Primes via Zeros: Interactive Proofs for Testing Primality of Natural Classes of Ideals

2025-03-25

AI Summary

This paper addresses the prime ideal testing problem in algebraic geometry: given an ideal $I$, determine whether its zero set $Z(I)$ is irreducible (i.e., whether $I$ is prime). While known to be EXPSPACE-hard in general—and equivalent to absolute irreducibility testing for principal ideals—the problem remains computationally challenging. We focus on two important classes: radical ideals and equidimensional Cohen–Macaulay ideals. Leveraging a synthesis of interactive proof systems, algebraic complexity theory, commutative algebra, and zero-set geometry, we achieve the first $Sigma_3^p cap Pi_3^p$ algorithm for primality testing under the Generalized Riemann Hypothesis (GRH), and the first unconditional PSPACE algorithm. Our approach yields exponential improvements in both time and space complexity for equidimensional Cohen–Macaulay ideals, substantially narrowing the theoretical complexity gap and approaching tight bounds.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

A central question in mathematics and computer science is the question of determining whether a given ideal $I$ is prime, which geometrically corresponds to the zero set of $I$, denoted $Z(I)$, being irreducible. The case of principal ideals (i.e., $m=1$) corresponds to the more familiar absolute irreducibility testing of polynomials, where the seminal work of (Kaltofen 1995) yields a randomized, polynomial time algorithm for this problem. However, when $m>1$, the complexity of the primality testing problem seems much harder. The current best algorithms for this problem are only known to be in EXPSPACE. In this work, we significantly reduce the complexity-theoretic gap for the ideal primality testing problem for the important families of ideals $I$ (namely, radical ideals and equidimensional Cohen-Macaulay ideals). For these classes of ideals, assuming the Generalized Riemann Hypothesis, we show that primality testing lies in $Sigma_3^p cap Pi_3^p$. This significantly improves the upper bound for these classes, approaching their lower bound, as the primality testing problem is coNP-hard for these classes of ideals. Another consequence of our results is that for equidimensional Cohen-Macaulay ideals, we get the first PSPACE algorithm for primality testing, exponentially improving the space and time complexity of prior known algorithms.

Problem

• Determining primality of radical and Cohen-Macaulay ideals
• Reducing complexity gap for ideal primality testing
• Improving space and time complexity for primality algorithms

Innovation

• Reduces complexity for radical and Cohen-Macaulay ideals
• Uses Generalized Riemann Hypothesis for primality testing
• Achieves PSPACE algorithm for Cohen-Macaulay ideals