论文标题
在第n个环形多项式的Vandermonde矩阵的条件数
On the condition number of the Vandermonde matrix of the nth cyclotomic polynomial
论文作者
论文摘要
最近，Blanco-Chacón证明了环学习与多项式学习之间的等效性，对于某些环形数字领域的某些家庭，通过为条件编号$ \ operatotorname {cond}（cond}（v_n）$ vandermonde matrix $ v_n $与$ n $ n $ n $ th cyclotom poly thcyclotom py cyclotom s cyclotomic cyclotom s $ vandermonde $ v_n $提供一些上限。我们证明了$ v_n $的单数值的一些结果，尤其是，我们确定$ n = 2^k p^\ ell $的$ \ operatoTorname {cond}（cond}（v_n）$，其中$ k，\ ell \ ell \ geq 0 $是integers，$ p $是一个奇数的质量数字。
Recently, Blanco-Chacón proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number $\operatorname{Cond}(V_n)$ of the Vandermonde matrix $V_n$ associated to the $n$th cyclotomic polynomial. We prove some results on the singular values of $V_n$ and, in particular, we determine $\operatorname{Cond}(V_n)$ for $n = 2^k p^\ell$, where $k, \ell \geq 0$ are integers and $p$ is an odd prime number.
