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Towards Deterministic Algorithms for Constant-Depth Factors of Constant-Depth Circuits

Mrinal Kumar, Varun Ramanathan, Ramprasad Saptharishi, Ben Lee Volk

TL;DR

A deterministic subexponential time algorithm that takes as input a multivariate polynomial f and a factor g, and outputs a list L of circuits that contains all irreducible factors of f computable by constant-depth circuits that contains all irreducible factors of f computable by constant-depth circuits.

Abstract

We design a deterministic subexponential time algorithm that takes as input a multivariate polynomial $f$ computed by a constant-depth circuit over rational numbers, and outputs a list $L$ of circuits (of unbounded depth and possibly with division gates) that contains all irreducible factors of $f$ computable by constant-depth circuits. This list $L$ might also include circuits that are spurious: they either do not correspond to factors of $f$ or are not even well-defined, e.g. the input to a division gate is a sub-circuit that computes the identically zero polynomial. The key technical ingredient of our algorithm is a notion of the pseudo-resultant of $f$ and a factor $g$, which serves as a proxy for the resultant of $g$ and $f/g$, with the advantage that the circuit complexity of the pseudo-resultant is comparable to that of the circuit complexity of $f$ and $g$. This notion, which might be of independent interest, together with the recent results of Limaye, Srinivasan and Tavenas, helps us derandomize one key step of multivariate polynomial factorization algorithms - that of deterministically finding a good starting point for Newton Iteration for the case when the input polynomial as well as the irreducible factor of interest have small constant-depth circuits.

Towards Deterministic Algorithms for Constant-Depth Factors of Constant-Depth Circuits

TL;DR

A deterministic subexponential time algorithm that takes as input a multivariate polynomial f and a factor g, and outputs a list L of circuits that contains all irreducible factors of f computable by constant-depth circuits that contains all irreducible factors of f computable by constant-depth circuits.

Abstract

We design a deterministic subexponential time algorithm that takes as input a multivariate polynomial computed by a constant-depth circuit over rational numbers, and outputs a list of circuits (of unbounded depth and possibly with division gates) that contains all irreducible factors of computable by constant-depth circuits. This list might also include circuits that are spurious: they either do not correspond to factors of or are not even well-defined, e.g. the input to a division gate is a sub-circuit that computes the identically zero polynomial. The key technical ingredient of our algorithm is a notion of the pseudo-resultant of and a factor , which serves as a proxy for the resultant of and , with the advantage that the circuit complexity of the pseudo-resultant is comparable to that of the circuit complexity of and . This notion, which might be of independent interest, together with the recent results of Limaye, Srinivasan and Tavenas, helps us derandomize one key step of multivariate polynomial factorization algorithms - that of deterministically finding a good starting point for Newton Iteration for the case when the input polynomial as well as the irreducible factor of interest have small constant-depth circuits.
Paper Structure (26 sections, 21 theorems, 37 equations, 2 algorithms)

This paper contains 26 sections, 21 theorems, 37 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Open Problems
  4. Proof Overview
  5. Notations and preliminaries
  6. Newton iteration
  7. Pseudo-quotients for preserving indivisibility
  8. Pseudo-resultant
  9. A starting point for Newton iteration
  10. Bounding the complexity of the denominator $P_{\mathrm{den}}(\mathbf{x})$ :
  11. Bounding the complexity of the numerator $P_{\mathrm{num}}(\mathbf{x},y)$ :
  12. The hitting set $\mathcal{H}$ suffices :
  13. Generating a list of candidate factors
  14. Proof of correctness of \ref{['algo:multiplicity-one-irreducible-factors']}
  15. Proof of \ref{['thm:intro:main']}
  16. ...and 11 more sections

Key Result

theorem 1.1

Let $\mathbb{Q}$ be the field of rational numbers and $\varepsilon > 0$, $\Delta_1, \Delta_2 \in \mathbb{N}$ be arbitrary constants. Then, there is a deterministic algorithm that takes as input an algebraic circuit $C \in (\Sigma\Pi)^{(\Delta_1)}$ of size $s$, bit-complexity $t$ and degree $D$, alon The size of the list $L$ as well as the running time of the algorithm are bounded above by $O(smDt)

Theorems & Definitions (37)

  • theorem 1.1: Subexponential list containing constant-depth factors of constant-depth circuits
  • lemma 3.0: Newton iteration
  • lemma 3.0: Computing minimal polynomials of approximate roots
  • definition 3.1: Pseudo-quotients, Forbes15
  • theorem 3.2: Divisibility testing to PIT Forbes15
  • definition 4.1: Pseudo-resultant
  • proof
  • lemma 4.3: Properties of pseudo-resultant
  • proof
  • lemma 4.4
  • ...and 27 more