We give an explicit formula of the inverse polynomial of a permutation polynomial of the form xrf(xs) over a finite field Fq where s | q − 1. This generalizes results in [6] where s = 1 or f = g q−1 s were considered respectively. We also apply our result to several interesting classes of permutation polynomials.

We give a full proof of the two dimensional Jacobian conjecture. We also give an algorithm to compute the inverse map of a polynomial map.

We prove that a finitely presented Rees quotient of a free inverse semigroup has polynomial growth if and only if it has bounded height. This occurs if and only if the set of nonzero reduced words has bounded Shirshov height and all nonzero reduced but not cyclically reduced words are nilpotent. This occurs also if and only if the set of nonzero geodesic words have bounded Shirshov height. We a...

The inverse conjecture for the Gowers norms U(V ) for finite-dimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers norm ‖f‖Ud(V ) if and only if it correlates with a phase polynomial φ = eF(P ) of degree at most d− 1, thus P : V → F is a polynomial of degree at most d− 1. In this paper, we develop a variant of the Furstenberg cor...

In a partial inverse combinatorial problem, given a partial solution, the goal is to modify data as small as possible such that there exists an optimal solution containing the given partial solution. In this paper, we study a constraint version of the partial inverse matroid problem in which the weight can only be increased. Two polynomial time algorithms are presented for this problem.

Matrix theory and its applications make wide use of the eigenprojections of square matrices. The paper demonstrated that the eigenprojection of a matrix A can be calculated with the use of any annihilating polynomial for A, where u ≥ indA. This enables one to establish the components and the minimum polynomial of A, as well as the Drazin inverse A.

A new algorithm is presented for the determination of the generalized inverse and the drazin inverse of a polynomial matrix. The proposed algorithms are based on the discrete Fourier transform and thus are computationally fast in contrast to other known algorithms. The above algorithms are implemented in the Mathematica programming language and illustrated via examples.

In this paper we introduce the notion of invertible dynamic system, we indicate a very general method to determine the inverse of such a system and we give evidence of the numerous applications of the subclass of dynamic systems defined by this notion. Key-Words: Invertible dynamic system, the inverse of a dynamic system, Lagrange interpolation polynomial.