Finding Integer Roots of Polynomials in Nearly Optimal Time

Abstract

Let $f\in\mathbb Z[x]$ be a polynomial of degree $n$ with integer coefficients of absolute value less than $2^b$. Using Pan's approximate factorization theorem, we give a deterministic algorithm that finds all distinct integer roots of $f$ in $\widetilde O(nb)$ bit operations. This matches the bit size of the input up to logarithmic factors and does not require $f$ to be square-free. Since we were unable to verify parts of the running time analysis of Pan's theorem, we also give a self-contained deterministic algorithm with running time $\widetilde O(nb+\min(n,b)^3b^2)$, which is $\widetilde O(nb)$ whenever $b\leq n^{1/4}$. This answers a question of Harvey and Hittmeir (2022), who asked whether integer roots can be found deterministically in quasilinear time, conditionally on Pan's theorem, and unconditionally for $b\leq n^{1/4}$. The fastest previous deterministic algorithm, due to Nir, takes $\widetilde O(n^{3/2}b)$ bit operations and works only for square-free polynomials. Instead of $p$-adic lifting, we combine approximate complex root finding with exact polynomial evaluation. We first approximate the complex roots of the polynomial formed by the top $\min(n,b)+1$ coefficients of $f$. Every integer root $r$ of $f$ with $|r|\geq 2$ lies within distance $2$ of some complex root of this truncated polynomial. Rounding the real parts of the approximate roots therefore gives a list of $O(\min(n,b))$ candidate integers that is guaranteed to contain all integer roots of $f$. Although a single candidate can have up to $b$ bits, a bound via the Mahler measure shows that the bit lengths of all candidates sum to $O(b)$. This allows us to test all candidates by exact multipoint evaluation in $\widetilde O(nb)$ bit operations.