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Michael E. Zieve

Publications and source records attributed to Michael E. Zieve.

At least 19 recordsLinked to original sources
arXiv2025

Determination of all complete mappings of F_{q^2} of the form aX^{3q}+bX^{2q+1}+cX^{q+2}+dX^3

For each prime power q, we determine all polynomials over F_{q^2} of the form f(X) := aX^{3q}+bX^{2q+1}+cX^{q+2}+dX^3 which induce complete mappings of F_{q^2}, in the sense that each of the functions x --> f(x) and x --> f(x)+x permutes F_{q^2}. This is the first result in the literature which classifies the complete mappings among some class of polynomials with arbitrarily large degree over finite fields of arbitrary characteristic. We also determine all permutation polynomials over F_{q^2} of the form X^{q+2}+bX^q+cX, and all permutations of (F_q)^2 induced by maps of the form (x,y) --> (x^3-exy^2-ax-by, y^3-cx-dy) where either e=0 or 3|q. The latter results add to the small number of results in the literature classifying all permutations induced by maps of prescribed forms.

math.NT↗
arXiv2025

Some classes of permutation pentanomials

For each prime p other than 3, and each power q=p^k, we present two large classes of permutation polynomials over F_{q^2} of the form X^r B(X^{q-1}) which have at most five terms, where B(X) is a polynomial with coefficients in {1,-1}. The special case p=2 of our results comprises a vast generalization of 76 recent results and conjectures in the literature. In case p>2, no instances of our permutation polynomials have appeared in the literature, and the construction of such polynomials had been posed as an open problem. Our proofs are short and involve no computations, in contrast to the proofs of many of the special cases of our results which were published previously.

math.NT↗
arXiv2024

Monodromy groups of indecomposable coverings of bounded genus

For each nonnegative integer $g$, we classify the ramification types and monodromy groups of indecomposable coverings of complex curves $f: X\to Y$ where $X$ has genus $g$, under the hypothesis that $n:=°(f)$ is sufficiently large and the monodromy group is not $A_n$ or $S_n$. This proves a conjecture of Guralnick and several conjectures of Guralnick and Shareshian.

math.AG↗
arXiv2024

Monodromy groups of product type

The combination of this paper and its companion complete the classification of monodromy groups of indecomposable coverings of complex curves $f:X\rightarrow \mathbb P^1$ of sufficiently large degree in comparison to the genus of $X$. In this paper we determine all such coverings with monodromy group $G\leq S_\ell\wr S_t$ of product type for $t\ge 2$.

math.AG↗
arXiv2023

On sums of powers of consecutive squares over finite fields, and sums of distinct values of polynomials

For each odd prime power q, and each integer k, we determine the sum of the k-th powers of all elements x in F_q for which both x and x+1 are squares in F_q^*. We also solve the analogous problem when one or both of x and x+1 is a nonsquare. We use these results to determine the sum of the elements of the image set f(F_q) for each f(X) in F_q[X] of the form X^4+aX^2+b, which resolves two conjectures by Finch-Smith, Harrington, and Wong.

math.NT↗
arXiv2023

Determination of hyperovals by lines through a few points

If S is a set of q+2 points in P^2(F_q) such that some point of S is not on any line containing two other points of S, then in suitable coordinates S has the form S_f:={(c:f(c):1) : c in F_q} U {(1:0:0),(0:1:0)} for some f(X) in F_q[X]. Let T be a subset of S_f which contains the two infinite points and at least 3+log_3(q)/4 finite points. We show that if there is no line passing through a point of T and two other points of S_f, and deg(f)<=q^{1/4}, then no three points of S_f are collinear, so that S_f is a hyperoval. We also determine all f(X) with deg(f)<=q^{1/4} for which S_f is a hyperoval, which strengthens a result that was proved by Caullery and Schmidt using entirely different methods.

math.CO↗
arXiv2023

Constructing permutation polynomials using generalized Redei functions

We determine all permutations in two large classes of polynomials over finite fields, where the construction of the polynomials in each class involves the denominators of a class of rational functions generalizing the classical Redei functions. Our results generalize eight recent results from the literature, and our proofs of our more general results are much shorter and simpler than the previous proofs of special cases.

math.NT↗
arXiv2023

Roots of certain polynomials over finite fields

We determine the roots in F_{q^3} of the polynomial X^{2q^k+1} + X + c for each positive integer k and each c in F_q, where q is a power of 2. We introduce a new approach for this type of question, and we obtain results which are more explicit than the previous results in this area. Our results resolve an open problem and a conjecture of Zheng, Kan, Zhang, Peng, and Li.

