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.

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.

Polynomial remainders and plane automorphisms

2003 ◽  
Vol 68 (1) ◽  
pp. 73-79
Author(s):  
Takis Sakkalis
Keyword(s):  
Jacobian Conjecture ◽  
Real Polynomial ◽  
Polynomial Automorphisms ◽  
Polynomial Map

This note relates polynomial remainders with polynomial automorphisms of the plane. It also formulates a conjecture, equivalent to the famous Jacobian Conjecture. The latter provides an algorithm for checking when a polynomial map is an automorphism. In addition, a criterion is presented for a real polynomial map to be bijective.

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 A Note on the Jacobian Conjecture

1990 ◽  
Vol 19 (303) ◽  
Author(s):  
Zhi-Li Zhang
Keyword(s):  
Simple Proof ◽  
Jacobian Conjecture ◽  
Necessary Condition ◽  
Polynomial Map ◽  
Formal Expansion ◽  
Sufficient And Necessary Condition

We extend a corollary in H. Bass, et al.: <em> The Jacobian Conjecture: Reduction of Degree and Formal Expansion of the Inverse</em>, yielding a sufficient and necessary condition for a polynomial map to have an inverse of the simplest form, and give a surprisingly simple proof for the Jacobian Conjecture in two variables of the case f_i = x_i-h_i, where h_i is homogeneous of degree &gt;= 2, i = 1,2.

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 Symmetric Jacobians

2014 ◽  
Vol 12 (6) ◽  
Author(s):  
Michiel Bondt
Keyword(s):  
Jacobian Conjecture ◽  
Arbitrary Degree ◽  
Polynomial Maps

AbstractThis article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such maps. For instance, we show that it suffices to prove the Jacobian conjecture for polynomial maps x + H over ℂ such that satisfies all symmetries of the square, where H is homogeneous of arbitrary degree d ≥ 3.

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