scholarly journals Thom Isotopy Theorem for Nonproper Maps and Computation of Sets of Stratified Generalized Critical Values

2019 ◽  
Vol 65 (1) ◽  
pp. 279-304
Author(s):  
Sĩ Tiệp Ðinh ◽  
Zbigniew Jelonek
Keyword(s):  
Dense Subset ◽  
Critical Values ◽  
Affine Variety ◽  
Singular Varieties ◽  
Polynomial Maps ◽  
Polynomial Map

AbstractLet $$X\subset {\mathbb {C}}^n$$ X ⊂ C n be an affine variety and $$f:X\rightarrow {\mathbb {C}}^m$$ f : X → C m be the restriction to X of a polynomial map $${\mathbb {C}}^n\rightarrow {\mathbb {C}}^m$$ C n → C m . We construct an affine Whitney stratification of X. The set K(f) of stratified generalized critical values of f can also be computed. We show that K(f) is a nowhere dense subset of $${\mathbb {C}}^m$$ C m which contains the set B(f) of bifurcation values of f by proving a version of the Thom isotopy lemma for nonproper polynomial maps on singular varieties.

Ont he monodromy representation of polynomial maps in n variables

2002 ◽  
Vol 39 (3-4) ◽  
pp. 361-367
Author(s):  
A. Némethi ◽  
I. Sigray
Keyword(s):  
Cohomology Group ◽  
Jacobian Conjecture ◽  
Natural Basis ◽  
Generic Fiber ◽  
Monodromy Representation ◽  
Polynomial Maps ◽  
Polynomial Map

For a   non-constant polynomial map f: Cn?Cn-1 we consider the monodromy representation on the cohomology group of its generic fiber. The main result of the paper determines its dimension and provides a natural basis for it. This generalizes the corresponding results of [2] or [10], where the case n=2 is solved. As  applications,  we verify the Jacobian conjecture for (f,g) when the generic fiber of f is either rational or elliptic. These are generalizations of the corresponding results of [5], [7], [8], [11] and [12], where the case  n=2 is treated.

An effective study of polynomial maps

2016 ◽  
Vol 16 (08) ◽  
pp. 1750141 ◽  
Author(s):  
Elżbieta Adamus ◽  
Paweł Bogdan ◽  
Teresa Crespo ◽  
Zbigniew Hajto
Keyword(s):  
Affine Space ◽  
Finite Sequence ◽  
Jacobian Conjecture ◽  
Equivalent Condition ◽  
Effective Algorithm ◽  
Polynomial Maps ◽  
Dimension Formula ◽  
Polynomial Map ◽  
Equivalent Statement

In this paper, using an effective algorithm, we obtain an equivalent statement to the Jacobian Conjecture. For a polynomial map [Formula: see text] on an affine space of dimension [Formula: see text] over a field of characteristic [Formula: see text], we define recursively a finite sequence of polynomial maps. We give an equivalent condition to the invertibility of [Formula: see text] as well as a formula for [Formula: see text] in terms of this finite sequence of polynomial maps. Some examples illustrate the effective aspects of our approach.

CONSTRUCTING CHAOTIC POLYNOMIAL MAPS

2009 ◽  
Vol 19 (02) ◽  
pp. 531-543 ◽  
Author(s):  
XU ZHANG ◽  
YUMING SHI ◽  
GUANRONG CHEN
Keyword(s):  
Logistic Map ◽  
Special Kind ◽  
Quadratic Polynomials ◽  
One Dimensional ◽  
Polynomial Maps ◽  
Polynomial Map ◽  
Expansion Theory ◽  
Chaotic Parameter ◽  
The Given ◽  
Nonzero Polynomial

This paper studies the construction of one-dimensional real chaotic polynomial maps. Given an arbitrary nonzero polynomial of degree m (≥ 0), two methods are derived for constructing chaotic polynomial maps of degree m + 2 by simply multiplying the given polynomial with suitably designed quadratic polynomials. Moreover, for m + 2 arbitrarily given different positive constants, a method is given to construct a chaotic polynomial map of degree 2m based on the coupled-expansion theory. Furthermore, by multiplying a real parameter to a special kind of polynomial, which has at least two different non-negative or nonpositive zeros, the chaotic parameter region of the polynomial is analyzed based on the snap-back repeller theory. As a consequence, for any given integer n ≥ 2, at least one polynomial of degree n can be constructed so that it is chaotic in the sense of both Li–Yorke and Devaney. In addition, two natural ways of generalizing the logistic map to higher-degree chaotic logistic-like maps are given. Finally, an illustrative example is provided with computer simulations for illustration.

scholarly journals The fourth dimension subgroups and polynomial maps, II

1978 ◽  
Vol 69 ◽  
pp. 1-7 ◽  
Author(s):  
Ken-Ichi Tahara
Keyword(s):  
Linear Equations ◽  
Integral Solution ◽  
Fourth Dimension ◽  
Direct Products ◽  
Kronecker Symbol ◽  
Cyclic Groups ◽  
Polynomial Maps ◽  
Polynomial Map

In our previous paper [3] we proved the following ([3, Theorem 16]) :THEOREM A. Let G be a 2-group of class 3. Let G2 and G/G2 be direct products of cyclic groups 〈yq〉 of order αq (1 ≦ q ≦ m), and of cyclic groups 〈hi〉 of order βi (1 ≦ i ≦ n) with β1 ≧ β2 ≧ · · · βn, respectively. Let xi be representatives of hi (1 ≦ i ≦ n), and put Then a homomorphism ψ:G3→T can be extended to a polynomial map from G to T of degree ≦ 4 if and only if there exists an integral solution in the following linear equations of Xiq (1 ≦ i ≦ n, 1 ≦ q ≦ m) with coefficients in T: (I)where δij is the Kronecker symbol for βi: i.e. δij = 1 or 0 according to βi = βj or βi > βj, respectively.

