Topology at infinity of polynominal mappings and Thom regularity condition

We consider polynomial mappings which have atypical fibres due to the asymptotic behavior at infinity. Fixing some proper extension of the polynomial mapping, we study the localizability at infinity of the variation of topology of fibres and the possibility of interpreting local results at infinity into global results. We prove local and global Bertini–Sard–Lefschetz type statements for noncompact spaces and nonproper mappings and we deduce results on the homotopy type or the connectivity of the fibres of polynomial mappings.     

Positive polynomials on Riesz spaces

We prove some properties of positive polynomial mappings between Riesz spaces, using finite difference calculus. We establish the polynomial analogue of the classical result that positive, additive mappings are linear. And we prove a polynomial version of the Kantorovich extension theorem.     

On the Class of Hölder Surfaces in Carnot—Carathéodory Spaces

Considering two classes of Hölder mappings of Carnot-Carathéodory spaces, graph mappings and smooth mappings in the Riemannian sense, we obtain an area formula for image surfaces. In particular, we describe the structure of the polynomial sub-Riemannian differential for graph mappings.

     

$$H_q$$ H q -Semiclassical orthogonal polynomials via polynomial mappings

The Ramanujan Journal - In this work we study orthogonal polynomials via polynomial mappings in the framework of the $$H_q$$ -semiclassical class. We consider two monic orthogonal polynomial...     

Asymptotic values of polynomial mappings of the real plane

Using algebraically constructible functions we give a generically effective method to detect asymptotic values of polynomial mappings with finite fibers defined on the real plane. By asymptotic variety we mean the set of points at which the polynomial mapping fails to be proper.     

Invariant theory and reversible-equivariant vector fields

In this paper we present results for the systematic study of reversible-equivariant vector fields-namely, in the simultaneous presence of symmetries and reversing symmetries-by employing algebraic techniques from invariant theory for compact Lie groups. The Hilbert-Poincaré series and their associated Molien formulae are introduced, and we prove the character formulae for the computation of dimensions of spaces of homogeneous anti-invariant polynomial functions and reversible-equivariant polynomial mappings. A symbolic algorithm is obtained for the computation of generators for the module of reversible-equivariant polynomial mappings over the ring of invariant polynomials. We show that this computation can be obtained directly from a well-known situation, namely from the generators of the ring of invariants and the module of the equivariants.     

Parameter space of one-parameter complex mappings

We find that the effective mappings of the one-parameter complex mappings of parameters in the neighborhood of the centers such that a critical point is also a periodic point have the forms of simplest polynomial mappings. The appearance of an infinite number of small copies of the Mandelbrot-like sets in the parameter space is a consequence of these effective mappings. Thus we are able to calculate the positions, the sizes, and the orientations of these small copies of the Mandelbrot-like set for any one-parameter complex mapping.     

Cycles of Polynomial Mappings in Several Variables over Rings of Integers in Finite Extensions of the Rationals,II

We prove (Theorem 1.1) an asymptotic for maximal cycle lengths of polynomial mappings in several variables over rings of integers in algebraic number fields. In addition (Proposition 3.1), we strengthen the shape of possible lengths of polynomial cycles over discrete valuation domains.     

Cycles of Polynomial Mappings in Several Variables over Rings of Integers in Finite Extensions of the Rationals, II

We prove (Theorem 1.1) an asymptotic for maximal cycle lengths of polynomial mappings in several variables over rings of integers in algebraic number fields. In addition (Proposition 3.1), we strengthen the shape of possible lengths of polynomial cycles over discrete valuation domains.     

Hermitian Symmetric Polynomials and CR Complexity

Properties of Hermitian forms are used to investigate several natural questions from CR geometry. To each Hermitian symmetric polynomial we assign a Hermitian form. We study how the signature pairs of two Hermitian forms behave under the polynomial product. We show, except for three trivial cases, that every signature pair can be obtained from the product of two indefinite forms. We provide several new applications to the complexity theory of rational mappings between hyperquadrics, including a stability result about the existence of non-trivial rational mappings from a sphere to a hyperquadric with a given signature pair.     

