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Derandomizing Multivariate Polynomial Factoring for Low Degree Factors

Pranjal Dutta, Amit Sinhababu, Thomas Thierauf

TL;DR

It is shown that the problem to compute all the constant degree irreducible factors of polynomial reduces in polynomial time to polynomial identity tests (PIT) for class $\mathcal{C}$ and divisibility tests of $f$ by constant degree polynomials.

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 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.

Derandomizing Multivariate Polynomial Factoring for Low Degree Factors

TL;DR

It is shown that the problem to compute all the constant degree irreducible factors of polynomial reduces in polynomial time to polynomial identity tests (PIT) for class and divisibility tests of by constant degree polynomials.

Abstract

For a polynomial from a class of polynomials, we show that the problem to compute all the constant degree irreducible factors of reduces in polynomial time to polynomial identity tests (PIT) for class and divisibility tests of by constant degree polynomials. We apply the result to several classes and obtain the constant degree factors in 1. polynomial time, for being polynomials that have only constant degree factors, 2. quasipolynomial time, for being sparse polynomials, 3. subexponential time, for 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 . 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 reduces in polynomial time to PIT for class , divisibility tests of by sparse polynomials, and irreducibility preserving bivariate projections for sparse polynomials. For 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.

Paper Structure

This paper contains 24 sections, 32 theorems, 52 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. General framework of factoring.
  3. PIT.
  4. Divisibility testing.
  5. Irreducibility projection.
  6. Special cases of multivariate factoring.
  7. Our results
  8. Constant-degree factors.
  9. Sparse factors.
  10. Preliminaries
  11. Computational problems and complexity measures
  12. Transformation to a monic polynomial
  13. Basics of factoring and interpolation and PIT
  14. Divisibility testing reduces to PIT
  15. Effective Hilbert's Irreducibility Theorem
  16. ...and 9 more sections

Key Result

Lemma 2.1

Let $p \in {\mathcal{P}}$ be a nonzero polynomial. A point ${\boldsymbol a} \in (\mathbb{F}\setminus\{0\})^n$ such that $p({\boldsymbol a}) \not= 0$ can be computed in time $nd\, \textup{T}_{{\rm PIT}({\mathcal{P}})}$.

Theorems & Definitions (48)

  • Lemma 2.1
  • proof
  • Lemma 2.2: Transformation to monic
  • proof
  • Lemma 2.3
  • Lemma 2.4: Trivariate Factorization
  • Lemma 2.5: Factor multiplicity
  • Theorem 2.6: Sparse PIT and interpolation klivans2001randomness
  • Theorem 2.7: PIT for constant depth circuits limaye2022superpolynomial
  • Lemma 2.8: Divisibility reduces to PIT forbes2015deterministic
  • ...and 38 more