On the number of solutions of certain diagonal equations over finite fields

We use character sums over finite fields to give formulas for the number of solutions of certain diagonal equations of the form$a_1{x}_1^{m_1}+a_2{x}_2^{m_2}+\cdots +a_n{x}_n^{m_n}=c.$ We also show that if the value distribution of character sums ${\sum }_{x\in {\mathbb{F}}_q}\chi (a{x}^m+bx)$, $a,b\in {\mathbb{F}}_q$, is known, then one can obtain the number of solutions of the system of equations$\{\begin{array}{cc}\hfill x_1+x_2+\cdots +x_n\hfill & \hfill =\alpha \hfill \\ \hfill x_1^m+x_2^m+\cdots +x_n^m\hfill & \hfill =\beta ,\hfill \end{array}$ for some particular m. We finally apply our results to induce some facts about Waring's problems and the covering radius of certain cyclic codes.     

On L-functions of certain exponential sums

Let ${\mathbb{F}}_q$ denote the finite field of order q of characteristic p. We study the p-adic valuations for zeros of L-functions associated with exponential sums of the following family of Laurent polynomials$f(x)=a_1{x}_{n+1}(x_1+\frac{1}{x_1})+\cdots +a_n{x}_{n+1}(x_n+\frac{1}{x_n})+a_{n+1}{x}_{n+1}+\frac{1}{x_{n+1}}$ where $a_i\in {\mathbb{F}}_q^{⁎}$, $i=1,2,\dots ,n+1$. When $n=2$, the estimate of the associated exponential sum appears in Iwaniecʼs work on small eigenvalues of the Laplace–Beltrami operator acting on automorphic functions with respect to the group ${\Gamma }_0(p)$, and Adolphson and Sperber gave complex absolute values for zeros of the corresponding L-function. Using the decomposition theory of Wan, we determine the generic Newton polygon (q-adic values of the reciprocal zeros) of the L-function. Working on the chain level version of Dworkʼs trace formula and using Wanʼs decomposition theory, we are able to give an explicit Hasse polynomial for the generic Newton polygon in low dimensions, i.e., $n⩽3$.     

Explicit min–max polynomials on the disc

Denote by ${\Pi }_{n+m?1}^2?\{{\sum }_{0\le i+j\le n+m?1}{c}_{i,j}{x}^i{y}^j:c_{i,j}\in \mathbb{R}\}$ the space of polynomials of two variables with real coefficients of total degree less than or equal to $n+m?1$. Let $b_0,b_1,\dots ,b_l\in \mathbb{R}$ be given. For $n,m\in \mathbb{N},n\ge l+1$ we look for the polynomial $b_0{x}^n{y}^m+b_1{x}^{n?1}{y}^{m+1}+?+b_l{x}^{n?l}{y}^{m+l}+q(x,y),q(x,y)\in {\Pi }_{n+m?1}^2$, which has least maximum norm on the disc and call such a polynomial a min–max polynomial. First we introduce the polynomial $2P_{n,m}(x,y)=x{G}_{n?1,m}(x,y)+y{G}_{n,m?1}(x,y)=2x^n{y}^m+q(x,y)$ and $q(x,y)\in {\Pi }_{n+m?1}^2$, where $G_{n,m}(x,y)?1/2^{n+m}(U_n\left(x\right)U_m\left(y\right)+U_{n?2}\left(x\right)U_{m?2}\left(y\right))$, and show that it is a min–max polynomial on the disc. Then we give a sufficient condition on the coefficients $b_j,j=0,\dots ,l,l$ fixed, such that for every $n,m\in \mathbb{N},n\ge l+1$, the linear combination ${\sum }_{\nu =0}^l{b}_{\nu }{P}_{n?\nu ,m+\nu }(x,y)$ is a min–max polynomial. In fact the more general case, when the coefficients $b_j$ and $l$ are allowed to depend on $n$ and $m$, is considered. So far, up to very special cases, min–max polynomials are known only for $x^n{y}^m$,$n,m\in {\mathbb{N}}_0$.     

The zero-one law for a complete orthonormal system

A complete orthonormal system of functions $\Theta ={\left\{{\theta }_n\right\}}_{n=1}^{\infty }\text{,}{\theta }_n\in L_{[0\text{,}1]}^{\infty }$ is constructed such that ${\sum }_{n=1}^{\infty }{a}_n{\theta }_n$ converges almost everywhere on $[0\text{,}1]$ if ${\left\{a_n\right\}}_{n=1}^{\infty }\in l^2$ and ${\sum }_{n=1}^{\infty }{a}_n{\theta }_n$ diverges a.e. for any ${\left\{a_n\right\}}_{n=1}^{\infty }?l^2$. We also show that for any complete ONS ${\left\{f_n\right\}}_{n=1}^{\infty }$ of functions defined on $[0\text{,}1]$ there exists a fixed non decreasing subsequence ${\left\{n_k\right\}}_{k=1}^{\infty }$ of natural numbers such that for any $f\in L_{[0\text{,}1]}^0$ and some sequence of coefficients ${\left\{b_n\right\}}_{n=1}^{\infty }$,

$\sum _{n=1}^{n_k}{b}_n{f}_n\to f\text{a.e. when}k\to \infty \text{.}$

To cite this article: K. Kazarian, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

On two Diophantine inequalities over primes

Let $1<c<37∕18,c\ne 2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in (N,2N]$, the Diophantine inequality $|p_1^c+p_2^c+p_3^c?R|<{log}^{?1}N$ is solvable in primes $p_1,p_2,p_3$. Moreover, we also investigate the problem of six primes and prove that the Diophantine inequality $|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c?N|<{log}^{?1}N$ is solvable in primes $p_1,p_2,p_3,p_4,p_5,p_6$ for sufficiently large real number $N$.     

Non-null-controllability of the Grushin operator in 2D

We are interested in the exact null controllability of the equation ${?}_{t}f?{?}_x^{2}f?x^2{?}_y^{2}f={\mathbb{1}}_{\omega }u$, with control u supported on ω. We show that, when ω does not intersect a horizontal band, the considered equation is never null-controllable. The main idea is to interpret the associated observability inequality as an $L^2$ estimate on polynomials, which Runge's theorem disproves. To that end, we study in particular the first eigenvalue of the operator $?{?}_x^2+{(nx)}^2$ with Dirichlet conditions on $(?1,1)$, and we show a quite precise estimation it satisfies, even when n is in some complex domain.     

Nonexistence and symmetry of solutions to some fractional laplacian equations in the upper half space

In this article, we consider the fractional Laplacian equation(-△)~(α/2)u = K(x)f(u), x ∈ R_+~n,u ≡ 0, x/∈R_+~n,where 0 α 2, R_+~n:= {x =(x_1, x_2, ···, x_n)|x n 0}. When K is strictly decreasing with respect to |x′|, the symmetry of positive solutions is proved, where x′=(x_1, x_2, ···, x_(n-1)) ∈R~(n-1). When K is strictly increasing with respect to x n or only depend on x n, the nonexistence of positive solutions is obtained.     

Sharp well-posedness of the cauchy problem for the higher-order dispersive equation

This current paper is devoted to the Cauchy problem for higher order dispersive equation u_t+ ?_x~(2n+1)u = ?_x(u?_x~nu) + ?_x~(n-1)(u_x~2), n ≥ 2, n ∈ N~+.By using Besov-type spaces, we prove that the associated problem is locally well-posed in H~(-n/2+3/4,-1/(2n))(R). The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H~(s,a)(R) with s -n/2+3/4 and all a∈R.