Ideals, Gröbner Bases, and PCPs
Prashanth Amireddy, Amik Raj Behera, Srikanth Srinivasan, Madhu Sudan, Sophus Valentin Willumsgaard
TL;DR
The paper resolves the longstanding goal of proving the PCP theorem with only one composition step by introducing a unified algebraic PCP framework that replaces the classic sum-check with zero-on-variety tests tied to Gröbner-bases concepts. Central to the construction is a new connection between Gröbner generating sets and zero-on-variety testing, enabling access to a broader class of varieties (including those near Hamming balls) with constant locality. The authors present a main theorem showing that 3-COLOR on a variety $V$ with extension degree $d$ and Gröbner complexity $k$ lies in $\mathsf{PCP}_{1,1/2}[c (k+m) \log q, \ell, c d \log q]$, with several explicit instantiations yielding varying randomness, query, and proof-size trade-offs, all composable via robust assignment testers. By composing two such PCPs, they obtain the PCP theorem with a single composition step, revealing deep connections between algebraic geometry (Gröbner theory) and complexity theory and opening avenues for further algebraic approaches in complexity reductions.
Abstract
All known proofs of the PCP theorem rely on multiple "composition" steps, where PCPs over large alphabets are turned into PCPs over much smaller alphabets at a (relatively) small price in the soundness error of the PCP. Algebraic proofs, starting with the work of Arora, Lund, Motwani, Sudan, and Szegedy use at least 2 such composition steps, whereas the "Gap amplification" proof of Dinur uses $Θ(\log n)$ such composition steps. In this work, we present the first PCP construction using just one composition step. The key ingredient, missing in previous work and finally supplied in this paper, is a basic PCP (of Proximity) of size $2^{n^ε}$, for any $ε> 0$, that makes $O_ε(1)$ queries. At the core of our new construction is a new class of alternatives to "sum-check" protocols. As used in past PCPs, these provide a method by which to verify that an $m$-variate degree $d$ polynomial $P$ evaluates to zero at every point of some set $S \subseteq \mathbb{F}_q^m$. Previous works had shown how to check this condition for sets of the form $S = H^m$ using $O(m)$ queries with alphabet $\mathbb{F}_q^d$ assuming $d \geq |H|$. Our work improves this basic protocol in two ways: First we extend it to broader classes of sets $S$ (ones closer to Hamming balls rather than cubes). Second, it reduces the number of queries from $O(m)$ to an absolute constant for the settings of $S$ we consider. Specifically when $S = (\{0,1\}^{m/c}_{\leq 1})^c$, we give such an alternate to the sum-check protocol with $O(1)$ queries with alphabet $\mathbb{F}_q^{O(c+d)}$, using proofs of size $q^{O(m^2/c)}$. Our new protocols use insights from the powerful theory of Gröbner bases to extend previously known protocols to these new settings with surprising ease. In doing so, they highlight why these theories from algebra may be of further use in complexity theory.