Related papers: Polynomial automorphisms, quantization and Jacobia…
After sketching recent advances and subtleties in classical relativistically covariant field theories, we give in this short Note some indications as to how the deformation quantization approach can be used to solve or at least give a…
We introduce a new method in the attempt to prove the Jacobian conjecture. In the complex dimension 2 case, we apply this method to prove some new results related the Jacobian conjecture.
It is the goal of this article to extend the notion of quantization from the standard interpretation focused on non-commuting observables defined starting from classical analogues, to the topological equivalents defined in terms of…
The Jacobian conjecture is an old unsolved problem in mathematics, which has been unsuccessfully attacked from many different angles. We add here another point of view pertaining to the so called formal inverse approach, that of…
The Jacobian Conjecture would follow if it were known that real polynomial maps with a unipotent Jacobian matrix are injective. The conjecture that this is true even for $C^1$ maps is explored here. Some results known in the polynomial case…
We present several versions of the Jacobian Conjecture in positive characteristic each of which if true would imply the Jacobian conjecture in characteristic 0. We test these characteristic p versions of the conjecture against several…
In this paper, we will first show that, the homogeneous polynomials which satisfy the Jacobian condition are injective on the lines that pass through the origin. Secondly, we will show that $F$ and $G'$ are paired, where $F$ is a Druzkowski…
Let k be a field of characteristic zero. Let phi be a k-endomorphism of the polynomial algebra k[x_1,...,x_n]. It is known that phi is an automorphism if and only if it maps irreducible polynomials to irreducible polynomials. In this paper…
We use the techniques of birational algebraic geometry and some combinatorial arguments related to weighted trees to study the structure of resolutions of compactifications of hypothetical counterexamples to the two-dimensional Jacobian…
Invited contribution to the Encyclopedia of Mathematical Physics (2nd edition), providing an overview over some main ideas and results in quantum cosmology. Key points: Canonical quantisation of homogeneous, isotropic cosmology; discussion…
An out of the box intellectual path exploring the foundations of quantum mechanics is discussed in some detail, in order to clarify why a possibly different way to look at the relevant fundamental questions can be identified and can support…
I use an instrumental approach to investigate some commonly made claims about interpretations of quantum mechanics, especially those that pertain questions of locality. The here presented investigation builds on a recently proposed taxonomy…
In this paper, we introduce the notion of Jacobi polynomials with multiple reference vectors of a code, and give the MacWilliams type identity for it. Moreover, we derive a formula to obtain the Jacobi polynomials using the Aronhold…
An important problem in computational arithmetic geometry is to find changes of coordinates to simplify a system of polynomial equations with rational coefficients. This is tackled by a combination of two techniques, called minimisation and…
We analyze a possible minimal counterexample to the Jacobian Conjecture $P,Q$ with $\gcd(deg(P),deg(Q))=16$ and show that its existence depends only on the existence of solutions for a certain Abel differential equation of the second kind.
Multi-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input…
The purpose of this short paper is to identify the mathematical essence of the superiorization methodology. This methodology has been developed in recent years while attempting to solve specific application-oriented problems. Consequently,…
We identify the Atkin polynomials in terms of associated Jacobi polynomials. Our identificationthen takes advantage of the theory of orthogonal polynomials and their asymptotics to establish many new properties of the Atkin polynomials.…
This is a brief review of the categorification of the Jones polynomial and its significance and ramifications in geometry, algebra, and low-dimensional topology.
The basic elements of the geometric approach to a consistent quantization formalism are summarized, with reference to the methods of the old quantum mechanics and the induced representations theory of Lie groups. A possible relationship…