In this paper we represent harmonic moments in the language of transfinite functions, that is projective limits of polynomials in infinitely many variables. We obtain also an explicit formula for the Jacobian of a generalized harmonic moment map. Mathematics Subject Classification (2000). 13B35; 30E05; 47A57.

In this paper we consider the Jacobian conjecture for a map f of complex affine spaces of dimension n. It is well-known that if f is proper then the conjecture will hold. Using topological arguments, specifically Smith theory, we show that the conjecture holds if and only if f is proper onto its image.

This paper contains sufficient conditions under which a map whose domain is a compact set is a bijection onto a given set. Relative to certain isoparametric finite element maps, one set of conditions involves the nonvanishing of the Jacobian; another the notion of overspill. An algorithm based on elimination is given for the numerical inversion of these maps.

The question about polynomial maps F : C → C, first raised by Keller [1] in 1939 for polynomials over the integers but now also raised for complex polynomials and, as such, known as The Jacobian Conjecture (JC), asks whether a polynomial map F with nonzero constant Jacobian determinant detF (x) need be a polyomorphism: Injective and also surjective with polynomial inverse. The known reductions ...

Developing foundational notions in the theory of Riemann surfaces, we prove the Riemann-Roch Theorem and Abel’s Theorem. These notions include sheaf cohomology, with particular focus on the zeroth and first cohomology groups, exact cohomology sequences induced by short exact sequences of sheaves, divisors, the Jacobian variety, and the Abel-Jacobi map. The general method of proof involves basic...

A non-zero constant Jacobian polynomial map F = (P,Q) : C −→ C 2 has a polynomial inverse if the component P is a simple polynomial, i.e. if, when P extended to a morphism p : X −→ P of a compactification X of C, the restriction of p to each irreducible component C of the compactification divisor D = X −C is either degree 0 or 1.

We verify the plane Jacobian conjecture for the rational polynomials: A polynomial map F = (P, Q) : C −→ C, P, Q ∈ C[x, y], is invertible if PxQy − PyQx ≡ const. 6= 0 and, in addition, P is a rational polynomial, i.e. the generic fiber of P is the 2-dimensional topological sphere with a finite number of punctures.

Consider a smooth, geometrically irreducible, projective curve of genus $g\ge 2$ defined over number field degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, theorem Faltings. We show that is bounded only in terms $g$, $d$ and Mordell–Weil rank curve's Jacobian, thereby answering affirmative question Mazur. In addition we obtain uniform bounds, $g$ $d$, fo...