Derandomizing Multivariate Polynomial Factoring for Low Degree Factors

Abstract: 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 \begin{enumerate} \item polynomial time, for $\mathcal{C}$ being polynomials that have only constant degree factors, \item quasipolynomial time, for $\mathcal{C}$ being sparse polynomials, \item subexponential time, for $\mathcal{C}$ being polynomials that have constant-depth circuits. \end{enumerate} 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.