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Register a new userThe Distance to a Squarefree Polynomial Over $mathbb{F}_2[x]$
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In this paper, we examine how far a polynomial in $mathbb{F}_2[x]$ can be from a squarefree polynomial. For any $epsilon>0$, we prove that for any polynomial $f(x)inmathbb{F}_2[x]$ with degree $n$, there exists a squarefree polynomial $g(x)inmathbb{F}_2[x]$ such that $mathrm{deg} (g) le n$ and $L_{2}(f-g)<(ln n)^{2ln(2)+epsilon}$ (where $L_{2}$ is a norm to be defined). As a consequence, the analogous result holds for polynomials $f(x)$ and $g(x)$ in $mathbb{Z}[x]$.
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We study an analogue of Serres modularity conjecture for projective representations $overline{rho}: operatorname{Gal}(overline{K} / K) rightarrow operatorname{PGL}_2(k)$, where $K$ is a totally real number field. We prove new cases of this conjecture when $k = mathbb{F}_5$ by using the automorphy lifting theorems over CM fields established in previous work of the authors.
We prove estimates for the level of distribution of the Mobius function, von Mangoldt function, and divisor functions in squarefree progressions in the ring of polynomials over a finite field. Each level of distribution converges to $1$ as $q$ goes to $infty$, and the power savings converges to square-root cancellation as $q$ goes to $infty$. These results in fact apply to a more general class of functions, the factorization functions, that includes these three. The divisor estimates have applications to the moments of $L$-functions, and the von Mangoldt estimate to one-level densities.
Permutation polynomials (PPs) of the form $(x^{q} -x + c)^{frac{q^2 -1}{3}+1} +x$ over $mathbb{F}_{q^2}$ were presented by Li, Helleseth and Tang [Finite Fields Appl. 22 (2013) 16--23]. More recently, we have constructed PPs of the form $(x^{q} +bx + c)^{frac{q^2 -1}{d}+1} -bx$ over $mathbb{F}_{q^2}$, where $d=2, 3, 4, 6$ [Finite Fields Appl. 35 (2015) 215--230]. In this paper we concentrate our efforts on the PPs of more general form [ f(x)=(ax^{q} +bx +c)^r phi((ax^{q} +bx +c)^{(q^2 -1)/d}) +ux^{q} +vx~~text{over $mathbb{F}_{q^2}$}, ] where $a,b,c,u,v in mathbb{F}_{q^2}$, $r in mathbb{Z}^{+}$, $phi(x)in mathbb{F}_{q^2}[x]$ and $d$ is an arbitrary positive divisor of $q^2-1$. The key step is the construction of a commutative diagram with specific properties, which is the basis of the Akbary--Ghioca--Wang (AGW) criterion. By employing the AGW criterion two times, we reduce the problem of determining whether $f(x)$ permutes $mathbb{F}_{q^2}$ to that of verifying whether two more polynomials permute two subsets of $mathbb{F}_{q^2}$. As a consequence, we find a series of simple conditions for $f(x)$ to be a PP of $mathbb{F}_{q^2}$. These results unify and generalize some known classes of PPs.
Determinants with Legendre symbol entries have close relations with character sums and elliptic curves over finite fields. In recent years, Sun, Krachun and his cooperators studied this topic. In this paper, we confirm some conjectures posed by Sun and investigate some related topics. For instance, given any integers $c,d$ with $d e0$ and $c^2-4d e0$, we show that there are infinitely many odd primes $p$ such that $$detbigg[left(frac{i^2+cij+dj^2}{p}right)bigg]_{0le i,jle p-1}=0,$$ where $(frac{cdot}{p})$ is the Legendre symbol. This confirms a conjecture of Sun.
Let $p$ be a prime, $k$ a positive integer and let $mathbb{F}_q$ be the finite field of $q=p^k$ elements. Let $f(x)$ be a polynomial over $mathbb F_q$ and $ainmathbb F_q$. We denote by $N_{s}(f,a)$ the number of zeros of $f(x_1)+cdots+f(x_s)=a$. In this paper, we show that $$sum_{s=1}^{infty}N_{s}(f,0)x^s=frac{x}{1-qx} -frac{x { M_f^{prime}}(x)}{qM_f(x)},$$ where $$M_f(x):=prod_{minmathbb F_q^{ast}atop{S_{f, m} e 0}}Big(x-frac{1}{S_{f,m}}Big)$$ with $S_{f, m}:=sum_{xin mathbb F_q}zeta_p^{{rm Tr}(mf(x))}$, $zeta_p$ being the $p$-th primitive unit root and ${rm Tr}$ being the trace map from $mathbb F_q$ to $mathbb F_p$. This extends Richmans theorem which treats the case of $f(x)$ being a monomial. Moreover, we show that the generating series $sum_{s=1}^{infty}N_{s}(f,a)x^s$ is a rational function in $x$ and also present its explicit expression in terms of the first $2d+1$ initial values $N_{1}(f,a), ..., N_{2d+1}(f,a)$, where $d$ is a positive integer no more than $q-1$. From this result, the theorems of Chowla-Cowles-Cowles and of Myerson can be derived.
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