In this paper, we discuss several additional properties a power linear Keller map may have. The Structural Conjecture by Drużkowski in [Dru] asserts that two such properties are equivalent, but we show that one of this properties is stronger than the other. We even show that the property of linear triangularizability is strictly in between. Furthermore, we give some positive results for small d...

The object of this note is to answer this question in the affirmative when F(x) is of class C2. (It can be mentioned that, even if F(x) is analytic, there need not exist such a map R of class C1 with nonvanishing Jacobian; [2].) The mapping (1.3) to be obtained below maps solution paths of (1.1) into those of (1.4) preserving parametrizations. The first part of the paper concerns the linearizat...

We say that an algebraic curve has split jacobian if its jacobian is isogenous to a product of elliptic curves. If X is a curve of genus 2, and /: X —► E a map from X to an elliptic curve, then X has split jacobian. It is not true that a complement to E in the jacobian of X is uniquely determined, but, under certain conditions, there is a canonical choice of elliptic curve E' and algebraic f:X—...

It was conjectured in [Nam73] that the Torelli map Mg → Ag associating to a curve its jacobian extends to a regular map from the DeligneMumford moduli space of stable curves Mg to the (normalization of the) Igusa blowup A cent g . A counterexample in genus g = 9 was found in [AB11]. Here, we prove that the extended map is regular for all g ≤ 8, thus completely solving the problem in every genus.

A polynomial map F = (P,Q) ∈ Z[x, y]2 with Jacobian JF := PxQy −PyQx ≡ 1 has polynomial inverse of integer coefficients if the complex plane curve P = 0 has infinitely many integer points. 2000 Mathematical Subject Classification: 14R15.

A natural question to ask is: which divisors of degree 0 do not arise from meromorphic functions? The answer is given in a theorem of Abel, which we will present here. Since each divisor up to linear equivalence also corresponds to an isomorphism class of line bundles of degree 0, we will also be able to use Abel’s theorem to classify degree 0 line bundles on X as points of a complex torus call...

Let k be a field of characteristic zero and F : k → k a polynomial map of the form F = x + H, where H is homogeneous of degree d ≥ 2. We show that the Jacobian Conjecture is true for such mappings. More precisely, we show that if JH is nilpotent there exists an invertible linear map T such that T−1HT = (0, h2(x1), h3(x1, x2)), where the hi are homogeneous of degree d. As a consequence of this r...

We begin with a brief history of the problem of determining the set of points of a curve that map to torsion points of the curve’s Jacobian. Let K be a number field, and suppose that X/K is an algebraic curve of genus g ≥ 2. Assume, furthermore, that X is embedded in its Jacobian variety J via a K-rational Albanese map i; thus there is a K-rational divisor D of degree one on X such that i = iD ...

A new method for function approximation and system identification based on the SelfOrganizing Map is presented. The standard Self-Organizing Map (SOM) is extended with Local Linear Mappings to enable the original algorithm to learn input-output relationships with reasonable accuracy. To every node in the SOM along with the input weight two output weights are assigned: one that stores the output...

We consider strictly log-concave measures, whose bounds degenerate at infinity. prove that the optimal transport map from Gaussian onto such a measure is locally Lipschitz, and eigenvalues of its Jacobian have controlled growth