257 research outputs found
On complete mappings and value sets of polynomials over finite fields
In this thesis we study several aspects of permutation polynomials over nite elds with odd characteristic. We present methods of construction of families of complete mapping polynomials; an important subclass of permutations. Our work on value sets of non-permutation polynomials focus on the structure of the spectrum of a particular class of polynomials. Our main tool is a recent classi cation of permutation polynomials of Fq, based on their Carlitz rank. After introducing the notation and terminology we use, we give basic properties of permutation polynomials, complete mappings and value sets of polynomials in Chapter 1. We present our results on complete mappings in Fq[x] in Chapter 2. Our main result in Section 2.2 shows that when q > 2n + 1, there is no complete mapping polynomial of Carlitz rank n, whose poles are all in Fq. We note the similarity of this result to the well-known Chowla-Zassenhaus conjecture (1968), proven by Cohen (1990), which is on the non-existence of complete mappings in Fp[x] of degree d, when p is a prime and is su ciently large with respect to d. In Section 2.3 we give a su cient condition for the construction of a family of complete mappings of Carlitz rank at most n. Moreover, for n = 4, 5, 6 we obtain an explicit construction of complete mappings. Chapter 3 is on the spectrum of the class Fq,n of polynomials of the form F(x) = f(x)+x, where f is a permutation polynomial of Carlitz rank at most n. Upper bounds for the cardinality of value sets of non-permutation polynomials of the xed degree d or xed index l were obtained previously, which depend on d or l respectively. We show, for instance, that the upper bound in the case of a subclass of Fq,n is q -2, i.e., is independent of n. We end this work by giving examples of complete mappings, obtained by our methods
On the difference between permutation polynomials over finite fields
The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990,
states that if , then there is no complete mapping polynomial
in \Fp[x] of degree . For arbitrary finite fields \Fq, a
similar non-existence result is obtained recently by I\c s\i k, Topuzo\u glu
and Winterhof in terms of the Carlitz rank of .
Cohen, Mullen and Shiue generalized the Chowla-Zassenhaus-Cohen Theorem
significantly in 1995, by considering differences of permutation polynomials.
More precisely, they showed that if and are both permutation
polynomials of degree over \Fp, with , then the
degree of satisfies , unless is constant. In this
article, assuming and are permutation polynomials in \Fq[x], we
give lower bounds for in terms of the Carlitz rank of
and . Our results generalize the above mentioned result of I\c s\i k et
al. We also show for a special class of polynomials of Carlitz rank that if is a permutation of \Fq, with , then
Counting Dyck paths by area and rank
The set of Dyck paths of length inherits a lattice structure from a
bijection with the set of noncrossing partitions with the usual partial order.
In this paper, we study the joint distribution of two statistics for Dyck
paths: \emph{area} (the area under the path) and \emph{rank} (the rank in the
lattice).
While area for Dyck paths has been studied, pairing it with this rank
function seems new, and we get an interesting -refinement of the Catalan
numbers. We present two decompositions of the corresponding generating
function: one refines an identity of Carlitz and Riordan; the other refines the
notion of -nonnegativity, and is based on a decomposition of the
lattice of noncrossing partitions due to Simion and Ullman.
Further, Biane's correspondence and a result of Stump allow us to conclude
that the joint distribution of area and rank for Dyck paths equals the joint
distribution of length and reflection length for the permutations lying below
the -cycle in the absolute order on the symmetric group.Comment: 24 pages, 7 figures. Connections with work of C. Stump
(arXiv:0808.2822v2) eliminated the need for 5 pages of proof in the first
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