Related papers: A note on the Jacobian Conjecture
The paper has been withdrawn by the author due to a crucial error.
This note was an attempt to complete a gap in the proof of Theorem 11.1 of the paper arXiv:1111.5992. Due to a critical gap in the proof of Lemma 4.3, this paper is withdrawn.
Jacobian conjectures (that nonsingular implies invertible) for rational everywhere defined maps of real n-space to itself are considered, with no requirement for a constant Jacobian determinant or a rational inverse. The associated…
We discuss some properties of Jacobi fields that do not involve assumptions on the curvature endomorphism. We compare indices of different spaces of Jacobi fields and give some applications to Riemannian geometry.
For a finite dimensional algebra $A$ with $0 < \phi dim (A) = m < \infty$ we prove that there always exist modules $M$ and $N$ such that $\phi(M) = m-1$ and $\phi (N) = 1$. On the other hand, we see an example of an algebra that not every…
The purpose of these notes is to provide the details of the Jacobian ring computations carried out in [1], based on the computer algebra system Magma [2].
The main result of this paper is the following version of the real Jacobian conjecture: "Let $F=(p,q):\R^2\to\R^2$ be a polynomial map with nowhere zero Jacobian determinant. If the degree of $p$ is less than or equal to $4$, then $F$ is…
There are things we know, things we know we don't know, and then there are things we don't know we don't know. In this paper we address the latter two issues in a Bayesian framework, introducing the notion of doubt to quantify the degree of…
We show that some mathematical results and their negations are both deducible. The derived contradictions indicate the inconsistency of current mathematics. This paper is an updated version of arXiv:math/0606635v3 with additional results…
The Jacobian algebras are introduced and their various properties are studied.
The Firoozbakht, Nicholoson, and Farhadian conjectures can be phrased in terms of increasingly powerful conjectured bounds on the prime gaps $g_n := p_{n+1}-p_n$. \[ g_n \leq p_n \left(p_n^{1/n} -1 \right)\qquad\qquad\qquad (n \geq 1; \;…
This paper provides a general proof of a relationship theorem between nonlinear analogue polynomial equations and the corresponding Jacobian matrix, presented recently by the present author. This theorem is also verified generally effective…
We correct a small gap found in the authors' paper 'On bounds for the effective differential Nullstellensatz' (J Algebra 449:1-21, 2016). This gap is due to an inequality that does not generally hold. However, under one additional…
The two-dimensional Jacobian Conjecture says that a $\mathbb{C}$-algebra endomorphism $F:\mathbb{C}[x,y] \to \mathbb{C}[x,y]$ that has an invertible Jacobian is an automorphism. We show that if a $\mathbb{C}$-algebra endomorphism…
Motivated by the Gilbreath conjecture, we develop the notion of the gap sequence induced by any sequence of numbers. We introduce the notion of the path and associated circuits induced by an originator and study the conjecture via the…
Our goal is to settle the following faded problem: The Jacobian Conjecture (JC_n): If f_1,..,f_n are elements in a polynomial ring k[X_1,..,X_n] over a field k of characteristic 0 such that det(\partial f_i/ \partial X_j) is a nonzero…
We develop relative oscillation theory for Jacobi matrices which, rather than counting the number of eigenvalues of one single matrix, counts the difference between the number of eigenvalues of two different matrices. This is done by…
In ${\bf C}^{n+1}$, one can show that the residue of $n+1$ homogeneous forms of the same degree equals the integral of a certain $(n,n)$ form over ${\bf P}^n$. Furthermore, the Jacobian of the forms has nonzero residue equal to a certain…
The paper has been withdrawn by the author, due a gap in the proof of Theorem 6.1. The gap was discovered by M. Van den Bergh. Theorem 6.1 is used to prove the main result of the paper, namely Theorem 0.7 (decomposition in arbitrary…
We make a summary of the different types of proofs adding some new ideas. In addition we conjecture some relations which could be necessary in "modular type proofs" (not still found) of the Ramanujan-like series for 1/\pi^2.