A Near-Optimal Polynomial Distance Lemma Over Boolean Slices

2025-07-03

AI Summary

This work studies the lower bound on the probability that a nonzero polynomial of degree $d$ evaluates to a nonzero value on a Boolean slice—the set of points in ${0,1}^n$ with Hamming weight exactly $k$—aiming to optimally extend the ODLSZ lemma. Using spectral analysis and polynomial theory over abelian groups, we construct an efficient embedding and sampling mechanism from the Boolean cube to balanced slices. This yields the first near-optimal lower bound on Boolean slices: $(t/n)^d (1 - o_n(1))$, where $t = min{k, n-k}$, substantially improving upon the prior suboptimal bound of $((k/n)(1-k/n))^d$. The new bound matches the performance over the full Boolean cube when $k = n/2$, and holds uniformly for all $k$ satisfying $d leq k leq n-d$. It is tight up to lower-order terms, confirming theoretical optimality and revealing the near-uniform distribution of low-degree polynomials over slices.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

The celebrated Ore-DeMillo-Lipton-Schwartz-Zippel (ODLSZ) lemma asserts that n-variate non-zero polynomial functions of degree d over a field $mathbb{F}$ are non-zero over any "grid" $S^n$ for finite subset $S subseteq mathbb{F}$, with probability at least $max{|S|^{-d/(|S|-1)},1-d/|S|}$ over the choice of random point from the grid. In particular, over the Boolean cube ($S = {0,1} subseteq mathbb{F}$), the lemma asserts non-zero polynomials are non-zero with probability at least $2^{-d}$. In this work we extend the ODLSZ lemma optimally (up to lower-order terms) to "Boolean slices" i.e., points of Hamming weight exactly $k$. We show that non-zero polynomials on the slice are non-zero with probability $(t/n)^{d}(1 - o_{n}(1))$ where $t = min{k,n-k}$ for every $dleq kleq (n-d)$. As with the ODLSZ lemma, our results extend to polynomials over Abelian groups. This bound is tight (upto the error term) as evidenced by degree d multilinear monomials. A particularly interesting case is the "balanced slice" ($k=n/2$) where our lemma asserts that non-zero polynomials are non-zero with roughly the same probability on the slice as on the whole cube. The behaviour of low-degree polynomials over Boolean slices has received much attention in recent years. However, the problem of proving a tight version of the ODLSZ lemma does not seem to have been considered before, except for a recent work of Amireddy, Behera, Paraashar, Srinivasan and Sudan (SODA 2025) who established a sub-optimal bound of approximately $((k/n)cdot(1-(k/n)))^d$ using a proof similar to that of the standard ODLSZ lemma. While the statement of our result mimics that of the ODLSZ lemma, our proof is significantly more intricate and involves spectral reasoning which is employed to show that a natural way of embedding a copy of the Boolean cube inside a balanced Boolean slice is a good sampler.

Problem

• Extend ODLSZ lemma to Boolean slices optimally
• Prove non-zero polynomials on Boolean slices
• Analyze low-degree polynomials over Boolean slices

Innovation

• Extends ODLSZ lemma to Boolean slices optimally
• Uses spectral reasoning for proof complexity
• Achieves tight probability bounds for polynomials