The paper describes an algebraic construction of the inversive difference field associated with a discrete-time rational nonlinear control system under the assumption that the system is submersive. We prove that a system is submersive iff its associated difference ideal is proper, prime and reflexive. Next, we show that Kähler differentials of the… (More)
In this paper, we give conditions under which the trinomials of the form x n + ax + b over finite field Fpm are not primitive and conditions under which there are no primitive trinomials of the form x n + ax + b over finite field Fpm. For finite field F4, We show that there are no primitive trinomials of the form x n + x + α, if n ≡ 1 mod 3 or n ≡ 0 mod 3… (More)
Let k be a commutative field, σ an automorphism of k, and δ a derivation on k with respect to σ. Ore in  defines a (univariate) polynomial ring k[∂; σ, δ], which is called an Ore or skew polynomial ring. An Ore polynomial ring is, in general, noncommutative. Its commutation rule is ∂r = σ(r)∂ + δ(r) for all r ∈ k. For example, C(t)[∂; 1, δ], where 1 maps… (More)
In this note, we present an answer to Exercise 9.3 in the Book Symbolic Integration I (second edition) by M. Bronstein, under an additional assumption that the real elementary extension in the exercise is purely transcendental. Our answer is based on a rather technical lemma derived from a naive attempt to do the exercise inductively.
In this paper, we explore the primitivity of trinomials over small finite fields. We extend the results of the primitivity of trinomials x n + ax + b over F4  to the general form x n + ax k + b. We prove that for given n and k, one of all the trinomials x n + ax k + b with b being the primitive element of F4 and a + b = 1 is primitive over F4 if and only… (More)
Recently, Kalikinkar Mandal and Guang Gong presented a family of nonlinear pseudo-random number generators using Welch-Gong Transformations in their paper . They also performed the cycle decomposition of the WG-NLFSR recurrence relations over different finite fields by computer simulations where the nonlinear recurrence relation is composed of a… (More)
We present a criterion for the similarity of length-two elements in a noncom-mutative principal ideal domain. The criterion enables us to develop an algorithm for determining whether B 1 A 1 and B 2 A 2 are similar, where A 1 , A 2 , B 1 , B 2 are first-order differential (difference) operators. The main step in the algorithm is to find a rational solution… (More)