[en] The aim of this paper is to show that Weierstrass preparation and division theorems hold for the ring of germs of superanalytic functions at a given point. This ring is the tensor product of the ring of germs of analytic functions at that point and a finite-dimensional complex Grassmann algebra. 7 refs

[en] This is an expository account which shows how the methods of non-standard analysis can be applied to prove the Nullstellensatz for germs of analytic functions. This method of proof was discovered originally by Abraham Robinson. The necessary concepts from model theory are described in some detail and the Nullstellensatz is proved by investigating the relation between the set of infinitesimal elements in the complex n-plane and the spectrum of the ring of germs of analytic functions. (author)

[en] Most known papers on spectral synthesis in complex domains are based on transitions from problems of spectral synthesis to equivalent problems of local description. As a rule, such a transition is carried out in the framework of some special assumptions and meets considerable difficulties. A general method developed in this paper enables one to verify the duality theorem in the setting of several complex variables, when each operator π_p(D), p=1,...,q, acts with respect to a single variable. These assumptions cover the case of a system of partial differential operators. The duality transition here breaks into three separate steps. Two of them are connected with classical results of the theory of analytic functions and only one relates to general duality theory. This allows one to speak about singling out the analytic component of the transition from a spectral synthesis problem to an equivalent problem of local description

[en] Let W be the generalized Sato-Levine invariant, that is, the unique Vassiliev invariant of order 3 for two-component links that is equal to zero on double torus links of type (1,k). It is proved that W-β-(k³-k)/6 where β is the invariant of order 3 proposed by Viro and Polyak in the form of representations of Gauss diagrams and k is the linking number

[en] A result on the approximation of a fixed system of analytic functions by translations of Hurwitz zeta-functions with transcendental parameter is established. This is an analogue of Voronin's theorem on the joint universality of the Dirichlet L-functions. Bibliography: 28 titles.

[en] A number of numerical methods proposed to compute complex roots of analytic, or more generally meromorphic, functions f(z) are compared critically and their basic limitations are pointed out. It is shown that these can be overcome via a suitable steepest descent algorithm for /f(z)/². The strategy proposed allows to find a zero (as well as its multiplicity) with a predetermined precision provided that the precision with which f(z) is computed is known. Some peculiar features of the algorithm (which does not require an explicit knowledge of f'(z)) are illustrated in a number of non trivial examples

[en] A hierarchy of extremal polynomials described in terms of real hyperelliptic curves of genus g≥0 is constructed. These polynomials depend on g integer-valued and g continuous parameters. The classical Chebyshev polynomials are obtained for g=0 and the Zolotarev polynomials for g=1

[en] In the present paper, we introduce two kinds of complex Bernstein–Stancu polynomials and complex Kantorovich–Stancu polynomials in movable compact disks. Their approximation properties for analytic functions in the movable compact disks are considered.

[en] The concept of the minimal polyanalytic polynomial was introduced by M. Huhtanen in connection with the generalized Lanczos process as applied to a normal matrix. The paper discusses the possibility of finding an equivalent of the characteristic polynomial in the set of polyanalytic polynomials.

[en] For N=2 or 3 it is shown that if E is the zero set of a holomorphic function in Usup(N) satisfying the separation condition of Alexander, viz., there exist r is an element of (0,1) and delta>0 such that |α-#betta#|>=delta whenever (z',α,z'') not= (z',#betta#,z'') are both in (Qsup(k-1)xUxQsup(N-k)) intersection E, where Q=(lambda is an element of C:r<|lambda|<1), then (a) E is the zero set of some F is an element of Hsup(infinity)(Usup(N)) and (b) for 0< p<=infinity, every g is an element of H(E) so that |g|sup(p) has a pluriharmonic majorant on E extends to a G is an element of Hsup(p)(Usup(N)). This generalizes earlier results of the author [Proc. Amer. Math. Soc., 60, 109-115 (1976)] and of Zarantonello [ibid., 78, 519-524 (1980)]. (author)