Let F: K n --> K n be a polynomial map where K R or C. Let us denote by J(F) the determinant of t... more Let F: K n --> K n be a polynomial map where K R or C. Let us denote by J(F) the determinant of the Jacobian of F. The Jacobian conjecture is the following statement: If J (F) never vanishes then the map F is injective. Originally, the conjecture was stated for K C with polynomials over Z by O. Keller [5]. When K C the assumption can be rewritten as J(F) C* (by the Fundamental Theorem of Algebra) and the conclusion can be rewritten as: F is invertible in the ring C[X Xn] [3]. For K R, injectivity of F implies its surjectivity [6,12] (a result that was generalized by A. Borel to real algebraic varieties). The conjecture for K C is still open for n > 2. The conjecture for K R, the so called real Jacobian conjecture, was recently shown to be false by S. Pinchuk 11 ]. The main purpose of this paper is to give a proof of the fact that there is no counterexample to the complex Jacobian conjecture of the type constructed by S. Pinchuk for the real case (Theorem 5). This proof depends on properties of the asymptotic values of polynomial maps. In Section 2 we will mention other results that are consequences of these properties and motivate further research of the asymptotic values of polynomial maps. A detailed study of the asymptotic values of polynomial maps is given in . 2. Global diffeomorphisms and asymptotic identities Let f" R ---> R n be a map. A point Xo R n is called an asymptotic value of f if there exists a curve or(t) (Xl(t) Xn(t)), 0 < < o, that extends to o such that lim f (a(t)) Xo. t--o tr(t) is called an asymptotic curve of f.
Let F = (P, Q) ∈ C[X, Y ] 2 be a polynomial mapping over the complex field C. Suppose that A mapp... more Let F = (P, Q) ∈ C[X, Y ] 2 be a polynomial mapping over the complex field C. Suppose that A mapping that satisfies the assumptions above is called a Keller mapping. In this paper we estimate the size of the co-image of F . We give a sufficient condition for surjectivity of Keller mappings in terms of its Phantom curve. This curve is closely related to the asymptotic variety of F .
A geometric inequality among three triangles, originating in circle packing problems, is introduc... more A geometric inequality among three triangles, originating in circle packing problems, is introduced. In order to prove it, we reduce the original formulation to the nonnegativity of a polynomial in four real indeterminates. Techniques based on sum of squares decompositions, semidefinite programming and symmetry reduction are then applied to provide an easily verifiable nonnegativity certificate.
In this paper we investigate properties of the Steiner symmetrization in the complex plane. We us... more In this paper we investigate properties of the Steiner symmetrization in the complex plane. We use two recursive dynamic processes in order to derive some sharp inequalities on analytic functions in the unit disk. We answer a question that was asked by Albert Baernstein II, regarding the coefficients of circular symmetrization. We mostly deal with the Steiner symmetrization G of an analytic function f in the unit disk U . We pose few problems we can not solve. An intriguing one is that of the inequality 2π 0
We prove the Jacobian Conjecture for the space of all the inner functions in the unit disc. 1 Kno... more We prove the Jacobian Conjecture for the space of all the inner functions in the unit disc. 1 Known facts Definition 1.1. Let B F be the set of all the finite Blaschke products defined on the unit disc D = {z ∈ C | |z| < 1}.
