arXiv subjects
Publications and source records attributed to Michael E. Zieve.
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.
We examine the different ways of writing a cover of curves $ϕ\colon C\to D$ over a field $K$ as a composition $ϕ=ϕ_n\circϕ_{n-1}\circ\dots\circϕ_1$, where each $ϕ_i$ is a cover of curves over $K$ of degree at least $2$ which cannot be written as the composition of two lower-degree covers. We show that if the monodromy group $\textrm{Mon}(ϕ)$ has a transitive abelian subgroup then the sequence $(\degϕ_i)_{1\le i\le n}$ is uniquely determined up to permutation by $ϕ$, so in particular the length $n$ is uniquely determined. We prove analogous conclusions for the sequences $(\textrm{Mon}(ϕ_i))_{1\le i\le n}$ and $(\textrm{Aut}(ϕ_i))_{1\le i\le n}$. Such a transitive abelian subgroup exists in particular when $ϕ$ is tamely and totally ramified over some point in $D(\overline{K})$, and also when $ϕ$ is a morphism of one-dimensional algebraic groups (or a coordinate projection of such a morphism). Thus, for example, our results apply to decompositions of polynomials of degree not divisible by $\textrm{char}(K)$, additive polynomials, elliptic curve isogenies, and Lattès maps.
Let G(q) be the group of permutations of the finite field F_q which is generated by those permutations that can be written as c-->ac^m+bc^n with 0<m<n<q and a,b in F_q with ab nonzero. We show that there are infinitely many q for which G(q) is the group of all permutations of F_q which fix 0. This resolves a conjecture of Vasilyev and Rybalkin.
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.
Carlitz proved that, for any prime power q other than 2, the group of all permutations of the finite field F_q is generated by the permutations induced by degree-one polynomials and x^{q-2}. His proof relies on a remarkable polynomial which appears to have been found by magic. We show here that no magic is required: there is a straightforward way to produce a simple polynomial which has the same remarkable properties as the complicated polynomial in Carlitz's proof. We also identify the crucial subtlety which allows such simple polynomials to exist, and discuss some consequences.
We construct Belyi maps having specified behavior at finitely many points. Specifically, for any curve C defined over Q-bar, and any disjoint finite subsets S, T in C(Q-bar), we construct a finite morphism f: C -> P^1 such that f ramifies at each point in S, the branch locus of f is {0,1, infty}, and f(T) is disjoint from {0,1, infty}. This refines a result of Mochizuki's. We also prove an analogous result over fields of positive characteristic, and in addition we analyze how many different Belyi maps f are required to imply the above conclusion for a single C and S and all sets T in C(Q-bar) \ S of prescribed cardinality.
The Mordell-Lang conjecture describes the intersection of a finitely generated subgroup with a closed subvariety of a semiabelian variety. Equivalently, this conjecture describes the intersection of closed subvarieties with the set of images of the origin under a finitely generated semigroup of translations. We study the analogous question in which the translations are replaced by algebraic group endomorphisms (and the origin is replaced by another point). We show that the conclusion of the Mordell-Lang conjecture remains true in this setting if either (1) the semiabelian variety is simple, (2) the semiabelian variety is $A^2$, where $A$ is a one-dimensional semiabelian variety, (3) the subvariety is a connected one-dimensional algebraic subgroup, or (4) each endomorphism has diagonalizable Jacobian at the origin. We also give examples showing that the conclusion fails if we make slight modifications to any of these hypotheses.
We give an explicit description of the F_{q^i}-rational points on the Fermat curve u^{q-1}+v^{q-1}+w^{q-1}=0 for each i=1,2,3. As a consequence, we observe that for any such point (u,v,w), the product uvw is a cube in F_{q^i}. We also describe the F_{q^2}-rational points on the Fermat surface u^{q-1}+v^{q-1}+w^{q-1}+x^{q-1}=0, and show that the product of the coordinates of any such points is a square.
Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They exist only in odd characteristic, but recently Zhou introduced an even characteristic analogue which has similar applications. In this paper we determine all planar functions on F_q of the form c-->uc^t, where q is a power of 2, t is an integer with 0<t<=q^{1/4}, and u is a nonzero element of F_q. This settles and sharpens a conjecture of Schmidt and Zhou.
We give conditions on the rational numbers a,b,c which imply that there are infinitely many triples (x,y,z) of rational numbers such that x+y+z=a+b+c and xyz=abc. We do the same for the equations x+y+z=a+b+c and x^3+y^3+z^3=a^3+b^3+c^3. These results rely on exhibiting families of positive-rank elliptic curves.
Recently Bassa, Garcia and Stichtenoth constructed a tower of function fields over GF(q^3) having many rational places relative to their genera. We show that, by removing the bottom field from this tower, we obtain the same tower we would obtain by removing certain fields from a tower constructed previously by Bezerra, Garcia and Stichtenoth.
We prove that if x^m + c*x^n permutes the prime field GF(p), where m>n>0 and c is in GF(p)^*, then gcd(m-n,p-1) > sqrt{p} - 1. Conversely, we prove that if q>=4 and m>n>0 are fixed and satisfy gcd(m-n,q-1) > 2q*(log log q)/(log q), then there exist permutation binomials over GF(q) of the form x^m + c*x^n if and only if gcd(m,n,q-1) = 1.
I present a construction of permutation polynomials based on cyclotomy, an additive analogue of this construction, and a generalization of this additive analogue which appears to have no multiplicative analogue. These constructions generalize recent results of J. Marcos from arXiv:0810.2738v1.
Every symplectic spread of PG(3,q), or equivalently every ovoid of Q(4,q), is shown to give rise to a certain family of permutation polynomials of GF(q) and conversely. This leads to an algebraic proof of the existence of the Tits-Luneburg spread of W(2^{2h+1}) and the Ree-Tits spread of W(3^{2h+1}), as well as to a new family of low-degree permutation polynomials over GF(3^{2h+1}). We prove the permutation property of the latter polynomials via an odd characteristic analogue of Dobbertin's approach to uniformly representable permutation polynomials over GF(2^n). These new permutation polynomials were later used by Ding, Wang, and Xiang in arXiv:math/0609586 to produce new skew Hadamard difference sets.
We determine the isogeny classes of abelian surfaces over F_q whose group of F_q-rational points has order divisible by q^2. We also solve the same problem for Jacobians of genus-2 curves.
For any elements b,c of a number field K, let G(b,c) denote the backwards orbit of b under the map f_c: C-->C given by f_c(x)=x^2+c. We prove an upper bound on the number of elements of G(b,c) whose degree over K is at most some constant B. This bound depends only on b, [K:Q], and B, and is valid for all b outside an explicit finite set. We also show that, for any N>3 and any b in K outside a finite set, there are only finitely many pairs of complex numbers (y,c) for which [K(y,c):K]<2^(N-3) and the value of the N-th iterate of f_c(x) at x=y is b. Moreover, the bound 2^(N-3) in this result is optimal.
Ritt studied the functional decomposition of a univariate complex polynomial f into prime (indecomposable) polynomials, f = u_1 o u_2 o ... o u_r. His main achievement was a procedure for obtaining any decomposition of f from any other by repeatedly applying certain transformations. However, Ritt's results provide no control on the number of times one must apply the basic transformations, which makes his procedure unsuitable for many theoretical and algorithmic applications. We solve this problem by giving a new description of the collection of all decompositions of a polynomial. Our results have been used by Ghioca, Tucker and Zieve (arXiv:0807.3576) to describe the polynomials f,g having orbits with infinite intersection; they have also been used by Medvedev and Scanlon to describe the affine curves invariant under a coordinatewise polynomial action.
We study the orbits of a polynomial f in C[X], namely the sets {e,f(e),f(f(e)),...} with e in C. We prove that if nonlinear complex polynomials f and g have orbits with infinite intersection, then f and g have a common iterate. More generally, we describe the intersection of any line in C^d with a d-tuple of orbits of nonlinear polynomials, and we formulate a question which generalizes both this result and the Mordell--Lang conjecture.