Derandomizing Multivariate Polynomial Factoring for Low Degree Factors

For a polynomial $f$ from a class $\mathcal{C}$ of polynomials, we show that the problem to compute all the constant degree irreducible factors of $f$ reduces in polynomial time to polynomial identity tests (PIT) for class $\mathcal{C}$ and divisibility tests of $f$ by constant degree polynomials. We apply the result to several classes $\mathcal{C}$ and obtain the constant degree factors in 1. polynomial time, for $\mathcal{C}$ being polynomials that have only constant degree factors, 2. quasipolynomial time, for $\mathcal{C}$ being sparse polynomials, 3. subexponential time, for $\mathcal{C}$ being polynomials that have constant-depth circuits. Result 2 and 3 were already shown by Kumar, Ramanathan, and Saptharishi with a different proof and their time complexities necessarily depend on black-box PITs for a related bigger class $\mathcal{C}’$. Our complexities vary on whether the input is given as a blackbox or whitebox. We also show that the problem to compute the sparse factors of polynomial from a class $\mathcal{C}$ reduces in polynomial time to PIT for class $\mathcal{C}$, divisibility tests of $f$ by sparse polynomials, and irreducibility preserving bivariate projections for sparse polynomials. For $\mathcal{C}$ being sparse polynomials, it follows that it suffices to derandomize irreducibility preserving bivariate projections for sparse polynomials in order to compute all the sparse irreducible factors efficiently. When we consider factors of sparse polynomials that are sums of univariate polynomials, a subclass of sparse polynomials, we obtain a polynomial time algorithm. This was already shown by Volkovich with a different proof.

💡 Research Summary

This paper addresses the problem of factoring multivariate polynomials when one is interested only in the irreducible factors of constant (i.e., bounded) degree. The authors develop a general reduction that shows, for any class 𝒞 of polynomials, the task of enumerating all constant‑degree irreducible factors of a given f ∈ 𝒞 can be performed in polynomial time using only two types of subroutines: (i) polynomial‑identity testing (PIT) for the top‑degree homogeneous component of polynomials in 𝒞, and (ii) divisibility tests of polynomials from 𝒞 by constant‑degree polynomials. When 𝒞 is closed under taking homogeneous components and partial derivatives, these two requirements collapse to (i) PIT for 𝒞 itself and (ii) divisibility tests within 𝒞 by constant‑degree polynomials.

The authors place this reduction within a standard five‑step factoring framework (monic transformation, bivariate projection, 2‑variate factorization, lifting, verification) but replace the traditionally random Hensel lifting step with a deterministic interpolation technique. The crucial random step—projecting to a bivariate polynomial while preserving irreducibility—is also derandomized for constant‑degree factors by exploiting Hilbert’s Irreducibility Theorem together with the availability of PIT and divisibility tests.

Applying the general theorem yields concrete algorithms for three important families:

1. Polynomials whose all factors have constant degree. Since PIT for such classes is known to be deterministic polynomial‑time, the entire factorization runs in polynomial time without any divisibility checks.

2. Sparse polynomials. Using the best known quasi‑polynomial‑time black‑box PIT for sparse circuits, the reduction gives a quasi‑polynomial‑time algorithm for constant‑degree factors of sparse polynomials.

3. Polynomials computed by constant‑depth circuits (AC⁰). Recent sub‑exponential‑time PIT results for this class lead to a sub‑exponential‑time algorithm for constant‑degree factors.

A notable aspect of the work is the careful distinction between white‑box and black‑box settings. For example, commutative read‑once oblivious branching programs (RO‑ABPs) admit a polynomial‑time white‑box PIT but only a quasi‑polynomial‑time black‑box PIT; consequently, linear factors of RO‑ABP‑computed polynomials can be extracted in polynomial time in the white‑box model but require quasi‑polynomial time in the black‑box model.

The paper then turns to the more challenging problem of extracting sparse irreducible factors of a polynomial f ∈ 𝒞. Here the reduction needs a third ingredient: an irreducibility‑preserving bivariate projection for sparse polynomials. The authors show that, assuming access to (i) PIT for 𝒞, (ii) divisibility tests of 𝒞 by constant‑degree polynomials, and (iii) such a projection, one can compute all sparse factors (with multiplicities) in polynomial time. While (iii) is currently only known in sub‑exponential time for general sparse polynomials, the authors identify a natural subclass—polynomials that are sums of univariate polynomials—where the projection can be performed deterministically in polynomial time. Consequently, all sparse factors of this subclass can be found efficiently, reproducing earlier results of Volkovich with a different proof technique.

The technical contributions can be summarized as follows:

• Derandomization of Hilbert’s Irreducibility step for constant‑degree factors, reducing it to homogeneous components and partial derivatives.
• Elimination of Hensel lifting by deterministic interpolation, simplifying the lifting phase and reducing dependence on circuit depth.
• Unified framework that expresses the complexity of constant‑degree factorization solely in terms of PIT and simple divisibility tests, applicable uniformly across dense, sparse, and constant‑depth circuit representations.
• Structural insight into sparse factorization, isolating the bivariate projection as the remaining obstacle and showing that its derandomization would immediately yield fully deterministic sparse factorization algorithms.
• Model‑sensitive analysis, highlighting how the availability of white‑box versus black‑box PIT directly influences the overall runtime for specific circuit classes.

In conclusion, the paper provides a clean, modular reduction from multivariate constant‑degree factorization to well‑studied derandomization problems. By doing so, it not only reproduces several known results with simpler arguments but also clarifies the exact bottlenecks for extending deterministic factorization to broader classes, especially sparse polynomials. The work paves the way for future research that targets the remaining bivariate projection problem, promising fully deterministic algorithms for sparse polynomial factorization.