-up allows one to consider Kac{Moody groups and algebras, thus significantly extending a traditional finite-dimensional universe.

We propose to study algebraic and geometric properties of polynomial maps on simple associative and Lie algebras and word maps on simple algebraic groups, being mainly interested in their images, i.e., aiming to solve equations in such algebras and groups. A seminal theorem by Armand Borel establishes the dominance of any non- indentity word map on any connected semisimple algebraic group, and the subsequent fascinating results of the past decade provide various results of similar flavour for finite simple groups. Furthermore, some important parallels between the group and algebra cases have been discovered, and substantial progress towards a long-standing conjecture of Irving Kaplansky has been achieved. We focus on the case where the relevant groups and algebras are defined over valued rings and fields, such as the ring of formal power series C[[t]], its field of fractions C((t)) (the field of Laurent series), the algebraic closure of the latter field Cfftgg (the field ^ of Puiseux series), the completion Cfftgg of the latter one, as well as close rings and fields such as the ring of Laurent polynomials C[t; t−1], the ring of polynomials C[t] and the field of rational functions C(t). On the one hand, better understanding of the behaviour of these maps over such special fields may give a way to the study of more general fields and rings, including those of arithmetic nature, such as Q , Z , F ((t)), etc. On the other hand, such p p q fields and rings appear in a natural way in various versions of deformation methods applied to relevant algebraic varieties over C, eventually giving way to important appli- cations. In this context, arising relationships with local analytic geometry and tropical geometry could be interesting in their own right. Finally, this set-up allows one to consider Kac{Moody groups and algebras, thus significantly extending a traditional finite-dimensional universe.

Доказательство гипотезы Капланского с точности до плотности

Решен случай матриц второго порядка имеются продвижения по третьему порядку

Доказательство гипотезы Капланского с точности до плотности