AI 中文总结
该研究提出一种更快的确定性整数求根算法，通过优化 $p$-进框架及候选验证算法，将无平方因子整数多项式求根的时间复杂度从 $\tilde{O}(n^2b)$ 改进为 $\tilde{O}(n^{3/2}b)$，实现了亚二次级的效率提升。
AI 中文摘要
我们针对次数为 $n$、无穷范数满足 $\lVert f\rVert_\infty<2^b$ 的无平方因子多项式 $f\in\mathbb Z[x]$，给出了一个用于寻找其所有整数根的确定性算法，运行时间为 $\tilde{O}(n^{3/2}b)$，改进了 Harvey 和 Hittmeir（《数论研究》，2022 年）提出的 $\tilde{O}(n^2b)$ 界。该算法遵循经典的 $p$-进框架：在素数 $p$ 模下寻找根，将其提升到 $p$ 的高次幂模，再验证提升后的候选解。主要新思想是不再寻找使 $f\bmod p$ 无平方因子的素数，而是寻找使 $p$ 模下重根的总重数较小的素数，这需要提升重根，我们用加权提升树处理。我们还给出了一个更快的确定性候选验证算法：给定 $n$ 个绝对值小于 $2^b$ 的候选整数，我们可在 $\tilde{O}(nb+\min(n^2,nb^2))$ 位运算中确定哪些是 $f$ 的根。这些要素共同构成了无平方因子情形下整数求根问题的首个确定性亚二次（关于 $n$）改进成果。
英文摘要
We give a deterministic algorithm for finding all integer roots of a square-free polynomial $f\in\mathbb Z[x]$ of degree $n$ with $\lVert f\rVert_\infty<2^b$. The running time is $$ \tilde{O}(n^{3/2}b), $$ improving the $\tilde{O}(n^2b)$ bound of Harvey and Hittmeir (Research in Number Theory, 2022). The algorithm follows the classical $p$-adic framework: find roots modulo a prime $p$, lift them modulo a high power of $p$, and verify the lifted candidates. The main new idea is to avoid searching for a prime for which $f\bmod p$ is square-free. Instead, we find a prime for which the total multiplicity of repeated roots modulo $p$ is small. This requires lifting repeated roots, which we handle using a weighted lifting tree. We also give a faster deterministic candidate-verification algorithm: given $n$ candidate integers smaller in absolute value than $2^b$, we decide which are roots of $f$ in $$ \tilde{O}(nb+\min(n^2,nb^2)) $$ bit operations. Together, these ingredients give the first deterministic subquadratic-in-$n$ improvement for integer root finding in the square-free case.
