Solutions of xqk+⋯+xq+x=ax^{q^k}+\cdots+x^{q}+x=a in GF2nGF{2^n}

Though it is well known that the roots of any affine polynomial over a finite field can be computed by a system of linear equations by using a normal base of the field, such solving approach appears to be difficult to apply when the field is fairly large. Thus, it may be of great interest to find an explicit representation of the solutions independently of the field base. This was previously done only for quadratic equations over a binary finite field. This paper gives an explicit representation of solutions for a much wider class of affine polynomials over a binary prime field

Complete solution over \GF{p^n} of the equation Xpk+1+X+a=0X^{p^k+1}+X+a=0

The problem of solving explicitly the equation $P_a(X):=X^{q+1}+X+a=0$ over the finite field \GF{Q}, where $Q=p^n$, $q=p^k$ and $p$ is a prime, arises in many different contexts including finite geometry, the inverse Galois problem \cite{ACZ2000}, the construction of difference sets with Singer parameters \cite{DD2004}, determining cross-correlation between $m$-sequences \cite{DOBBERTIN2006} and to construct error correcting codes \cite{Bracken2009}, cryptographic APN functions \cite{BTT2014,Budaghyan-Carlet_2006}, designs \cite{Tang_2019}, as well as to speed up the index calculus method for computing discrete logarithms on finite fields \cite{GGGZ2013,GGGZ2013+} and on algebraic curves \cite{M2014}. Subsequently, in \cite{Bluher2004,HK2008,HK2010,BTT2014,Bluher2016,KM2019,CMPZ2019,MS2019,KCM19}, the \GF{Q}-zeros of $P_a(X)$ have been studied. In \cite{Bluher2004}, it was shown that the possible values of the number of the zeros that $P_a(X)$ has in \GF{Q} is $0$, $1$, $2$ or $p^{\gcd(n, k)}+1$. Some criteria for the number of the \GF{Q}-zeros of $P_a(x)$ were found in \cite{HK2008,HK2010,BTT2014,KM2019,MS2019}. However, while the ultimate goal is to explicit all the \GF{Q}-zeros, even in the case $p=2$, it was solved only under the condition $\gcd(n, k)=1$ \cite{KM2019}. In this article, we discuss this equation without any restriction on $p$ and $\gcd(n,k)$. In \cite{KCM19}, for the cases of one or two \GF{Q}-zeros, explicit expressions for these rational zeros in terms of $a$ were provided, but for the case of $p^{\gcd(n, k)}+1$ \GF{Q}- zeros it was remained open to explicitly compute the zeros. This paper solves the remained problem, thus now the equation $X^{p^k+1}+X+a=0$ over \GF{p^n} is completely solved for any prime $p$, any integers $n$ and $k$

Phase transition to chimera state in two populations of oscillators interacting via a common external environment

We consider two populations of coupled oscillators, interacting with each other through a common external environment. The external environment is synthesized by the contributions from all oscillators of both populations. Such indirect coupling via an external medium arises naturally in many fields, e.g., dynamical quorum sensing in coupled biological and chemical systems. We analyze the existence and stability of a variety of stationary states on the basis of the Ott-Antonsen reduction method, which reveals that the interaction via an external environment gives rise to unusual collective behaviors such as the uniform drifting, non-uniform drifting and chimera states. We present a complete bifurcation diagram, which provides the underlying mechanism of the phase transition towards chimera state with the route of incoherence ${}\rightarrow{}$ uniform drift ${}\rightarrow{}$ non-uniform drift ${}\rightarrow{}$ chimera