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Let n be a positive integer and let $\mathbb{F} _{q^n}$ be the finite field with $q^n$ elements, where q is a prime power. We introduce a natural action of the projective semilinear group on the set of monic irreducible polynomials over the finite field $\mathbb{F} _{q^n}$. Our main results provide information on the characterisation and number of fixed points.
Symplectic finite semifields can be used to construct nonlinear binary codes of Kerdock type (i.e., with the same parameters of the Kerdock codes, a subclass of Delsarte–Goethals codes). In this paper, we introduce nonbinary Delsarte–Goethals codes of parameters
$(q^{m+1}\ ,\ q^{m(r+2)+2}\ ,\ {\frac{q-1}{q}(q^{m+1}-q^{\frac{m+1}{2}+r})})$
over a Galois field of order
$q=2^l$
, for all
$0\le r\le\frac{m-1}{2}$
, with m ≥ 3 odd, and show the connection of this construction to finite semifields.
Let
$r,n>1$
be integers and
$q$
be any prime power
$q$
such that
$r\mid q^{n}-1$
. We say that the extension
$\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$
possesses the line property for
$r$
-primitive elements property if, for every
$\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}\in \mathbb{F}_{q^{n}}^{\ast }$
such that
$\mathbb{F}_{q^{n}}=\mathbb{F}_{q}(\unicode[STIX]{x1D703})$
, there exists some
$x\in \mathbb{F}_{q}$
such that
$\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D703}+x)$
has multiplicative order
$(q^{n}-1)/r$
. We prove that, for sufficiently large prime powers
$q$
,
$\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$
possesses the line property for
$r$
-primitive elements. We also discuss the (weaker) translate property for extensions.
We prove that sums of length about $q^{3/2}$ of Hecke eigenvalues of automorphic forms on $\operatorname{SL}_{3}(\mathbf{Z})$ do not correlate with $q$-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky–Nelson, corresponding to multiplicative Dirichlet characters, and applies, in particular, to trace functions of small conductor modulo primes.
Such a sequence is eventually periodic and we denote by
$P(n)$
the maximal period of such sequences for given odd
$n$
. We prove a lower bound for
$P(n)$
by counting certain partitions. We then estimate the size of these partitions via the multiplicative order of two modulo
$n$
.
We obtain a new lower bound on the size of the value set $\mathscr{V}(f)=f(\mathbb{F}_{p})$ of a sparse polynomial $f\in \mathbb{F}_{p}[X]$ over a finite field of $p$ elements when $p$ is prime. This bound is uniform with respect to the degree and depends on some natural arithmetic properties of the degrees of the monomial terms of $f$ and the number of these terms. Our result is stronger than those that can be extracted from the bounds on multiplicities of individual values in $\mathscr{V}(f)$.
A polynomial $f$ over a finite field $\mathbb{F}_{q}$ can be classified as a permutation polynomial by the Hermite–Dickson criterion, which consists of conditions on the powers $f^{e}$ for each $e$ from $1$ to $q-2$, as well as the existence of a unique solution to $f(x)=0$ in $\mathbb{F}_{q}$. Carlitz and Lutz gave a variant of the criterion. In this paper, we provide an alternate proof to the theorem of Carlitz and Lutz.
We give an algorithmic generalisation of Dickson’s method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson’s idea is to formulate from Hermite’s criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree $d$. Previous classifications of PPs of degree at most 6 were essentially deduced from manual analysis of these polynomial equations, but this approach is no longer viable for $d>6$. Our idea is to calculate some radicals of ideals generated by the polynomials, implemented by a computer algebra system. Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree 8 over an arbitrary finite field of odd order $q>8$. Such PPs exist if and only if $q\in \{11,13,19,23,27,29,31\}$ and are explicitly listed in normalised form.
We obtain a new sum–product estimate in prime fields for sets of large cardinality. In particular, we show that if $A\subseteq \mathbb{F}_{p}$ satisfies $|A|\leq p^{64/117}$ then $\max \{|A\pm A|,|AA|\}\gtrsim |A|^{39/32}.$ Our argument builds on and improves some recent results of Shakan and Shkredov [‘Breaking the 6/5 threshold for sums and products modulo a prime’, Preprint, 2018, arXiv:1806.07091v1] which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy $E^{+}(P)$ of some subset $P\subseteq A+A$. Our main novelty comes from reducing the estimation of $E^{+}(P)$ to a point–plane incidence bound of Rudnev [‘On the number of incidences between points and planes in three dimensions’, Combinatorica38(1) (2017), 219–254] rather than a point–line incidence bound used by Shakan and Shkredov.