F ∈ Aut(C n ), is that det J(F ) ∈ C × . The Jacobian Conjecture speculates the validity of the i... more F ∈ Aut(C n ), is that det J(F ) ∈ C × . The Jacobian Conjecture speculates the validity of the inverse statement. Thus the Jacobian Conjecture is This is true for dimension n = 1 but it is wide open for dimension n ≥ 2. The original version of the conjecture was stated by Ott Keller in 1939 . Keller worked over the integers Z and with polynomial mappings F : Z n → Z n that satisfy the unimodular Jacobian condition det J(F ) = 1 or (-1). Since that time, much research has been done on the so-called Keller Problem, giving rise to great many beautiful ideas and theories. Notable results are the degree reduction theorems . These are based on K-theoretic principles that allow us, for example, to reduce the proof of the conjecture to the seemingly simple case of mappings of the form F = (X 1 , . . . , X n ) + H(X 1 , . . . , X n ), where (X 1 , . . . , X n ) is the identity mapping and H(X 1 , . . . , X n ) is a cubic homogeneous mapping. However, one should prove the Jacobian Conjecture for such cubic mappings in an arbitrary dimension. We also note that the conjecture is known to be true in any dimension and for mappings of degree deg F ≤ 2, . So it seems that "we are almost there". Yet the degree 3 case seems to be out of our reach at least for the present. Some experts in this field tend to believe that the general conjecture (n ≥ 2) might be false but that the two-dimensional case is possibly true. We mention here two pivotal two dimensional results. The first is a theorem of Moh , which asserts the validity of the conjecture for n = 2 and degree d = deg F ≤ 100 or so. This is a difficult result. The second result is the ingenious counterexample of Pinchuk [17] to the so-called Real Jacobian Conjecture ; namely, if F : R 2 → R 2 is a real polynomial mapping that satisfies the real Jacobian condition det J F (X, Y ) = 0 ∀ (X, Y ) ∈ R 2 , then F -1 exists. We note that in this case det J F need not be a non-zero constant, but that the conclusion is also weaker, namely, F -1 does not need to be a polynomial mapping. This natural real version of the original Jacobian Conjecture was open for sometime until in 1993 Pinchuk found a counterexample. His original clever construction gave rise to a degree 35 counterexample, but almost immediately it was reduced to a degree 25 counterexample. It is interesting to mention that the minimal degree of a counterexample is still not known, but we do have some lower bounds (d ≥ 7). Another approach to solving the Jacobian Conjecture is via a thorough analysis of the structure of the semigroup of all the normalized étale mappings on C n .This approach is outlined in the papers of Kambyashi [7,
Let X be a topological space. The semigroup of all the étale mappings of X (the local homeomorphi... more Let X be a topological space. The semigroup of all the étale mappings of X (the local homeomorphisms X → X), is denoted by et(X). If G ∈ et(X) then the G-right (left) composition operator on et(X) is defined by: the semigroup of the (normalized) Keller mappings and outlines a new method of a possible attack on this open problem (in preparation). The construction uses the left composition operator and the injectivity problem is essential. In this paper we will completely solve the injectivity problems of the two composition operators for (normalized) Keller mappings. We will also solve the much easier surjectivity problem of these composition operators.
Let F = (P, Q) ∈ C[X, Y] 2 be a polynomial mapping over the complex field C. Suppose that A mappi... more Let F = (P, Q) ∈ C[X, Y] 2 be a polynomial mapping over the complex field C. Suppose that A mapping that satisfies the assumptions above is called a Keller mapping. In this paper we estimate the size of the co-image of F. We give a sufficient condition for surjectivity of Keller mappings in terms of its Phantom curve. This curve is closely related to the asymptotic variety of F.
This paper describes a part of the factorization theory of the family of all the entire functions... more This paper describes a part of the factorization theory of the family of all the entire functions with non vanishing derivatives. In particular it proves that this family of mappings contains primes. This assures that this family of entire functions has two non degenerate fractal representations.
In this paper we investigate properties of the Steiner symmetrization in the complex plane. We us... more In this paper we investigate properties of the Steiner symmetrization in the complex plane. We use two recursive dynamic processes in order to derive some sharp inequalities on analytic functions in the unit disk. We answer a question that was asked by Albert Baernstein II, regarding the coefficients of circular symmetrization. We mostly deal with the Steiner symmetrization G of an analytic function f in the unit disk U. We pose few problems we can not solve. An intriguing one is that of the inequality ∫_0^2π |f(re^iθ)|^pdθ<∫_0^2π |G(re^iθ)|^pdθ, 0<p<∞ which is true for p=2 (we prove) but can not be true for too large p. What is the largest such exponent or its supremum?