scholarly journals Corrigendum: ‘On certain algebraic curves related to polynomial maps, Compositio Math. 103 (1996), 319–350’

2010 ◽  
Vol 147 (1) ◽  
pp. 332-334 ◽  
Author(s):  
Patrick Morton
Keyword(s):  
Finite Field ◽  
Laurent Series ◽  
Algebraic Curves ◽  
Periodic Points ◽  
Series Expansions ◽  
Rational Field ◽  
Polynomial Maps ◽  
Polynomial Map ◽  
Hensel’S Lemma ◽  
Compositio Math

AbstractAn argument is given to fill a gap in a proof in the author’s article On certain algebraic curves related to polynomial maps, Compositio Math. 103 (1996), 319–350, that the polynomial Φn(x,c), whose roots are the periodic points of period n of a certain polynomial map x→f(x,c), is absolutely irreducible over the finite field of p elements, provided that f(x,1) has distinct roots and that the multipliers of the orbits of period n are also distinct over $\mathbb { F}_p$. Assuming that Φn(x,c) is reducible in characteristic p, we show that Hensel’s lemma and Laurent series expansions of the roots can be used to obtain a factorization of Φn(x,c) in characteristic 0, contradicting the absolute irreducibility of this polynomial over the rational field.

scholarly journals Irreducibility Properties of Keller Maps

2016 ◽  
Vol 23 (04) ◽  
pp. 663-680 ◽  
Author(s):  
Michiel de Bondt ◽  
Dan Yan
Keyword(s):  
Linear Combination ◽  
Characteristic Zero ◽  
Jacobian Matrix ◽  
Jacobian Conjecture ◽  
Extra Condition ◽  
Irreducible Polynomials ◽  
Polynomial Maps ◽  
Polynomial Map ◽  
Keller Map

Jędrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well. In connection with the above classification of invertible polynomial maps and the Jacobian Conjecture, we study irreducibility properties of several types of Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith, we generalize the result of Bakalarski that the components of cubic homogeneous Keller maps with a symmetric Jacobian matrix (over ℂ and hence any field of characteristic zero) are irreducible. Furthermore, we show that the Jacobian Conjecture can even be reduced to any of these types with the extra condition that each affinely linear combination of the components of the polynomial map is irreducible. This is somewhat similar to reducing the planar Jacobian Conjecture to the so-called (planar) weak Jacobian Conjecture by Kaliman.

On the volume and the number of lattice points of some semialgebraic sets

2015 ◽  
Vol 26 (10) ◽  
pp. 1550078 ◽  
Author(s):  
Ha Huy Vui ◽  
Tran Gia Loc
Keyword(s):  
Open Subset ◽  
Newton Polyhedron ◽  
Lattice Points ◽  
Semialgebraic Sets ◽  
Newton Polyhedra ◽  
Polynomial Maps ◽  
Polynomial Map

Let f = (f1,…,fm) : ℝn→ ℝmbe a polynomial map; we consider the set Gf(r) = {x ∈ ℝn: |fi(x)| ≤ r, i = 1,…,m}. We show that if f satisfies the Mikhailov–Gindikin condition then:(i) Volume Gf(r) ≍ rθ( ln r)n-k-1,(ii) Cardinal [Formula: see text], as r → ∞,where the exponents θ, k, θ′, k′ are determined explicitly in terms of the Newton polyhedra of f. Moreover, the polynomial maps satisfying the Mikhailov–Gindikin condition form an open subset of the set of polynomial maps having the same Newton polyhedron.

scholarly journals Stability of Real Parametric Polynomial Discrete Dynamical Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-13
Author(s):  
Fermin Franco-Medrano ◽  
Francisco J. Solis
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Special Function ◽  
Real Polynomial ◽  
Polynomial Maps ◽  
Polynomial Map ◽  
Exact Boundary ◽  
The Stability ◽  
Real Polynomial Maps

We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameterλand generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to realmth degree real polynomial maps. In essence, we give conditions for the stability of the fixed points of any real polynomial map with real fixed points. In order to do this, we have introduced the concept ofcanonical polynomial mapswhich are topologically conjugate to any polynomial map of the same degree with real fixed points. The stability of the fixed points of canonical polynomial maps has been found to depend solely on a special function termedProduct Position Functionfor a given fixed point. The values of this product position determine the stability of the fixed point in question, when it bifurcates and even when chaos arises, as it passes through what we have termedstability bands. The exact boundary values of these stability bands are yet to be calculated for regions of type greater than one for polynomials of degree higher than three.

Sums of Lyapunov exponents for some polynomial maps of ${\bf C}^2$

1998 ◽  
Vol 18 (3) ◽  
pp. 613-630 ◽  
Author(s):  
MATTIAS JONSSON
Keyword(s):  
Lyapunov Exponents ◽  
Homogeneous Part ◽  
Polynomial Maps ◽  
Polynomial Map

We give a formula for the sum of the Lyapunov exponents of a nondegenerate polynomial map $f$ of ${\bf C}^{2}$ close to $(z,w)\to (z^d,w^d)$. The formula only involves the behavior of $f$ at infinity. In particular, it follows that the sum only depends on the homogeneous part of $f$ of degree $d$.

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