On some conditions for a polynomial map with constant Jacobian to be invertible

1983 ◽  
Vol 40 (1) ◽  
pp. 415-420
Author(s):  
G. Angerm�ller
Keyword(s):  
Polynomial Map

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.

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.

scholarly journals Algebraically Solvable Problems: Describing Polynomials as Equivalent to Explicit Solutions

10.37236/734 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Combinatorial Nullstellensatz ◽  
Algebraic Solution ◽  
Explicit Solutions ◽  
Polynomial Map ◽  
Combinatorial Number Theory ◽  
Special Cases ◽  
The Matrix ◽  
Grid Points ◽  
Algebraic Solutions ◽  
Solvable Problems

The main result of this paper is a coefficient formula that sharpens and generalizes Alon and Tarsi's Combinatorial Nullstellensatz. On its own, it is a result about polynomials, providing some information about the polynomial map $P|_{\mathfrak{X}_1\times\cdots\times\mathfrak{X}_n}$ when only incomplete information about the polynomial $P(X_1,\dots,X_n)$ is given.In a very general working frame, the grid points $x\in \mathfrak{X}_1\times\cdots\times\mathfrak{X}_n$ which do not vanish under an algebraic solution – a certain describing polynomial $P(X_1,\dots,X_n)$ – correspond to the explicit solutions of a problem. As a consequence of the coefficient formula, we prove that the existence of an algebraic solution is equivalent to the existence of a nontrivial solution to a problem. By a problem, we mean everything that "owns" both, a set ${\cal S}$, which may be called the set of solutions; and a subset ${\cal S}_{\rm triv}\subseteq{\cal S}$, the set of trivial solutions.We give several examples of how to find algebraic solutions, and how to apply our coefficient formula. These examples are mainly from graph theory and combinatorial number theory, but we also prove several versions of Chevalley and Warning's Theorem, including a generalization of Olson's Theorem, as examples and useful corollaries.We obtain a permanent formula by applying our coefficient formula to the matrix polynomial, which is a generalization of the graph polynomial. This formula is an integrative generalization and sharpening of:1. Ryser's permanent formula.2. Alon's Permanent Lemma.3. Alon and Tarsi's Theorem about orientations and colorings of graphs.Furthermore, in combination with the Vigneron-Ellingham-Goddyn property of planar $n$-regular graphs, the formula contains as very special cases:4. Scheim's formula for the number of edge $n$-colorings of such graphs.5. Ellingham and Goddyn's partial answer to the list coloring conjecture.

ON RESULTANT CRITERIA AND FORMULAS FOR THE INVERSION OF A POLYNOMIAL MAP

2001 ◽  
Vol 29 (8) ◽  
pp. 3327-3339
Author(s):  
S. Hachaichi-Mesnager
Keyword(s):  
Polynomial Map

scholarly journals A new inversion formula for a polynomial map in two variables

1991 ◽  
Vol 76 (2) ◽  
pp. 119-120 ◽  
Author(s):  
Kossivi Adjamagbo ◽  
Arno van den Essen
Keyword(s):  
Inversion Formula ◽  
Polynomial Map

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.

POLYNOMIAL MAP FACTORIZATION OF SYMPLECTIC MAPS

2003 ◽  
Vol 14 (06) ◽  
pp. 847-854 ◽  
Author(s):  
GOVINDAN RANGARAJAN
Keyword(s):  
Integration Method ◽  
Symplectic Integration ◽  
Stability Studies ◽  
Symplectic Maps ◽  
Long Term Stability ◽  
Polynomial Map ◽  
Symplectic Map ◽  
Integration Algorithms ◽  
Three Degree Of Freedom

Long-term stability studies of nonlinear Hamiltonian systems require symplectic integration algorithms which are both fast and accurate. In this paper, we study a symplectic integration method wherein the symplectic map representing the Hamiltonian system is refactorized using polynomial symplectic maps. This method is analyzed for the three degree of freedom case. Finally, we apply this algorithm to study a large particle storage ring.

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