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Rational degree is polynomially related to degree

Matt Kovacs-Deak, Daochen Wang, Rain Zimin Yang

TL;DR

This work resolves the long-standing question of whether the degree of a Boolean function is polynomially related to its rational degree by proving $deg(f) \leq 2 \mathrm{rdeg}(f)^4$ for all $f$, with a stronger bound in terms of sign and nondeterministic degrees. The authors develop a framework connecting rational degree to nondeterministic degree via minimum block sensitivity and use a hitting-set argument within a decision-tree construction to derive upper bounds on decision-tree depth, leading to the quartic bound. They also present an effective Hypercube Nullstellensatz, conjecture refined bounds on influence and approximate nondeterministic degree, and discuss the broader implications for polynomial relations among Boolean complexity measures. The results illuminate the structural interplay between algebraic representations and combinatorial properties, with consequences for quantum postselection, Nullstellensatz-style certificates, and classical complexity separations. Overall, the paper closes a major open problem and lays groundwork for further tightening of the relationships among degree, rational degree, and related complexity notions.

Abstract

We prove that $\mathrm{deg}(f) \leq 2 \, \mathrm{rdeg}(f)^4$ for every Boolean function $f$, where $\mathrm{deg}(f)$ is the degree of $f$ and $\mathrm{rdeg}(f)$ is the rational degree of $f$. This resolves the second of the three open problems stated by Nisan and Szegedy, and attributed to Fortnow, in 1994.

Rational degree is polynomially related to degree

TL;DR

This work resolves the long-standing question of whether the degree of a Boolean function is polynomially related to its rational degree by proving for all , with a stronger bound in terms of sign and nondeterministic degrees. The authors develop a framework connecting rational degree to nondeterministic degree via minimum block sensitivity and use a hitting-set argument within a decision-tree construction to derive upper bounds on decision-tree depth, leading to the quartic bound. They also present an effective Hypercube Nullstellensatz, conjecture refined bounds on influence and approximate nondeterministic degree, and discuss the broader implications for polynomial relations among Boolean complexity measures. The results illuminate the structural interplay between algebraic representations and combinatorial properties, with consequences for quantum postselection, Nullstellensatz-style certificates, and classical complexity separations. Overall, the paper closes a major open problem and lays groundwork for further tightening of the relationships among degree, rational degree, and related complexity notions.

Abstract

We prove that for every Boolean function , where is the degree of and is the rational degree of . This resolves the second of the three open problems stated by Nisan and Szegedy, and attributed to Fortnow, in 1994.
Paper Structure (11 sections, 14 theorems, 43 equations)

This paper contains 11 sections, 14 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Boolean functions
  4. Technical tools
  5. Rational degree lower bound
  6. Implications and open problems
  7. Effective Hypercube Nullstellensatz
  8. Gotsman-Linial conjecture
  9. Approximate nondeterministic degree
  10. Improving polynomial relations
  11. Rational degree and quantum postselection

Key Result

Theorem 1

Let $P \in \mathbb{R}[X]$. Let $a_1,a_2,b_1,b_2 \in \mathbb{R}$ be such that $a_1 < a_2$ and $b_1 < b_2$. Suppose $P(x) \in [b_1,b_2]$ for all $x\in [a_1,a_2]$. Then, for all $x\in [a_1,a_2]$,

Theorems & Definitions (41)

  • Definition 1: Degree
  • Definition 2: Rational degree
  • Definition 3: Sign degree
  • Definition 4: Nondeterministic degree
  • proof
  • proof : Proof sketch
  • Definition 5: Block sensitivity
  • Definition 6: Hitting set
  • Definition 7: Decision tree complexity
  • Theorem 1: Markov
  • ...and 31 more