scholarly journals Monodromy at Infinity of Polynomial Maps and Newton Polyhedra (with an Appendix by C. Sabbah)

2012 ◽  
Author(s):  
Y. Matsui ◽  
K. Takeuchi
Keyword(s):  
Newton Polyhedra ◽  
Polynomial Maps ◽  
Monodromy At Infinity

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.

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.

scholarly journals Algebraic independence of multipliers of periodic orbits in the space of polynomial maps of one variable

2014 ◽  
Vol 36 (4) ◽  
pp. 1156-1166 ◽  
Author(s):  
IGORS GORBOVICKIS
Keyword(s):  
Periodic Orbits ◽  
Moduli Space ◽  
Algebraic Independence ◽  
Complex Polynomials ◽  
Generic Point ◽  
Algebraic Functions ◽  
Polynomial Maps ◽  
Affine Subspaces

We consider the space of complex polynomials of degree $n\geq 3$ with $n-1$ distinct marked periodic orbits of given periods. We prove that this space is irreducible and the multipliers of the marked periodic orbits, considered as algebraic functions on that space, are algebraically independent over $\mathbb{C}$. Equivalently, this means that at its generic point the moduli space of degree-$n$ polynomial maps can be locally parameterized by the multipliers of $n-1$ arbitrary distinct periodic orbits. We also prove a similar result for a certain class of affine subspaces of the space of complex polynomials of degree $n$.

Polynomial maps in two variables with smoothness on one line

2017 ◽  
Vol 46 (4) ◽  
pp. 1534-1538
Author(s):  
Hang Yang ◽  
Xiankun Du
Keyword(s):  
Polynomial Maps

Exponential Sums and Newton Polyhedra: Cohomology and Estimates

1989 ◽  
Vol 130 (2) ◽  
pp. 367 ◽  
Author(s):  
Alan Adolphson ◽  
Steven Sperber
Keyword(s):  
Exponential Sums ◽  
Newton Polyhedra

Asymptotics and Newton polyhedra

1988 ◽  
pp. 233-262
Author(s):  
V. I. Arnold ◽  
S. M. Gusein-Zade ◽  
A. N. Varchenko
Keyword(s):  
Newton Polyhedra

scholarly journals Real Polynomial Maps and Singular Open Books at Infinity

2016 ◽  
Vol 118 (1) ◽  
pp. 57 ◽  
Author(s):  
Raimundo N. Araújo Dos Santos ◽  
Ying Chen ◽  
Mihai Tibăr
Keyword(s):  
Open Book ◽  
Real Polynomial ◽  
Open Books ◽  
Polynomial Maps ◽  
Real Polynomial Maps

We provide significant conditions under which we prove the existence of stable open book structures at infinity, i.e. on spheres $S^{m-1}_R$ of large enough radius $R$. We obtain new classes of real polynomial maps $\mathsf{R}^m \to \mathsf{R}^p$ which induce such structures.

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