Jacobian Conjecture as a Problem on Integral Points on Affine Curves

We generalize Siegel’s theorem on integral points on affine curves to integral points of bounded degree, giving a complete characterization of affine curves with infinitely many integral points of degree $d$ or less over some number field. Generalizing Picard’s theorem, we prove an analogous result characterizing complex affine curves admitting a nonconstant holomorphic map from a degree $d$ (or less) analytic cover of $\mathbb{C}$.

Download Full-text

Affine curves with infinitely many integral points

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-02-06841-7 ◽

2002 ◽

Vol 131 (5) ◽

pp. 1357-1359 ◽

Cited By ~ 2

Author(s):

Dimitrios Poulakis

Keyword(s):

Integral Points ◽

Affine Curves

Download Full-text

Ont he monodromy representation of polynomial maps in n variables

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.39.2002.3-4.7 ◽

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.

Download Full-text

On planar embeddings of pseudoplanar affine curves of genus ≤1

Journal of Algebra ◽

10.1016/j.jalgebra.2021.03.003 ◽

2021 ◽

Vol 577 ◽

pp. 32-44

Author(s):

Shashikant Mulay

Keyword(s):

Affine Curves ◽

Planar Embeddings

Download Full-text

On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

Journal of Inequalities and Applications ◽

10.1186/s13660-019-2230-4 ◽

2019 ◽

Vol 2019 (1) ◽

Author(s):

WonTae Hwang ◽

Kyunghwan Song

Keyword(s):

Zeta Function ◽

Rational Number ◽

Riemann Zeta Function ◽

Rational Numbers ◽

Critical Strip ◽

Integer Part ◽

Integral Points ◽

The Riemann Zeta Function ◽

Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

Download Full-text