Related papers: Some algorithms related to the Jacobian Conjecture
We develop a probabilistic approach to the celebrated Jacobian conjecture, which states that any Keller map (i.e. any polynomial mapping $F\colon \mathbb{C}^n \to \mathbb{C}^n$ whose Jacobian determinant is a nonzero constant) has a…
In this paper a recursive algorithm is presented for evaluating multivariate Pad\'e approximants (of the rectangular type described in the work of Lutterodt) which is analogous to the Jacobi formula for univariate Pad\'e approximants. This…
This paper investigates the problem of bounding possible output from a counterfactual query given a set of observational data. While various works of literature have described methodologies to generate efficient algorithms that provide an…
We show how to efficiently compute functions on jacobian varieties and their quotients. We deduce a quasi-optimal algorithm to compute $(l,l)$ isogenies between jacobians of genus two curves.
We design an algorithm for finding counterfactuals with strong theoretical guarantees on its performance. For any monotone model $f : X^d \to \{0,1\}$ and instance $x^\star$, our algorithm makes \[ {S(f)^{O(\Delta_f(x^\star))}\cdot \log…
Any practical attempt to solve the Regge equations, these being a large system of non-linear algebraic equations, will almost certainly employ a Newton-Raphson like scheme. In such cases it is essential that efficient algorithms be used…
Realistic physical implementations of quantum computers can entail tradeoffs which depart from the ideal model of quantum computation. Although these tradeoffs have allowed successful demonstration of certain quantum algorithms, a crucial…
Quadratic Programming (QP) is the well-studied problem of maximizing over {-1,1} values the quadratic form \sum_{i \ne j} a_{ij} x_i x_j. QP captures many known combinatorial optimization problems, and assuming the unique games conjecture,…
This paper is a preliminary report on our search for new good examples of Hall's Conjecture. We present a new algorithm that will detect all good examples within a given search space. We have implemented the algorithm, and our executions…
The Jacobi ensemble is one of the classical ensembles of random matrix theory. Prominent in applications are properties of the eigenvalues at the spectrum edge, specifically the distribution of the largest (e.g. Roy's largest root test in…
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…
Based on the results people have obtained, we try to prove the Jacobian conjecture, but there is a gap in the proof.
We derive a sequence of measures whose corresponding Jacobi matrices have special properties and a general mapping of an open quantum system onto 1D semi infinite chains with only nearest neighbour interactions. Then we proceed to use the…
This paper mainly studies the gradient-based Jacobi-type algorithms to maximize two classes of homogeneous polynomials with orthogonality constraints, and establish their convergence properties. For the first class of homogeneous…
The conjugacy problem for a finitely generated group $G$ is the two-variable problem of deciding for an arbitrary pair $(u,v)$ of elements of $G$, whether or not $u$ is conjugate to $v$ in $G$. We construct examples of finitely generated,…
In this paper, we consider a family of Jacobi-type algorithms for simultaneous orthogonal diagonalization problem of symmetric tensors. For the Jacobi-based algorithm of [SIAM J. Matrix Anal. Appl., 2(34):651--672, 2013], we prove its…
The two dimensional Jacobian Conjecture says that a morphism $f:\mathbb{C}[x,y]\to \mathbb{C}[x,y]$ having an invertible Jacobian, is invertible. We show that a morphism $f$ having an invertible Jacobian is invertible, in each of the…
When the Euclidean algorithm produces a symmetric sequence of quotients, we give explicit formulas for the remainders that allow the analysis of two families of quadratic forms in the remainders.
The increasing use of machine learning in practice and legal regulations like EU's GDPR cause the necessity to be able to explain the prediction and behavior of machine learning models. A prominent example of particularly intuitive…
Most nonlinear partial differential equation (PDE) solvers require the Jacobian matrix associated to the differential operator. In PETSc, this is typically achieved by either an analytic derivation or numerical approximation method such as…