This paper develops further the theory of the automorphic group of non-constant entire functions.... more This paper develops further the theory of the automorphic group of non-constant entire functions. This theory has already a long history that essentially started with two remarkable papers of Tatsujirô Shimizu that were published in 1931. The elements φ(z) of the group are defined by the automorphic equation f (φ(z)) = f (z), were f (z) is entire. Tatsujirô Shimizu also refers to the functions of this group as those functions that are determined by f -1 • f . He proved many remarkable properties of those automorphic functions. He indicated how they induce a beautiful geometric structure on the complex plane. Those structures were termed by Tatsujirô Shimizu, the system of normal polygonal domains, and the more refined system of the fundamental domains of f (z). The last system if exists tiles up the complex plane with remarkable geometric tiles that are conformally mapped to one another by the automorphic functions. In the Ph.D thesis of the author, those tiles were also called the system of the maximal domains of f (z). One can not avoid noticing the many similarities between this automorphic group and its accompanying geometric structures and analytic properties, and the more tame discrete groups that appear in the theory of hyperbolic geometry and also the arithmetic groups in number theory. This paper pursues further the theory initiated by Tatsujirô Shimizu, towards understanding global properties of the automorphic group, rather than just understanding the properties of the individual automorphic functions. We hope to be able in sequel papers to generalize arithmetic and analytic tools such as the Selberg trace formula, to this new setting. Contents 1 Some background and the contribution of Tatsujirô Shimizu 4 14 Consequences to Aut(f ) that follow from the classical theory of entire functions 15 The relations between scattering theory and automorphic functions 16 Local groups 17 The sums of the k'th derivatives of all the elements of the automorphic group Aut(f ), for any f ∈ E of order 0 < ρ < 1 2 , k = 1, 2, 3, . . . 18 The circular density of the orbits of the automorphic group Aut(f ), for any f ∈ E of order 0 < ρ < 1 2 19 The Vieta formulas for Aut(f ), f ∈ E of order 0 ≤ ρ < 1 20 Embedding the automorphic group within a larger group 21 Relations between the construction of the direct system of the automorphic groups, and Weierstrass products 22 Continuity properties of the automorphic groups 23 Amenability of the automorphic group
We consider a possible approach to the Jacobian conjecture. We define the Jacobian variety of Cn ... more We consider a possible approach to the Jacobian conjecture. We define the Jacobian variety of Cn of degree d, denoted by J(n, d), whose points parametrize the set of all the n-Jacobian tuples of total degree at most d normalized to map 0 ∈ Cn onto itself. We use the term ”variety” as a not necessarily irreducible algebraic set. The set Aut0,d(C) of all the polynomial automorphisms of Cn of total degree at most d that map 0 onto itself corresponds to a subset of the Jacobian variety of degree d. This subset is in fact a subvariety, i.e. it is a Zariski closed subset of the variety. This might be significant to settle the Jacobian conjecture. For instance, if the jacobian variety of degree d is irreducible, then it reduces the problem to computing the dimensions of these two varieties. The conjecture is true if and only if the two dimensions are equal. Otherwise, almost any point on the Jacobian variety will serve as a counterexample to the Jacobian conjecture. Using results of Magnus...
In this paper we investigate properties of the Steiner symmetrization in the complex plane. We us... more In this paper we investigate properties of the Steiner symmetrization in the complex plane. We use two recursive dynamic processes in order to derive some sharp inequalities on analytic functions in the unit disk. We answer a question that was asked by Albert Baernstein II, regarding the coefficients of circular symmetrization. We mostly deal with the Steiner symmetrization $G$ of an analytic function $f$ in the unit disk $U$. We pose few problems we can not solve. An intriguing one is that of the inequality $$ \int_{0}^{2\pi} |f(re^{i\theta})|^{p}d\theta\le\int_{0}^{2\pi} |G(re^{i\theta})|^{p}d\theta,\,\,0<p<\infty $$ which is true for $p=2$ (we prove) but can not be true for too large $p$. What is the largest such exponent or its supremum?
