ANALYTIC, DIFFERENTIAL-GEOMETRIC AND ALGEBRAIC PROPERTIES OF SMOOTH FUNCTION OVER POLYNUMBERS
2011jnw | Pavlov D.G., Kokarev S.S.  // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia; RSEC Logos, Yaroslavl, Russia, geom2004@mail.ru

The paper is a brief review of results on the theory of differentiable functions of polynumbers variable Pn → Pn and of its applications. We define derivative of a function of polynumbers variable, basing on special classification of degenerated (i. e. irreversible) polynumbers and on the theorem stating general form of R-linear mapping Pn → Pn. Then we define holomorphic function of polynumbers variable as subclass of differentiable functions by the set of differential conditions (polynumbers analog of Cauchy-Riemannian conditions), which in isotropic basis have the form: kdf = 0, (k = 1, . . . , n−1) where kd= Ckd, C - conjugation in algebra Pn. Some generalized classes of holomorphic functions Gnka1 ,ka2 ,...,kar are defined by monomic differential equations, which can be classified by the set of vectors of non-negative integer n-dimensional lattice Zn+ . The question of holomorphic continuation of some smooth function from submanifolds of Pn to Pn is discussed. We derive polynumbers version of Cauchy theorem and Cauchy integral formulae together with possible multidimensional generalization the first one. Using symmetric Berwald-Moor form we develop symmetric analog of differential forms calculus (Symmetric product, Hodge star and external differential). We analyze transformation properties of derivatives of scalar polynumbers functions and of those geometrical objects, that can be constructed from these derivatives. In particular, we construct real scalar invariants, appropriate for Lagrangian formalism in polynumbers field theory. Basing on supports algebra we formulate tangent construction, playing important role in physical interpreting of polynumbers field theory. The formula for Levi-Civita connections coefficients concordant with Berwald-Moore form and formula for volume form based on n-root metric of Finsler type in even dimensions are derived. Also we consider some deformational aspects of smooth function of polynumbers variable and prove the statement, that any R-algebra can be embedded into space of bilinear forms over Pn. The paper can be treated as preliminary sketch of general theory of functions of polynumbers variable (TFPV).

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