Let $K$ be a field that admits a cyclic Galois extension of degree $n\geq 2$. The symmetric group $S_{n}$ acts on $K^{n}$ by permutation of coordinates. Given a subgroup $G$ of $S_{n}$ and $u\in K^{n}$, let $V_{G}(u)$ be the $K$-vector space spanned by the orbit of $u$ under the action of $G$. In this paper we show that, for a special family of groups $G$ of affine type, the dimension of $V_{G}(u)$ can be computed via the greatest common divisor of certain polynomials in $K[x]$. We present some applications of our results to the cases $K=\mathbb{Q}$ and $K$ finite.
We present the geometry behind counting twin prime polynomials in $\mathbb{F}_{q}[T]$ in general. We compute cohomology and explicitly count points by means of a twisted Lefschetz trace formula applied to these parametrizing varieties for cubic twin prime polynomials. The elliptic curve $X^{3}=Y(Y-1)$ occurs in the geometry, and thus counting cubic twin prime polynomials involves the associated modular form. In theory, this approach can be extended to higher degree twin primes, but the computations become harder.
The formula we get in degree 3 is compatible with the Hardy–Littlewood heuristic on average, agrees with the prediction for $q\equiv 2$ (mod 3), but shows anomalies for $q\equiv 1$ (mod 3).
We prove a function field analogue of Maynard’s celebrated result about primes with restricted digits. That is, for certain ranges of parameters $n$ and $q$, we prove an asymptotic formula for the number of irreducible polynomials of degree $n$ over a finite field $\mathbb{F}_{q}$ whose coefficients are restricted to lie in a given subset of $\mathbb{F}_{q}$.
We discuss 1-factorizations of complete graphs that “match” a given Hadamard matrix. We prove the existence of these factorizations for two families of Hadamard matrices: Walsh matrices and certain Paley matrices.
We obtain a non-trivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short intervals, which in turn is based on a new estimate on bilinear sums of Kloosterman sums. These results are analogues of those obtained by various authors for Kloosterman sums modulo a prime. However, the underlying technique is different and allows us to obtain non-trivial results starting from much shorter ranges.
for an additive character
$\unicode[STIX]{x1D712}$
over
$\mathbb{F}_{q}$
and a polynomial
$Q\in \mathbb{F}_{q}[x_{0},\ldots ,x_{n-1}]$
of degree at most 2 in the coefficients
$x_{0},\ldots ,x_{n-1}$
of
$f=\sum _{i<n}x_{i}t^{i}$
. As in the integers, it is reasonable to expect that, due to the random-like behaviour of
$\unicode[STIX]{x1D707}$
, such sums should exhibit considerable cancellation. In this paper we show that the correlation (1) is bounded by
$O_{\unicode[STIX]{x1D716}}(q^{(-1/4+\unicode[STIX]{x1D716})n})$
for any
$\unicode[STIX]{x1D716}>0$
if
$Q$
is linear and
$O(q^{-n^{c}})$
for some absolute constant
$c>0$
if
$Q$
is quadratic. The latter bound may be reduced to
$O(q^{-c^{\prime }n})$
for some
$c^{\prime }>0$
when
$Q(f)$
is a linear form in the coefficients of
$f^{2}$
, that is, a Hankel quadratic form, whereas, for general quadratic forms, it relies on a bilinear version of the additive-combinatorial Bogolyubov theorem.
We improve some previously known deterministic algorithms for finding integer solutions $x,y$ to the exponential equation of the form $af^{x}+bg^{y}=c$ over finite fields.
The cardinality of the set of
$D\leqslant x$
for which the fundamental solution of the Pell equation
$t^{2}-Du^{2}=1$
is less than
$D^{1/2+\unicode[STIX]{x1D6FC}}$
with
$\unicode[STIX]{x1D6FC}\in [\frac{1}{2},1]$
is studied and certain lower bounds are obtained, improving previous results of Fouvry by introducing the
$q$
-analogue of van der Corput method to algebraic exponential sums with smooth moduli.
In this paper, we investigate the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as ‘Birch sums’. Our main theorem gives upper and lower bounds (of nearly the same order of magnitude) for the distribution of large values of this maximum, that hold in a wide uniform range. This improves a recent result of Kowalski and Sawin. The proofs use a blend of probabilistic methods, harmonic analysis techniques, and deep tools from algebraic geometry. The results can also be generalized to other types of
$\ell$
-adic trace functions. In particular, the lower bound of our result also holds for partial sums of Kloosterman sums. As an application, we show that there exist
$x\in [1,p]$
and
$a\in \mathbb{F}_{p}^{\times }$
such that
$|\sum _{n\leqslant x}\exp (2\unicode[STIX]{x1D70B}i(n^{3}+an)/p)|\geqslant (2/\unicode[STIX]{x1D70B}+o(1))\sqrt{p}\log \log p$
. The uniformity of our results suggests that this bound is optimal, up to the value of the constant.