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Computational Arithmetic Geometry I: Sentences Nearly in the Polynomial Hierarchy

J. Maurice Rojas

TL;DR

This work analyzes the border between decidability and complexity for Diophantine problems in low dimensions, showing that generic instances of an exists-forall-exists prefix fall into $\mathbf{coNP}$ while GRH yields $\mathbf{PP^{NP^{NP}}}$ routines for zero-dimensional rational feasibility. It develops a framework connecting modular reductions, toric resultants, and Galois theory to relate polynomial systems over $\mathbb{C},\mathbb{Q},\mathbb{Z}/p\mathbb{Z}$, deriving explicit prime-density bounds and conditional complexity results. The paper also provides unconditional $\mathbf{PSPACE}$ bounds for emptiness/finiteness and describes conditional amendments to $\mathbf{AM}$ and $\mathbf{P^{NP^{NP}}}$ when GRH holds, together with effective quantitative tools (height/discriminant bounds, primitive-element theorems) that may be of independent interest. Overall, it builds a bridge between arithmetic geometry and computational complexity, with practical implications for cryptographic constructions that avoid certain Diophantine instances and a detailed toolkit for analyzing diophantine predicates via algebraic geometry.

Abstract

We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: (I) Given a polynomial f in Z[v,x,y], decide the sentence \exists v \forall x \exists y f(v,x,y)=0, with all three quantifiers ranging over N (or Z). (II) Given polynomials f_1,...,f_m in Z[x_1,...,x_n] with m>=n, decide if there is a rational solution to f_1=...=f_m=0. We show that, for almost all inputs, problem (I) can be done within coNP. The decidability of problem (I), over N and Z, was previously unknown. We also show that the Generalized Riemann Hypothesis (GRH) implies that, for almost all inputs, problem (II) can be done via within the complexity class PP^{NP^NP}, i.e., within the third level of the polynomial hierarchy. The decidability of problem (II), even in the case m=n=2, remains open in general. Along the way, we prove results relating polynomial system solving over C, Q, and Z/pZ. We also prove a result on Galois groups associated to sparse polynomial systems which may be of independent interest. A practical observation is that the aforementioned Diophantine problems should perhaps be avoided in the construction of crypto-systems.

Computational Arithmetic Geometry I: Sentences Nearly in the Polynomial Hierarchy

TL;DR

This work analyzes the border between decidability and complexity for Diophantine problems in low dimensions, showing that generic instances of an exists-forall-exists prefix fall into while GRH yields routines for zero-dimensional rational feasibility. It develops a framework connecting modular reductions, toric resultants, and Galois theory to relate polynomial systems over , deriving explicit prime-density bounds and conditional complexity results. The paper also provides unconditional bounds for emptiness/finiteness and describes conditional amendments to and when GRH holds, together with effective quantitative tools (height/discriminant bounds, primitive-element theorems) that may be of independent interest. Overall, it builds a bridge between arithmetic geometry and computational complexity, with practical implications for cryptographic constructions that avoid certain Diophantine instances and a detailed toolkit for analyzing diophantine predicates via algebraic geometry.

Abstract

We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: (I) Given a polynomial f in Z[v,x,y], decide the sentence \exists v \forall x \exists y f(v,x,y)=0, with all three quantifiers ranging over N (or Z). (II) Given polynomials f_1,...,f_m in Z[x_1,...,x_n] with m>=n, decide if there is a rational solution to f_1=...=f_m=0. We show that, for almost all inputs, problem (I) can be done within coNP. The decidability of problem (I), over N and Z, was previously unknown. We also show that the Generalized Riemann Hypothesis (GRH) implies that, for almost all inputs, problem (II) can be done via within the complexity class PP^{NP^NP}, i.e., within the third level of the polynomial hierarchy. The decidability of problem (II), even in the case m=n=2, remains open in general. Along the way, we prove results relating polynomial system solving over C, Q, and Z/pZ. We also prove a result on Galois groups associated to sparse polynomial systems which may be of independent interest. A practical observation is that the aforementioned Diophantine problems should perhaps be avoided in the construction of crypto-systems.

Paper Structure

This paper contains 10 sections, 16 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction and Main Results
  2. Dimensions One and Two
  3. Dimension Zero
  4. Background Tools
  5. Genus Zero Varieties and the Proof of Theorem \ref{['MAIN:PEPPER']}
  6. Prime Distribution: Proving Theorem \ref{['MAIN:START']}
  7. Proving Assertion (2) of Theorem \ref{['MAIN:START']}
  8. Proving Assertion (1) of Theorem \ref{['MAIN:START']}
  9. The Proof of Theorem \ref{['MAIN:RIEMANN']}
  10. The Proof of Theorem \ref{['thm:galois']}

Key Result

Theorem 1

Fix the Newton polytope $P$ of a polynomial $f\!\in\!\mathbb{Z}[v,x,y]$ and suppose that $P$ has at least one integral point in its interior. Assume further that we measure input size via the dense encoding. Then, for a generic choice of coefficients depending only on $P$, we can decide whether $\ex

Theorems & Definitions (38)

  • Theorem 1
  • Remark 1
  • Example 1
  • Definition 1
  • Definition 2
  • Theorem 2
  • Remark 2
  • Remark 3
  • Theorem 3
  • Remark 4
  • ...and 28 more