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Lower bounds for some decision problems over C

Gregorio Malajovich

TL;DR

The work addresses lower bounds for decision problems over the complex numbers in the Ostrowsky model, focusing on zeros of explicit polynomials and distinguishing uniform from non-uniform settings. It leverages valuation theory and Newton diagrams to translate root-structure properties into computational lower bounds via the canonical-path method and multiplicative complexity arguments. The authors present a uniform lower bound for a zero-testing problem $q^{d}(t)=0$ and two non-uniform lower bounds for $p^{d}(t)=0$, with explicit hard instances and a general lemma linking rapid root-valuation growth to hardness. These results imply intrinsic difficulty of explicit polynomial-zero decision problems over $\mathbb{C}$ in both uniform and non-uniform models, with connections to classes like $\mathcal{P}/\text{poly}$.

Abstract

Lower bounds for some explicit decision problems over the complex numbers are given.

Lower bounds for some decision problems over C

TL;DR

The work addresses lower bounds for decision problems over the complex numbers in the Ostrowsky model, focusing on zeros of explicit polynomials and distinguishing uniform from non-uniform settings. It leverages valuation theory and Newton diagrams to translate root-structure properties into computational lower bounds via the canonical-path method and multiplicative complexity arguments. The authors present a uniform lower bound for a zero-testing problem and two non-uniform lower bounds for , with explicit hard instances and a general lemma linking rapid root-valuation growth to hardness. These results imply intrinsic difficulty of explicit polynomial-zero decision problems over in both uniform and non-uniform models, with connections to classes like .

Abstract

Lower bounds for some explicit decision problems over the complex numbers are given.

Paper Structure

This paper contains 4 sections, 2 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Background and notations
  3. Uniform lower bounds
  4. Non-uniform lower bounds

Key Result

Proposition 1

Suppose that $\zeta_1, \cdots, \zeta_d$ are the roots of a univariate polynomial $g \in K[x]$. Let the roots of $g$ be ordered so that and let the increasing sequence $i_j$ assume the values $0$, $d$ and all the values of $i$ where: Then the sharp corners of the Newton diagram are precisely the points of the form $(i_j, \nu(g_{i_j}))$ for all $j$. Moreover, the slope of the segment $[(i_{j-1}, \

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof : Proof of Proposition \ref{['prop1']}
  • proof : Proof of Lower bound \ref{['low3']}
  • Lemma 1
  • proof : Proof of Lemma \ref{['lower1']}
  • proof : Proof of Lower Bound \ref{['low2']}