Related papers: On the Jacobian Question

Combining a modular approach to the $abc$ conjecture developed by the second author with the classical method of linear forms in logarithms, we obtain improved unconditional bounds for two classical problems. First, for Szpiro's conjecture…

We establish an invertibility criterion for free polynomials and free functions evaluated on some tuples of matrices. We show that if the derivative is nonsingular on some domain closed with respect to direct sums and similarity, the…

We describe a three precision variant of Newton's method for nonlinear equations. We evaluate the nonlinear residual in double precision, store the Jacobian matrix in single precision, and solve the equation for the Newton step with…

Using the author's inversion formula for automorphisms of the Weyl algebras with polynomial coefficients and the bound on its degree a slightly shorter (algebraic) proof is given of the result of A. Belov-Kanel and M. Kontsevich that the…

A polynomial map $F=(P,Q)\in \Z [x,y]^2$ with Jacobian $JF:=P_xQ_y-P_yQ_x\equiv 1$ has a polynomial inverse of integer coefficients if the complex plane curve P=0 has infinitely many integer points.

We propose a randomized polynomial time algorithm for computing nontrivial zeros of quadratic forms in 4 or more variables over $\mathbb{F}_q(t)$, where $\mathbb{F}_q$ is a finite field of odd characteristic. The algorithm is based on a…

Consider a quadratic polynomial $f\left(\xi_{1},\dots,\xi_{n}\right)$ of independent Bernoulli random variables. What can be said about the concentration of $f$ on any single value? This generalises the classical Littlewood--Offord problem,…

We prove two criteria for direct sum decomposability of homogeneous polynomials. For a homogeneous polynomial with a non-zero discriminant, we interpret direct sum decomposability of the polynomial in terms of factorization properties of…

A particular case of the Jacobian conjecture is considered and for small dimensional cases a computational approach is offered

We propose the method for obtaining invariants of arbitrary representations of Lie groups that reduces this problem to known problems of linear algebra. The basis of this method is the idea of a special extension of the representation…

Multivariate versions of classical orthogonal polynomials such as Jacobi, Hahn, Laguerre and Meixner are reviewed and their connection explored by adopting a probabilistic approach. Hahn and Meixner polynomials are interpreted as posterior…

The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…

This report concerns the inverse problem of estimating a spacially dependent coefficient of a partial differential equation from observations of the solution at the boundary. Such a problem can be formulated as an optimal control problem…

It is well known that the Newton method may not converge when the initial guess does not belong to a specific quadratic convergence region. We propose a family of new variants of the Newton method with the potential advantage of having a…

A classical theorem of Wendroff shows that one may reconstructs a sequence of orthogonal polynomials on the real line from two non-constant polynomials of consecutive degrees whose zeros strictly interlace on the real line. In this note we…

In previous work of the authors, we investigated the Born and inverse Born series for a scalar wave equation with linear and nonlinear terms, the nonlinearity being cubic of Kerr type [8]. We reported conditions which guarantee convergence…

The Jacobian Conjecture has been reduced to the symmetric homogeneous case. In this paper we give an inversion formula for the symmetric case and relate it to a combinatoric structure called the Grossman-Larson Algebra. We use these tools…

This paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This construction uses a single linear differential form defined from the…

Considered herein is a modified Newton method for the numerical solution of nonlinear equations where the Jacobian is approximated using a complex-step derivative approximation. We show that this method converges for sufficiently small…

Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…