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arXiv · 2009.13306

On local linear convexity generalized to commutative algebras

T. M. Osipchuk
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Abstract

A commutative associative algebra A with an identity over the field of real numbers which has a basis, where all elements are invertible, is considered in the work. Moreover, among matrixes consisting of the structure constants of A, there is at least one that is non-degenerate. The notion of linearly convex domains in the multi-dimensional complex space and some of their properties are generalized to the space that is the Cartesian product of n algebras A. Namely, the separate necessary and sufficient conditions of the local A-linear convexity of domains with smooth boundary in the space are obtained in terms of nonnegativity and positivity of formal quadratic differential form in A, respectively.

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T. M. Osipchuk. 2020-09-28. On local linear convexity generalized to commutative algebras. https://arxiv.org/abs/2009.13306

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