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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).\n  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. 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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).\n  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. 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