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

Abhibhav Garg, Rafael Oliveira, Nitin Saxena

TL;DR

This work tackles the problem of testing primality for ideals generated by polynomials, focusing on radical and equidimensional Cohen–Macaulay classes. By combining effective base-change theorems, Bertini-type reductions, and Koiran-style interactive protocols under GRH, the authors place ideal primality testing in $coAM$ and hence in $\Sigma_{3}^{p} \cap \Pi_{3}^{p}$ for radical and equidimensional CM ideals, approaching known lower bounds. They also obtain a PSPACE algorithm for primality testing in the CM setting, and provide a detailed height-and-degree framework to manage arithmetic-expressibility across fields, ultimately enabling observable witnesses via Lang-Weil bounds and prime-density arguments. The results substantially tighten the complexity gap for primality testing in these natural classes and open avenues for extending such protocols beyond the current assumptions and classes. The methods have potential implications for computational algebraic geometry, complexity theory, and interactive proof systems in number-theoretic/algebraic contexts.

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 $Σ_3^p \cap Π_3^p$. 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.

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

TL;DR

This work tackles the problem of testing primality for ideals generated by polynomials, focusing on radical and equidimensional Cohen–Macaulay classes. By combining effective base-change theorems, Bertini-type reductions, and Koiran-style interactive protocols under GRH, the authors place ideal primality testing in and hence in for radical and equidimensional CM ideals, approaching known lower bounds. They also obtain a PSPACE algorithm for primality testing in the CM setting, and provide a detailed height-and-degree framework to manage arithmetic-expressibility across fields, ultimately enabling observable witnesses via Lang-Weil bounds and prime-density arguments. The results substantially tighten the complexity gap for primality testing in these natural classes and open avenues for extending such protocols beyond the current assumptions and classes. The methods have potential implications for computational algebraic geometry, complexity theory, and interactive proof systems in number-theoretic/algebraic contexts.

Abstract

A central question in mathematics and computer science is the question of determining whether a given ideal is prime, which geometrically corresponds to the zero set of , denoted , being irreducible. The case of principal ideals (i.e., ) 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 , 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 (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 . 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.

Paper Structure

This paper contains 33 sections, 57 theorems, 27 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Summary of contributions.
  3. Proof overview
  4. Dimension of $V_{p}$.
  5. Irreducibility of $V_{p}$.
  6. Reducibility of $V_{p}$ and ${\mathbb F}_{p}$-definability of some of its components.
  7. Putting it all together
  8. Case 1: $V$ has an isolated point as a solution.
  9. Case 2: $V$ has two components of dimension $1$.
  10. Acknowledgements
  11. Preliminaries
  12. Notation
  13. Results from complexity theory
  14. Algebraic circuits
  15. Results from linear algebra
  16. ...and 18 more sections

Key Result

Theorem 1.2

Let $C$ be an algebraic circuit of size $s$ with integer constants that computes $f_{1}, \dots, f_{m} \in {\mathbb Z}\left[{x_{1}, \dots, x_{n}}\right]$. If $I := \left({f_{1}, \dots, f_{m}}\right)$ is either radical or equidimensional Cohen-Macaulay, and if the dimension of $I$ is given, then the c

Theorems & Definitions (88)

  • Theorem 1.2: Interactive protocols for primality
  • Theorem 1.3: Primality testing in PH
  • Lemma 2.1
  • Corollary 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Proposition 2.7: coNP-hardness of ideal primality testing
  • proof
  • ...and 78 more