127 research outputs found
Decompositions of Laurent polynomials
In the 1920's, Ritt studied the operation of functional composition g o h(x)
= g(h(x)) on complex rational functions. In the case of polynomials, he
described all the ways in which a polynomial can have multiple `prime
factorizations' with respect to this operation. Despite significant effort by
Ritt and others, little progress has been made towards solving the analogous
problem for rational functions. In this paper we use results of Avanzi--Zannier
and Bilu--Tichy to prove analogues of Ritt's results for decompositions of
Laurent polynomials, i.e., rational functions with denominator a power of x.Comment: 31 page
Factorizations of certain bivariate polynomials
We determine the factorization of X*f(X)-Y*g(Y) over K[X,Y] for all
squarefree additive polynomials f,g in K[X] and all fields K of odd
characteristic. This answers a question of Kaloyan Slavov, who needed these
factorizations in connection with an algebraic-geometric analogue of the Kakeya
problem.Comment: 5 page
Permutation polynomials induced from permutations of subfields, and some complete sets of mutually orthogonal latin squares
We present a general technique for obtaining permutation polynomials over a
finite field from permutations of a subfield. By applying this technique to the
simplest classes of permutation polynomials on the subfield, we obtain several
new families of permutation polynomials. Some of these have the additional
property that both f(x) and f(x)+x induce permutations of the field, which has
combinatorial consequences. We use some of our permutation polynomials to
exhibit complete sets of mutually orthogonal latin squares. In addition, we
solve the open problem from a recent paper by Wu and Lin, and we give simpler
proofs of much more general versions of the results in two other recent papers.Comment: 13 pages; many new result
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