Let $p$ be a prime and $n$ be a positive integer, and consider $f_b(X)=X+(X^p-X+b)^{-1}\in \Bbb F_p(X)$, where $b\in\Bbb F_{p^n}$ is such that $\text{Tr}_{p^n/p}(b)\ne 0$. It is known that (i) $f_b$ permutes $\Bbb F_{p^n}$ for $p=2,3$ and all $n\ge 1$; (ii) for $p>3$ and $n=2$, $f_b$ permutes $\Bbb F_{p^2}$ if and only if $\text{Tr}_{p^2/p}(b)=\pm 1$; and (iii) for $p>3$ and $n\ge 5$, $f_b$ does not permute $\Bbb F_{p^n}$. It has been conjectured that for $p>3$ and $n=3,4$, $f_b$ does not permute $\Bbb F_{p^n}$. We prove this conjecture for sufficiently large $p$.
A formula discovered by L. Carlitz in 1935 finds an interesting application in permutation rational functions of finite fields. It allows us to determine all rational functions of degree three that permute the projective line $\Bbb P^1(\Bbb F_q)$ over $\Bbb F_q$, a result previously obtained by Ferraguti and Micheli through a different method. It also allows us to determine all rational functions of degree four that permute $\Bbb P^1(\Bbb F_q)$ under a certain condition. (A complete determination of all rational functions of degree four that permute $\Bbb P^1(\Bbb F_q)$ without any condition w…
Let $k$ be a positive integer and $S_{2k}={\tt x}+{\tt x}^4+...+{\tt x}^{4^{2k-1}}\in\Bbb F_2[{\tt x}]$. It was recently conjectured that ${\tt x}+S_{2k}^{4^{2k}}+S_{2k}^{4^k+3}$ is a permutation polynomial of $\Bbb F_{4^{3k}}$. In this note, the conjecture is confirmed and a generalization is obtained.
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…
Let $μ_{q+1}$ denote the set of $(q+1)$-th roots of unity in $\mathbb{F}_{q^2 }$. We construct permutation polynomials over $\mathbb{F}_{q^2}$ by using rational functions of any degree that induce bijections either on $μ_{q+1}$ or between $μ_{q+1}$ and $\mathbb{F}_q \cup \{\infty\}$. In particular, we generalize results from Zieve.
Let $\Bbb F_q$ be the finite field with $q$ elements and let $p=\text{char}\,\Bbb F_q$. It was conjectured that for integers $e\ge 2$ and $1\le a\le pe-2$, the polynomial $X^{q-2}+X^{q^2-2}+\cdots+X^{q^a-2}$ is a permutation polynomial of $\Bbb F_{q^e}$ if and only if (i) $a=2$ and $q=2$, or (ii) $a=1$ and $\text{gcd}(q-2,q^e-1)=1$. In the present paper we confirm this conjecture.
A permutation rational function $f\in \mathbb{F}_q(x)$ is a rational function that induces a bijection on $\mathbb{F}_q$, that is, for all $y\in\mathbb{F}_q$ there exists exactly one $x\in\mathbb{F}_q$ such that $f(x)=y$. Permutation rational functions are intimately related to exceptional rational functions, and more generally exceptional covers of the projective line, of which they form the first important example. In this paper, we show how to efficiently generate many permutation rational functions over large finite fields using isogenies of elliptic curves, and discuss some cryptographic …
Let $q$ be a prime power, $\mathbb F_q$ be the finite field of order $q$ and $\mathbb F_q(x)$ be the field of rational functions over $\mathbb F_q$. In this paper we classify all rational functions $φ\in \mathbb F_q(x)$ of degree 3 that induce a permutation of $\mathbb P^1(\mathbb F_q)$. Our methods are constructive and the classification is explicit: we provide equations for the coefficients of the rational functions using Galois theoretical methods and Chebotarev Density Theorem for global function fields. As a corollary, we obtain that a permutation rational function of degree 3 permutes $\…
Let $p$ be a prime and $n$ be a positive integer. Let $f_b(X)=X+(X^p-X+b)^{-1}$, where $b\in\Bbb F_{p^n}$ is such that $\text{Tr}_{p^n/p}(b)\ne 0$. In 2008, Yuan et al. \cite{Yuan-Ding-Wang-Pieprzyk-FFA-2008} showed that for $p=2,3$, $f_b$ permutes $\Bbb F_{p^n}$ for all $n\ge 1$. Using the Hasse-Weil bound, we show that when $p>3$ and $n\ge 5$, $f$ does not permute $\Bbb F_{p^n}$. For $p>3$ and $n=2$, we prove that $f_b$ permutes $\Bbb F_{p^2}$ if and only if $\text{Tr}_{p^2/p}(b)=\pm 1$. We conjecture that for $p>3$ and $n=3,4$, $f_b$ does not permute $\Bbb F_{p^n}$.
In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the $p$-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements.
Let $\mathbb F_q$ denote the finite field with $q$ elements. In this paper we use the relationship between suitable polynomials and number of rational points on algebraic curves to give the exact number of elements $a\in \mathbb F_q$ for which the binomial $x^n(x^{(q-1)/r} + a)$ is a permutation polynomial in the cases $r = 2$ and $r = 3$.
We prove a function field version of Chowla's conjecture on the autocorrelation of the Möbius function in the limit of a large finite field of characteristic 2.