There are three types of results in this paper. The first, extending a representation theorem on ... more There are three types of results in this paper. The first, extending a representation theorem on a conformal mapping that omits two values of equal modulus. This was due to Brickman and Wilken. They constructed a representation as a convex combination with two terms. Our representation constructs convex combinations with unlimited number of terms. In the limit one can think of it as an integration over a probability space with the uniform distribution. The second result determines the sign of ℜ L(z0(f(z))2) up to a remainder term which is expressed using a certain integral that involves the Löwner chain induced by f(z), for a support point f(z) which maximizes ℜ L. Here L is a continuous linear functional on H(U), the topological vector space of the holomorphic functions in the unit disk U = {z ∈ ℂ | |z| < 1}. Such a support point is known to be a slit mapping and f(z0) is the tip of the slit ℂ − f(U). The third demonstrates some properties of support points of the subspace Sn of...
Let F: K n --> K n be a polynomial map where K R or C. Let us denote by J(F) the determinant of t... more Let F: K n --> K n be a polynomial map where K R or C. Let us denote by J(F) the determinant of the Jacobian of F. The Jacobian conjecture is the following statement: If J (F) never vanishes then the map F is injective. Originally, the conjecture was stated for K C with polynomials over Z by O. Keller [5]. When K C the assumption can be rewritten as J(F) C* (by the Fundamental Theorem of Algebra) and the conclusion can be rewritten as: F is invertible in the ring C[X Xn] [3]. For K R, injectivity of F implies its surjectivity [6,12] (a result that was generalized by A. Borel to real algebraic varieties). The conjecture for K C is still open for n > 2. The conjecture for K R, the so called real Jacobian conjecture, was recently shown to be false by S. Pinchuk 11 ]. The main purpose of this paper is to give a proof of the fact that there is no counterexample to the complex Jacobian conjecture of the type constructed by S. Pinchuk for the real case (Theorem 5). This proof depends on properties of the asymptotic values of polynomial maps. In Section 2 we will mention other results that are consequences of these properties and motivate further research of the asymptotic values of polynomial maps. A detailed study of the asymptotic values of polynomial maps is given in . 2. Global diffeomorphisms and asymptotic identities Let f" R ---> R n be a map. A point Xo R n is called an asymptotic value of f if there exists a curve or(t) (Xl(t) Xn(t)), 0 < < o, that extends to o such that lim f (a(t)) Xo. t--o tr(t) is called an asymptotic curve of f.
The theory of pseudo circle packings is developed. It generalizes the theory of circle packings. ... more The theory of pseudo circle packings is developed. It generalizes the theory of circle packings. It allows the realization of almost any graph embedding by a geometric structure of circles. The corresponding Thurston's relaxation mapping is defined and is used to prove the existence and the rigidity of the pseudo circle packing. It is shown that iterates of this mapping, starting from an arbitrary point, converge to its unique positive fixed point. The coordinates of this fixed point give the radii of the packing. A key property of the relaxation mapping is its superadditivity. The proof of that is reduced to show that a certain real polynomial in four variables and of degree 20 is always nonnegative. This in turn is proved by using recently developed algorithms from real algebraic geometry. Another important ingredient in the development of the theory is the use of nonnegative matrices and the corresponding Perron-Frobenius theory.
Let F = (P, Q) ∈ C[X, Y ] 2 be a polynomial mapping over the complex field C. Suppose that A mapp... more Let F = (P, Q) ∈ C[X, Y ] 2 be a polynomial mapping over the complex field C. Suppose that A mapping that satisfies the assumptions above is called a Keller mapping. In this paper we estimate the size of the co-image of F . We give a sufficient condition for surjectivity of Keller mappings in terms of its Phantom curve. This curve is closely related to the asymptotic variety of F .
Uploads
Papers by Ronen Peretz