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Regarding non-unique factorization of integer-valued polynomials over a discrete valuation domain (R , M) with finite residue field, it is known that there exist absolutely irreducible elements, that is, irreducible elements all of whose powers factor uniquely, and non-absolutely irreducible elements. We completely and constructively characterize the absolutely irreducible elements among split integer-valued polynomials. They correspond bijectively to finite sets, which we call balanced , characterized by a combinatorial property regarding the distribution of their elements among residue classes of powers of M. For each such balanced set as the set of roots of a split polynomial, there exists a unique vector of multiplicities and a unique constant so that the corresponding product of monic linear factors times the constant is an absolutely irreducible integer-valued polynomial. This also yields sufficient criteria for integer-valued polynomials over Dedekind domains to be absolutely irreducible. [ABSTRACT FROM AUTHOR]

Let n be a positive integer, q = 2 n , and let F q be the finite field with q elements. For each positive integer m , let D m ( X ) be the Dickson polynomial of the first kind of degree m with parameter 1. Assume that m > 1 is a divisor of q + 1 . We study the existence of α ∈ F q ⁎ such that D m ( α ) = D m ( α − 1 ) = 0 . We also explore the connections of this question to an open question by Wiedemann and a game called “Button Madness”. [ABSTRACT FROM AUTHOR]

The subset sum problem is an important theoretical problem with many applications, such as in coding theory, cryptography, graph theory and other fields. One of the many aspects of this problem is to answer the solvability of the k -subset sum problem. It has been proven to be NP-hard in general. However, if the evaluation set has some special algebraic structure, it is possible to obtain some good conclusions. Zhu, Wan and Keti proposed partial results of this problem over two special kinds of evaluation sets. We generalize their conclusions in this paper, and propose asymptotical results of the solvability of the k -subset sum problem by using estimates of additive character sums over the evaluation set, together with the Brun sieve and the new sieve proposed by Li and Wan. We also apply the former two examples as application of our results. [ABSTRACT FROM AUTHOR]

In this paper, we completely determine all necessary and sufficient conditions such that the polynomial f (x) = x 3 + a x q + 2 + b x 2 q + 1 + c x 3 q , where a , b , c ∈ F q ⁎ , is a permutation quadrinomial of F q 2 over any finite field of odd characteristic. This quadrinomial has been studied first in [25] by Tu, Zeng and Helleseth, later in [24] Tu, Liu and Zeng revisited these quadrinomials and they proposed a more comprehensive characterization of the coefficients that results with new permutation quadrinomials, where c h a r (F q) = 2 and finally, in [16] , Li, Qu, Li and Chen proved that the sufficient condition given in [24] is also necessary and thus completed the solution in even characteristic case. In [6] Gupta studied the permutation properties of the polynomial x 3 + a x q + 2 + b x 2 q + 1 + c x 3 q , where c h a r (F q) = 3 , 5 and a , b , c ∈ F q ⁎ and proposed some new classes of permutation quadrinomials of F q 2 . In particular, in this paper we classify all permutation polynomials of F q 2 of the form f (x) = x 3 + a x q + 2 + b x 2 q + 1 + c x 3 q , where a , b , c ∈ F q ⁎ , over all finite fields of odd characteristic and obtain several new classes of such permutation quadrinomials. [ABSTRACT FROM AUTHOR]