We study sparse polynomials with bounded individual degree and their factors, obtaining the following structural and algorithmic results. 1. A deterministic polynomial-time algorithm to find all sparse divisors of a sparse polynomial of bounded individual degree, together with the first upper bound on the number of non-monomial irreducible factors of such polynomials. 2. A poly(n,s^dlogℓ)-time algorithm that recovers ℓ irreducible s-sparse polynomials of individual degree at most d from blackbox access to their (not necessarily sparse) product. This partially resolves a question of Dutta-Sinhababu-Thierauf (RANDOM 2024). In particular, if ℓ=O(1) the algorithm runs in polynomial time. 3. Deterministic algorithms for factoring a product of s-sparse polynomials of individual degree d from blackbox access. Over fields of characteristic zero or sufficiently large characteristic the runtime is poly(n,s^d^3log n); over arbitrary fields it is poly(n,(d^2)!,s^d^5log n). This improves Bhargava-Saraf-Volkovich (JACM 2020), which gives poly(n,s^d^7log n) time for a single sparse polynomial. For a single sparse input we obtain poly(n,s^d^2log n) time. 4. Given blackbox access to a product of factors of sparse polynomials of bounded individual degree, we give a deterministic polynomial-time algorithm to find all irreducible sparse multiquadratic factors with multiplicities. This generalizes the algorithms of Volkovich (RANDOM 2015, 2017) and extends the complete-power test of Bisht-Volkovich (CC 2025). To handle arbitrary fields we introduce a notion of primitive divisors that removes characteristic assumptions from most of our algorithms.
We develop catalytic algorithms for fundamental problems in algorithm design that run in polynomial time, use only 𝒪(log(n)) workspace, and use sublinear catalytic space matching the best-known space bounds of non-catalytic algorithms running in polynomial time. First, we design a polynomial time algorithm for directed s-t connectivity using n / 2^Θ(√(log n)) catalytic space, which matches the state-of-the-art time-space bounds in the non-catalytic setting [Barnes et al., 1998], and improves the catalytic space usage of the best known algorithm [Cook and Pyne, 2026]. Furthermore, using only 𝒪(log(n)) random bits we get a randomized algorithm whose running time nearly matches the fastest time bounds known for space-unrestricted algorithms. Second, we design polynomial time algorithms for the problems of computing Edit Distance, Longest Common Subsequence, and the Discrete Fréchet Distance, again using n / 2^Θ(√(log n)) catalytic space. This again matches non-catalytic time-space frontier for Edit Distance and Least Common Subsequence [Kiyomi et al., 2021].
