In this paper, we construct new families of imaginary and real quadratic fields that are $p$-rational. In the imaginary case, we prove that for any positive integer $k$ and any integer $m$, the imaginary quadratic field $\mathbb{Q}\left(\sqrt{-(kp+m)}\right)$ is $p$-rational for sufficiently large primes $p$. The proof...
We study the generic nullity of generalized commutator operators \[L_{\mathbf{A}}(X)=s_{k+1}(A_1,\cdots,A_k,X)\] on matrix algebras, where $s_{k+1}$ denotes the standard polynomial. Dixon and Pressman conjectured an explicit formula for the generic nullity of $L_{\mathbf{A}}$, and Brassil and Reichstein proved the conj...
Let $k=\mathbb{F}_q$, $E=\mathbb{F}_{q^n}$ and $\mathrm{Tr}=\mathrm{Tr}_{E/k}$. For $r\ge 2$, $a\in k^{\times}$ and $x\in E^{\times}$, let $\mathrm{N}(E,r,x,a)$ be the number of $r$-tuples $(x_1,\cdots,x_r)$ in $(E^{\times})^r$ satisfying $x_1\cdots x_r=x$ and $\mathrm{Tr}(x_1+\cdots+x_r)=a$. We prove $\left|\mathrm{N}...
We study a half-interval distribution problem for polynomial residues modulo an odd prime $p$: how often the fractional part of $\varphi(x)/p$ lies in the upper half of the unit interval as $x$ ranges over $1\leq x<p/2$. Using finite Fourier expansions together with the Weil bound, we prove an asymptotic formula $\#\le...
Xue-Jun Guo, Chen Lin, Zhe-Feng Xu· 0 citations
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