math.NT↗
arXiv2023

Low-degree permutation rational functions over finite fields

We determine all degree-4 rational functions f(X) in F_q(X) which permute P^1(F_q), and answer two questions of Ferraguti and Micheli about the number of such functions and the number of equivalence classes of such functions up to composing with degree-one rational functions. We also determine all degree-8 rational functions f(X) in F_q(X) which permute P^1(F_q) in case q is sufficiently large, and do the same for degree 32 in case either q is odd or f(X) is a nonsquare. Further, for most other positive integers n<4096, for each sufficiently large q we determine all degree-n rational functions f(X) in F_q(X) which permute P^1(F_q) but which are not compositions of lower-degree rational functions in F_q(X). Some of these results are proved by using a new Galois-theoretic characterization of additive (linearized) polynomials among all rational functions, which is of independent interest.

math.NT↗
arXiv2022

A note on the paper arXiv:2207.13335

We show that all of the "new" permutation polynomials in the recent paper arXiv:2207.13335 (H. Song et al.) are in fact known. We also present a new type of question in this area.

math.NT↗
arXiv2022

Determination of a class of permutation quadrinomials

We determine all permutation polynomials over F_{q^2} of the form X^r A(X^{q-1}) where, for some Q which is a power of the characteristic of F_q, the integer r is congruent to Q+1 (mod q+1) and all terms of A(X) have degrees in {0, 1, Q, Q+1}. We then use this classification to resolve eight conjectures and open problems from the literature, and we show that the simplest special cases of our result imply 58 recent results from the literature. Our proof makes a novel use of geometric techniques in a situation where they previously did not seem applicable, namely to understand the arithmetic of high-degree rational functions over small finite fields, despite the fact that in this situation the Weil bounds do not provide useful information.

math.NT↗
arXiv2021

Tangent-Chebyshev rational maps and Redei functions

Recently Lima and Campello de Souza introduced a new class of rational functions over odd-order finite fields, and explained their potential usefulness in cryptography. We show that these new functions are conjugate to the classical family of Redei rational functions, so that the properties of the new functions follow from properties of Redei functions. We also prove new properties of these functions, and introduce analogous functions in characteristic 2, while also introducing a new version of trigonometry over finite fields of even order, which is of independent interest.

cs.CR↗
arXiv2021

The relative Riemann-Hurwitz formula

For any nonconstant f,g in C(x) such that the numerator H(x,y) of f(x)-g(y) is irreducible, we compute the genus of the normalization of the curve H(x,y)=0. We also prove an analogous formula in arbitrary characteristic when f and g have no common wildly ramified branch points, and generalize to (possibly reducible) fiber products of nonconstant morphisms of curves f:A-->D and g:B-->D.

math.AG↗
arXiv2021

A new family of exceptional rational functions

For each odd prime power q, we construct an infinite sequence of rational functions f(X) in F_q(X), each of which is exceptional, which means that for infinitely many n the map c-->f(c) induces a bijection of P^1(F_{q^n}). Moreover, each of our functions f(X) is indecomposable, which means that it cannot be written as the composition of lower-degree rational functions in F_q(X). In case q is not a power of 3, these are the first known examples of indecomposable exceptional rational functions f(X) over F_q which have non-solvable monodromy groups and have arbitrarily large degree. These are also the first known examples of wildly ramified indecomposable exceptional rational functions f(X), other than linear changes of polynomials.

math.NT↗
arXiv2020

Extensions of absolute values on two subfields

We describe the absolute values on a field which simultaneously extend absolute values on two subfields. We also give a common generalization of many versions of Abhyankar's lemma on ramification indices, which is both widely applicable and easy to state. We then apply these results to count points on the fibered product of two curve morphisms C --> X and D --> X which lie over prescribed points on C and D.

math.AC↗
arXiv2020

Functional equations in polynomials

We determine all F,G in C[X] of degree at least 2 for which the semigroup generated by F and G under composition is not the free semigroup on the letters F and G. We also solve the same problem for F,G in X^2 C[[X]], and prove partial analogues over arbitrary fields.

math.DS↗
arXiv2014

Some planar monomials in characteristic 2

Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They were originally defined only in odd characteristic, but recently Zhou introduced a definition in even characteristic which yields similar applications. In this paper we show that certain functions over $\mathbb{F}_{2^r}$ are planar, which proves a conjecture of Schmidt and Zhou. The key to our proof is a new result about the $\mathbb{F}_{q^3}$-rational points on the degree-$(q-1)$ Fermat curve $x^{q-1}+y^{q-1}=z^{q-1}$.

math.CO↗