Discovering the roots: Uniform closure results for algebraic classes under factoring

Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the number of roots $r$ is small but the multiplicities are exponentially large. Our method sets up a linear system in $r$ unknowns and iteratively builds the roots as formal power series. For an algebraic circuit $f(x_1,\ldots,x_n)$ of size $s$ we prove that each factor has size at most a polynomial in: $s$ and the degree of the squarefree part of $f$. Consequently, if $f_1$ is a $2^{\Omega(n)}$-hard polynomial then any nonzero multiple $\prod_{i} f_i^{e_i}$ is equally hard for arbitrary positive $e_i$’s, assuming that $\sum_i \text{deg}(f_i)$ is at most $2^{O(n)}$. It is an old open question whether the class of poly($n$)-sized formulas (resp. algebraic branching programs) is closed under factoring. We show that given a polynomial $f$ of degree $n^{O(1)}$ and formula (resp. ABP) size $n^{O(\log n)}$ we can find a similar size formula (resp. ABP) factor in randomized poly($n^{\log n}$)-time. Consequently, if determinant requires $n^{\Omega(\log n)}$ size formula, then the same can be said about any of its nonzero multiples. As part of our proofs, we identify a new property of multivariate polynomial factorization. We show that under a random linear transformation $\tau$, $f(\tau\overline{x})$ completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. This with allRootsNI are the techniques that help us make progress towards the old open problems, supplementing the large body of classical results and concepts in algebraic circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 & Burgisser, FOCS 2001).

💡 Research Summary

The paper studies the problem of factoring multivariate polynomials within algebraic circuit complexity classes. It introduces a matrix‑valued recurrence, called allRootsNI, which generalizes the classical Newton iteration from a single simple root to all roots of a polynomial simultaneously. By treating the roots as formal power‑series and solving a linear system in the number of distinct roots r, the authors obtain a method that converges quadratically and works even when multiplicities are exponentially large.

Using this tool, the authors prove several new uniform closure results. The first main theorem shows that if a polynomial f can be computed by a circuit of size s and its square‑free part rad(f) has degree d, then every factor of f has a circuit of size polynomial in s + d. This improves on Kaltofen’s classic VP‑factoring result, which does not extend to the exponential‑degree regime because it requires solving systems with a number of unknowns equal to the individual degrees of variables.

A second theorem deals with “modular division”: any degree‑d factor of a circuit‑computed polynomial can be expressed as A/B modulo x_i^{d+1}, where both A and B have circuits of size polynomial in s·d. Although B may be non‑invertible, this representation still yields a polynomial‑size description of the factor.

The paper then focuses on the uniform closure of three important subclasses: VF (formulas), VBP (algebraic branching programs), and VNP (the nondeterministic analogue of VP). The authors prove that VF(n log n), VBP(n log n) and VNP(n log n) are all closed under factoring. Moreover, they give a randomized algorithm that, given a formula or ABP of size n^{O(log n)} and polynomial degree, outputs a non‑trivial factor as a formula or ABP of comparable size in poly(n log n) time. The algorithm works by applying a random linear transformation τ to the input polynomial, which puts the polynomial in a “general position” where it completely factors via power‑series roots. The allRootsNI recurrence then computes the power‑series expansions of all roots, and a linear system recovers the coefficients of the desired factor. Because the system dimension is r (the number of distinct roots) and r is at most polynomial in n for the considered classes, the overall runtime stays polynomial.

An important corollary is a hardness‑preserving property: if a polynomial f₁ is 2^{Ω(n)}‑hard and the total degree of the factors in a product ∏ f_i^{e_i} is at most 2^{O(n)}, then any non‑zero multiple of f₁ remains 2^{Ω(n)}‑hard. This follows directly from the size bound on factors in terms of the square‑free part’s degree.

The paper also discusses field requirements (characteristic zero, algebraically closed fields such as ℂ or ℚ) and notes that the results extend to other fields provided the characteristic exceeds the degree of the input polynomial.

In summary, the contributions are: (1) a novel all‑roots Newton iteration that yields formal‑power‑series representations of all roots; (2) polynomial‑size circuit bounds for factors in terms of the original circuit size and the degree of the square‑free part; (3) uniform closure of VF, VBP, and VNP under factoring, together with an explicit randomized poly‑time algorithm; and (4) a new hardness‑preserving transformation for high‑degree circuits. The work bridges numerical root‑finding techniques with algebraic complexity theory, opening avenues for further research on eliminating non‑unit divisions, handling high‑degree square‑free parts, and connecting these results to polynomial identity testing.