We design a deterministic algorithm that, given blackbox access to the product $f=\prod_{i=1}^{\ell}{h_i}$ of $\ell$ irreducible $s$-sparse $n$-variate polynomials of bounded individual degree $d$, over fields of characteristic zero, and more generally over fields of sufficiently large positive characteristic, recovers the $h_i$'s and their multiplicities in time $\mathrm{poly}(n,(s\ell d)^d)$. For any constant $d>2$, this is the first \emph{polynomial-time} algorithm for this problem, resolving for the bounded-individual-degree regime an open question of Dutta, Sinhababu, and Thierauf (Random 2026). The previous best deterministic algorithm, due to the authors, runs in $\mathrm{poly}(n,d^d,s^{d\log \ell},\ell^d)$ time, which is quasi-polynomial in $\ell$.

The improvement is enabled by a new sparse rational interpolation theorem in the bounded-individual-degree setting, based on reconstructing the denominator from its logarithmic derivatives. Given blackbox access to rational functions $a_1/b,\ldots,a_N/b$ where the $a_j$'s and $b$ are $s$-sparse of individual degree at most $d$ with $\gcd(a_1,\ldots,a_N,b)=1$, we recover the $a_j$'s and $b$ in time $\mathrm{poly}(n,d!,s^d,N)$. In contrast to the rational interpolation theorem of Chuyoon-Shpilka (2026), our algorithm reconstructs the denominator directly and does not require a precomputed list of its irreducible factors.