Time-delay coordinates and polynomial mappings

The use of time-delay coordinates to reconstruct mappings is well known and provides an important practical tool in studying real-world problems. In this note we formulate the underlying mathematical analysis in the natural context of polynomial mappings and real analytic systems. This is particularly well adapted to systems defined by simple algebraic equations where, unlike in the general case, we do not require techniques from differential geometry.     

A nonlinear Pietsch domination theorem

Pietsch’s domination theorem, which is known for linear, multilinear and polynomial mappings, is extended to a larger class of nonlinear mappings.     

Polynomial sub-Riemannian differentiability on Carnot groups

The polynomial sub-Riemannian differentiability of classes of mappings of Carnot groups and graphs is proved. Examples of polynomial sub-Riemannian differentials preserving Hausdorff dimension are given.     

The polynomial sub-Riemannian differentiability of some Hölder mappings of Carnot groups

The polynomial sub-Riemannian differentiability is established for the large classes of Hölder mappings in the sub-Riemannian sense, namely, the classes of smooth mappings, their graphs, and the graphs of Lipschitz mappings in the sub-Riemannian sense defined on nilpotent graded groups. We also describe some special bases that carry the sub-Riemannian structure of the preimage to the image.     

Testing sets for properness of polynomial mappings

In this paper we characterize testing sets for properness of polynomial mappings . We also study the set of points at which such mappings are not proper. As the first application we give a proof of the Complementary Conjecture of McKay and Wang (Conjecture 12 in [16]). The second application is an answer to the problem of Kraft- Russell about a characterization of The third application is (given in [13]) a solution of the Problem of Van den Essen and Shpilrain about endomorphism of the ring The fourth application is a theorem about extensions of affine varieties. Received: 27 February 1998 / in revised form: 10 January 1999     

Composite Julia sets generated by infinite polynomial arrays

A definition of a generalized filled-in Julia set generated by an infinite array of proper polynomial mappings in  is introduced. It is shown that such Julia sets depend analytically on the defining polynomial mappings.     

On the Schwarz symmetry principle in a model case

In this article, we prove that smooth CR diffeomorphisms between two real analytic holomorphically nondegenerate hypersurfaces, one of which is rigid and polynomial, extend to be locally biholomorphic. It turns out that the result can be generalized to not totally degenerate mappings, in the sense of Baouendi and Rothschild.

     

On the Noether exponent and other applications of local intersection index

A connection between the index of intersection defined in local analytic geometry and the Noether exponent for germs of holomorphic mappings is established. Also a generalization of estimates for the ?ojasiewicz exponent at infinity for polynomial mappings defined on algebraic varieties is commented using local intersection theory methods.     

On the Dirichlet–Riquier problem for biharmonic equations

In the paper, it is proved that the distribution of a measurable polynomial on an infinite-dimensional space with log-concave measure is absolutely continuous if the polynomial is not equal to a constant almost everywhere. A similar assertion is proved for analytic functions and for some other classes of functions. Properties of distributions of norms of polynomial mappings are also studied. For the space of measurable polynomial mappings of a chosen degree, it is proved that the L ¹-norm with respect to a log-concave measure is equivalent to the L ¹-norm with respect to the restriction of the measure to an arbitrarily chosen set of positive measure.     

近于凸映照子族全部项齐次展开式的精确估计

本文建立了Cn中单位多圆柱上近于凸映照子族和一类近于准凸映照全部项齐次展开式的精确估计.与此同时,作为推论给出了Cn中单位多圆柱上近于凸映照子族和一类近于准凸映照精确的增长定理和精确的偏差定理上界估计.所得主要结论表明Cn中单位多圆柱上关于近于凸映照子族和一类近于准凸映照的Bieberbach猜想成立,而且与单复变数的经典结论相一致.