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Michael Zieve

Publications and source records attributed to Michael Zieve.

10 recordsLinked to original sources
arXiv2022

A note on the paper arXiv:2112.14547

We give historical remarks related to arXiv:2112.14547 ("A New Method of Construction of Permutation Trinomials with Coefficients 1", by Guo et al.). In particular, we show that the "new" permutation polynomials in that paper are actually well known. In addition we give a simpler derivation of these permutation polynomials than had been given previously, which demonstrates the general method of producing permutation polynomials that was introduced in arXiv:1310.0776.

math.CO↗
arXiv2016

Permutation polynomials of the form x+c*Tr(x^k)

Let F_{q^n} be the field of order q^n, and let Tr be the trace map from F_{q^n} to its q-element subfield. We exhibit nine sequences of polynomials of the form f(x):=x+c*Tr(x^k), with c in F_{q^n}, such that for each polynomial the function F_{q^n}-->F_{q^n} given by c-->f(c) is a permutation of F_{q^n}. We also computed all permutation polynomials of this form over finite fields of size less than 5000, and found that our examples comprise all examples with n>1 except for some simple cases where the polynomial induces a homomorphism of the additive group of F_{q^n}, along with a few sporadic examples. One intriguing feature is that our proofs of the different sequences use various different methods, including a new variant of Dobbertin's method among others.

math.NT↗
arXiv2014

Uniform Boundedness of S-Units in Arithmetic Dynamics

Let K be a number field and let S be a finite set of places of K which contains all the Archimedean places. For any f(z) in K(z) of degree d at least 2 which is not a d-th power in \bar{K}(z), Siegel's theorem implies that the image set f(K) contains only finitely many S-units. We conjecture that the number of such S-units is bounded by a function of |S| and d (independently of K and f). We prove this conjecture for several classes of rational functions, and show that the full conjecture follows from the Bombieri--Lang conjecture.

math.NT↗
arXiv2013

Two questions on polynomial decomposition

Given a univariate polynomial f(x) over a ring R, we examine when we can write f(x) as g(h(x)) where g and h are polynomials of degree at least 2. We answer two questions of Gusic regarding when the existence of such g and h over an extension of R implies the existence of such g and h over R. We also pose two new questions along these lines.

math.AC↗
arXiv2013

Planar functions and perfect nonlinear monomials over finite fields

The study of finite projective planes involves planar functions, namely, functions f : F_q --> F_q such that, for each nonzero a in F_q, the function c --> f(c+a) - f(c) is a bijection on F_q. Planar functions are also used in the construction of DES-like cryptosystems, where they are called perfect nonlinear functions. We determine all planar functions on F_q of the form c --> c^t, under the assumption that q >= (t-1)^4. This implies two conjectures of Hernando, McGuire and Monserrat. Our arguments also yield a new proof of a conjecture of Segre and Bartocci from 1971 about monomial hyperovals in finite Desarguesian projective planes.

math.CO↗
arXiv2011

On hyperbolic fixed points in ultrametric dynamics

Let K be a complete ultrametric field. We give lower and upper bounds for the size of linearization discs for power series over K near hyperbolic fixed points. These estimates are maximal in the sense that there exist examples where these estimates give the exact size of the corresponding linearization disc. In particular, at repelling fixed points, the linearization disc is equal to the maximal disc on which the power series is injective.

math.DS↗
arXiv2008

Everywhere ramified towers of global function fields

We consider a tower of function fields F_0 < F_1 < ... over a finite field such that every place of every F_i ramified in the tower and the sequence genus(F_i)/[F_i:F_0] has a finite limit. We also construct a tower in which every place ramifies and the sequence N_i/[F_i:F_0] has a positive limit, where N_i is the number of degree-one places of F_i. These towers answer questions posed by Stichtenoth.

math.NT↗
arXiv2007

Analogues of the Jordan-Holder theorem for transitive G-sets

Let G be a transitive group of permutations of a finite set X, and suppose that some element of G has at most two orbits on X. We prove that any two maximal chains of groups between G and a point-stabilizer of G have the same length, and the same sequence of relative indices between consecutive groups (up to permutation). We also deduce the same conclusion when G has a transitive quasi-Hamiltonian subgroup.

math.GR↗
arXiv2007

Equivalence of sparse circulants: the bipartite Ádám problem

We consider n-by-n circulant matrices having entries 0 and 1. Such matrices can be identified with sets of residues mod n, corresponding to the columns in which the top row contains an entry 1. Let A and B be two such matrices, and suppose that the corresponding residue sets S_A and S_B have size at most 3. We prove that the following are equivalent: (1) there are integers u,v mod n, with u a unit, such that S_A = uS_B + v; (2) there are permutation matrices P,Q such that A=PBQ. Our proof relies on some new results about vanishing sums of roots of unity. We give examples showing this result is not always true for denser circulants, as well as results showing it continues to hold in some situations. We also explain how our problem relates to the Adam problem on isomorphisms of circulant directed graphs.

math.CO↗