The conjecture due to Bertrand and Rodriguez Villegas asserts that the 1-norm of the nonzero element in an exterior power of the units of a number field has a certain lower bound. We prove this conjecture for the exterior square case when the number field is a real multi-quadratic Galois extension of any degree of the rationals.
Define $θ(x)=(x-1)/3$ if $x\geq 1$, and $θ(x)=2x/(1-x)$ if $x<1$. We conjecture that the orbit of every positive rational number ends in 0. In particular, there does not exist any positive rational fixed point for a map in the semigroup $Ω$ generated by the maps $3x+1$ and $x/(x+2)$. In this paper, we prove that the asymptotic density of the set of elements in $Ω$ that have rational fixed points is zero.
Hegarty conjectured for $n\neq 2, 3, 5, 7$ that $\mathbb{Z}/n\mathbb{Z}$ has a permutation which destroys all arithmetic progressions mod $n$. For $n\ge n_0$, Hegarty and Martinsson demonstrated that $\mathbb{Z}/n\mathbb{Z}$ has an arithmetic-progression destroying permutation. However $n_0\approx 1.4\times 10^{14}$ and thus resolving the conjecture in full remained out of reach of any computational techniques. However, this paper using constructions modeled after those used by Elkies and Swaminathan for the case of $\mathbb{Z}/p\mathbb{Z}$ with $p$ being prime, establish the conjecture in ful…
In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the generalized $abc$-conjecture holds in K, then there exist at least $c\,\log X$ prime numbers $p \leq X$ for which K is $p$-rational, here $c$ is some nonzero constant depending on K. The real quadratic case was recently suggested by Böckle-Guiraud-Kalyanswamy-Khare.
In this short note we show that the uniform abc-conjecture over number fields puts strong restrictions on the coordinates of rational points on elliptic curves. For the proof we use a variant of the uniform abc-conjecture over number fields formulated by Mochizuki. As an application, we generalize a result of Silverman on elliptic non-Wieferich primes.
In 1923 Schur considered the following problem. Let f(X) be a polynomial with integer coefficients that induces a bijection on the residue fields Z/pZ for infinitely many primes p. His conjecture, that such polynomials are compositions of linear and Dickson polynomials, was proved by M. Fried in 1970. Here we investigate the analogous question for rational functions, also we allow the base field to be any number field. As a result, there are many more rational functions for which the analogous property holds. Some infinite series come from rational isogenies of elliptic curves and deformations…
In this paper, we examine how well a rational point P on an algebraic variety X can be approximated by other rational points. We conjecture that if P lies on a rational curve, then the best approximations to P on X can be chosen to lie along a rational curve. We prove this conjecture for a wide range of examples, and for a great many more examples we deduce our conjecture from Vojta's Main Conjecture.
Let $q>2$ be a prime power and $f={\tt x}^{q-2}+t{\tt x}^{q^2-q-1}$, where $t\in\Bbb F_q^*$. It was recently conjectured that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following holds: (i) $t=1$, $q\equiv 1\pmod 4$; (ii) $t=-3$, $q\equiv \pm1\pmod{12}$; (iii) $t=3$, $q\equiv -1\pmod 6$. We confirm this conjecture in the present paper.
The purpose of this note is give some evidence in support of conjectures of Poonen, and Morton and Silverman, on the periods of rational numbers under the iteration of quadratic polynomials. In particular, Poonen conjectured that there are at most 9 periodic points defined over the rational numbers for any map in the family x^2 + c for c rational. We verify this conjecture for c values up to height 10^8. For quadratic number fields, we provide evidence that the upper bound on the exact period of Q-rational periodic point is 6.
We classify the finite primitive permutation groups which have a cyclic subgroup with two orbits. This extends classical topics in permutation group theory, and has arithmetic consequences. By a theorem of C. L. Siegel, affine algebraic curves with infinitely many integral points are parametrized by rational functions whose monodromy groups have this property. We classify the possibilities of these monodromy groups, and give an application to Hilbert's irreducibility theorem.
Fix a rational basepoint b and a rational number c. For the quadratic dynamical system f_c(x) = x^2+c, it has been shown that the number of rational points in the backward orbit of b is bounded independent of the choice of rational parameter c. In this short note we investigate the dependence of the bound on the basepoint b, assuming a strong form of the Mordell Conjecture.
Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, four classes of monomial complete permutation polynomials and one class of trinomial complete permutation polynomials are presented, one of which confirms a conjecture proposed by Wu et al. (Sci. China Math., to appear. Doi: 10.1007/s11425-014-4964-2). Furthermore, we give two classes of trinomial permutation polynomials, and make some progress on a conjecture about the differential uniformity of power permutation polynomials proposed by B…
We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa's conjecture on exponential sums, with the log-canonical threshold in the exponent of the estimates. We show that this covers optimally all situations of the conjectures for non-rational singularities, by comparing the log canonical threshold with a local notion of the motivic oscillation index.
We construct classes of permutation polynomials over F_{Q^2} by exhibiting classes of low-degree rational functions over F_{Q^2} which induce bijections on the set of (Q+1)-th roots of unity in F_{Q^2}. As a consequence, we prove two conjectures about permutation trinomials from a recent paper by Tu, Zeng, Hu and Li.
