The asymptotic variety of a Pinchuk map as a polynomial curve

The asymptotic variety of a counterexample of Pinchuk type to the strong real Jacobian conjecture is explicitly described by low degree polynomials.     

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.     

The asymptotic lift of a completely positive map

Starting with a unit-preserving normal completely positive map acting on a von Neumann algebra—or more generally a dual operator system—we show that there is a unique reversible system (i.e., a complete order automorphism α of a dual operator system N) that captures all of the asymptotic behavior of L, called the asymptotic lift of L. This provides a noncommutative generalization of the Frobenius theorems that describe the asymptotic behavior of the sequence of powers of a stochastic n×n matrix. In cases where M is a von Neumann algebra, the asymptotic lift is shown to be a W^∗-dynamical system (N,Z), and we identify (N,Z) as the tail flow of the minimal dilation of L. We are also able to identify the Poisson boundary of L as the fixed algebra Nα. In general, we show the action of the asymptotic lift is trivial iff L is slowly oscillating in the sense that

     

On the ratio of values of a polynomial

For given positive integersa andb, the equationa(x + 1)… (x + k) =b(y+1)… (y + k) in positive integers is considered. More general equations are also considered.     

The Bloch-Ogus spectral sequence of a real algebraic variety

For ann-dimensional nonsingular real varietyX, we study the local-global spectral sequence

The intersection number of real polynomial mappings

Let X=V(f₁,…,fn−m)⊂Rn be a compact real algebraic set and g:X→R2m be a continuous function. If the diagonal in X×X is isolated in the set of self-intersection points of g, we define the intersection number of g. In the case where X is a manifold and g is an immersion it is the intersection number defined by Whitney. In the case where g is a polynomial mapping, we present an effective formula for this number.     

The etale homotopy theory of a geometric fibration

A geometric fibration, f:XY, is a smooth map of schemes which locally on Y admits a smooth, relative compactification. The etale homotopy type of the geometric fibre when completed away from the residue characteristics of Y, , is shown to be weakly homotopy equivalent to the completion of the Hurewicz fibre of the etale homotopy type of f,F(f _et ^r )^{^}. This implies a homotopy sequence for f. A key topological fact is verified in the appendix: for any pointed Hurewicz fibre triple FEB, the action of _l(F) on the homotopy type of various covering spaces of F extends to an action of _l(E).Partially supported by the N.S.F., I.H.E.S., and University of Warwick.     

On ramification values of a polynomial mapping

 Let be affine algebraic varieties of dimension n-1. There is a proper polynomial mapping such that the set of its ramification values contains hypersurfaces , which are birationally equivalent to Received: 21 December 2001 Mathematics Subject Classification (2000): 14 D 06, 14 Q 20     

Reaching generalized critical values of a polynomial

Let $f: \mathbb K ^n \rightarrow \mathbb K $ be a polynomial, $\mathbb K =\mathbb R , \,\mathbb C $ . We give an algorithm to compute the set of generalized critical values. The algorithm uses a finite dimensional space of rational arcs along which we can reach all generalized critical values of $f$ .     

On the generalized critical values of a polynomial mapping

 Let be a polynomial dominant mapping and let deg f _i ≤d. We prove that the set K(f) of generalized critical values of f is contained in the algebraic hypersurface of degree at most D=(d+s(m−1)(d−1)) ⁿ , where . This implies in particular that the set B(f) of bifurcations points of f is contained in the hypersurface of degree at most D=(d+s(m−1)(d−1)) ⁿ . We give also an algorithm to compute the set K(f) effectively. Received: 11 June 2001 / Revised version: 1 July 2002 Published online: 24 January 2003 The author is partially supported by the KBN grant 2 PO3A 017 22. Mathematics Subject Classification (2000): 14D06, 14Q20, 51N10, 51N20, 15A04     

The Albanese map of the Fano surface of a real M-cubic threefold

The restriction to the set of real points of the Albanese map of the Fano surface of a real M-cubic threefold is considered. Some topological properties of this map are proved.     

Geometry of real polynomial mappings

In this paper we study the set of points at which a real polynomial mapping is not proper. Received: 7 May 1999; in final form: 18 September 2000 / Published online: 25 June 2001     

On a certain geometric mean of the values of a polynomial

The maximum of the geometric mean of the values of a polynomial in the vertices of a regularkgon inscribed into the unit circle is greater than or equal to its Mahler measure. It also tends to the Mahler measure ask tends to infinity. We give quantitative versions of this statement: the upper bounds for the ratio of these two quantities. Partially supported by the Lithuanian State Science and Studies Foundation. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 1, pp. 17–27, January–March, 2000.