Related papers: Some algorithms related to the Jacobian Conjecture
In the field of the Jacobian conjecture it is well-known after Druzkowski that from a polynomial "cubic-homogeneous" mapping we can build a higher-dimensional "cubic-linear" mapping and the other way round, so that one of them is invertible…
In this paper we are concerned with a Gordan-type theorem involving an arbitrary number of inequality functions. We not only state its validity under a weak convexity assumption on the functions, but also show it is an optimal result. We…
Computing the probability of a formula given the probabilities or weights associated with other formulas is a natural extension of logical inference to the probabilistic setting. Surprisingly, this problem has received little attention in…
In this short note we present a family of counterexamples to the King's conjecture.
We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group $H \leq \mathrm{SL}(n,…
The Jacobian algebras are introduced and their various properties are studied.
The discrete logarithm problem in Jacobians of curves of high genus $g$ over finite fields $\FF_q$ is known to be computable with subexponential complexity $L_{q^g}(1/2, O(1))$. We present an algorithm for a family of plane curves whose…
Non-conjugate Gaussian processes (NCGPs) define a flexible probabilistic framework to model categorical, ordinal and continuous data, and are widely used in practice. However, exact inference in NCGPs is prohibitively expensive for large…
In this paper, we introduce a method for approximating the solution to inference and optimization tasks in uncertain and deterministic reasoning. Such tasks are in general intractable for exact algorithms because of the large number of…
Conjectures involving infinite families of restricted partition congruences can be difficult to verify for a number of individual cases, even with a computer. We demonstrate how the machinery of Radu's algorithm may be modified and employed…
Polynomial solving algorithms are essential to applied mathematics and the sciences. As such, reduction of their complexity has become an incredibly important field of topological research. We present a topological approach to constructing…
An effective numerical method is presented for optimizing model parameters that can be applied to any type of system of non-linear equations and any number of data-points, which does not require explicit formulation of the objective…
Given a model in algebraic statistics and some data, the likelihood function is a rational function on a projective variety. Algebraic algorithms are presented for computing all critical points of this function, with the aim of identifying…
In this article we demonstrate how algorithmic probability theory is applied to situations that involve uncertainty. When people are unsure of their model of reality, then the outcome they observe will cause them to update their beliefs. We…
We show that the algorithm for computing an element of the Clarke generalized Jacobian of a max-type function proposed by Zheng-da Huang and Guo-chun Ma can be extended to a much wider class of functions representable as a difference of…
One presents many Concatenated and Operation Sequences, P-Q Relationships, Digital Sequences, Magic Squares, Prime Conjectures, k-Divisibility and Strong Divisibility Sequences, Geometric Conjectures, Proposed problems.
This paper describes a class of probabilistic approximation algorithms based on bucket elimination which offer adjustable levels of accuracy and efficiency. We analyze the approximation for several tasks: finding the most probable…
Evaluation of counterfactual queries (e.g., "If A were true, would C have been true?") is important to fault diagnosis, planning, and determination of liability. In this paper we present methods for computing the probabilities of such…
We present a one-parameter family of models for square contingency tables that interpolates between the classical quasisymmetry model and its Pearsonian analogue. Algebraically, this corresponds to deformations of toric ideals associated…
This article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such…