Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Massively Parallel Reductions in Multivariate Polynomial Systems: Bridging the Symbolic Preprocessing Gap on GPGPU Architectures

Chandrasekhar Gokavarapu

TL;DR

The paper identifies symbolic preprocessing as the primary bottleneck in scalable Gröbner basis computation on GPUs, reframing S-polynomial reductions as kernel computations of a structured sparse Macaulay matrix over $\mathbb{F}_p$. It proposes a GPU-centric two-pass compilation (dynamic-to-static) that converts dynamic symbolic data into static, write-once layouts via FBSP, monomial-key encodings, and bulk joins, enabling efficient numeric kernels. It introduces a principled architecture combining FBSP with register-resident modular arithmetic, and compares Elimination (PSGE) and Krylov (Block Wiedemann/Lanczos) strategies as the numeric phase, with correctness guarantees for SP/FBSP and an implementation blueprint. The work bridges algebraic syzygy theory with SIMT hardware constraints, providing a formal framework and concrete compiler-like steps toward massively parallel reductions in multivariate polynomial systems, along with open problems on sparsity control and signature-aware scheduling for GPU execution.

Abstract

Gröbner basis computation over multivariate polynomial rings remains one of the most powerful yet computationally hostile primitives in symbolic computation. While modern algorithms (Faugère-type F4 and signature-based F5) reduce many instances to large sparse linear algebra over finite fields, their dominant cost is not merely elimination but the symbolic preprocessing that constructs Macaulay-style matrices whose rows encode shifted reducers. This phase is characterized by dynamic combinatorics (monomial discovery, sparse row assembly, and deduplication) and is typically memory-latency bound, resisting naive parallelization. This article develops a rigorous synthesis that reframes S-polynomial reduction as syzygy discovery: row construction is a structured map from module relations to the kernel of a massive, sparse, highly non-random Macaulay matrix A over Fp. Building on this viewpoint, we propose a GPU-targeted architecture that (i) converts dynamic symbolic data structures into static, two-pass allocations via prefix-sum planning; (ii) enforces coalesced memory access through structure-of-arrays polynomial layouts and sorted monomial dictionaries; and (iii) integrates finite-field arithmetic kernels (Montgomery/Barrett-style reduction) at register granularity. On the linear-algebra side, we explore the transition from classical Gaussian elimination to parallel structured Gaussian elimination (PSGE) and to Krylov-type kernel solvers (Block Wiedemann/Lanczos) that better match GPU throughput while controlling fill-in. The result is a principled bridge between algebraic syzygy theory and SIMT hardware constraints, isolating the true bottleneck and providing a pathway to massively parallel reductions for multivariate polynomial systems.

Massively Parallel Reductions in Multivariate Polynomial Systems: Bridging the Symbolic Preprocessing Gap on GPGPU Architectures

TL;DR

The paper identifies symbolic preprocessing as the primary bottleneck in scalable Gröbner basis computation on GPUs, reframing S-polynomial reductions as kernel computations of a structured sparse Macaulay matrix over . It proposes a GPU-centric two-pass compilation (dynamic-to-static) that converts dynamic symbolic data into static, write-once layouts via FBSP, monomial-key encodings, and bulk joins, enabling efficient numeric kernels. It introduces a principled architecture combining FBSP with register-resident modular arithmetic, and compares Elimination (PSGE) and Krylov (Block Wiedemann/Lanczos) strategies as the numeric phase, with correctness guarantees for SP/FBSP and an implementation blueprint. The work bridges algebraic syzygy theory with SIMT hardware constraints, providing a formal framework and concrete compiler-like steps toward massively parallel reductions in multivariate polynomial systems, along with open problems on sparsity control and signature-aware scheduling for GPU execution.

Abstract

Gröbner basis computation over multivariate polynomial rings remains one of the most powerful yet computationally hostile primitives in symbolic computation. While modern algorithms (Faugère-type F4 and signature-based F5) reduce many instances to large sparse linear algebra over finite fields, their dominant cost is not merely elimination but the symbolic preprocessing that constructs Macaulay-style matrices whose rows encode shifted reducers. This phase is characterized by dynamic combinatorics (monomial discovery, sparse row assembly, and deduplication) and is typically memory-latency bound, resisting naive parallelization. This article develops a rigorous synthesis that reframes S-polynomial reduction as syzygy discovery: row construction is a structured map from module relations to the kernel of a massive, sparse, highly non-random Macaulay matrix A over Fp. Building on this viewpoint, we propose a GPU-targeted architecture that (i) converts dynamic symbolic data structures into static, two-pass allocations via prefix-sum planning; (ii) enforces coalesced memory access through structure-of-arrays polynomial layouts and sorted monomial dictionaries; and (iii) integrates finite-field arithmetic kernels (Montgomery/Barrett-style reduction) at register granularity. On the linear-algebra side, we explore the transition from classical Gaussian elimination to parallel structured Gaussian elimination (PSGE) and to Krylov-type kernel solvers (Block Wiedemann/Lanczos) that better match GPU throughput while controlling fill-in. The result is a principled bridge between algebraic syzygy theory and SIMT hardware constraints, isolating the true bottleneck and providing a pathway to massively parallel reductions for multivariate polynomial systems.
Paper Structure (85 sections, 9 theorems, 44 equations, 1 figure)

This paper contains 85 sections, 9 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction: Complexity, Structure, and Hardware Mismatch
  2. Thesis.
  3. Contributions (conceptual).
  4. Algebraic Foundations: From Buchberger to Syzygies, $F_4$, and $F_5$
  5. Monomials, leading data, and $S$-polynomials
  6. Syzygy module viewpoint and signatures
  7. $F_4$ as batched module reduction; $F_5$ as signature-constrained reduction
  8. Reduction as Kernel Computation of a Sparse Macaulay Matrix
  9. From shifted reducers to a coefficient operator
  10. Kernel vectors as (truncated) syzygies
  11. Interpretation for elimination and reduction
  12. Parallelism in Multivariate Polynomial System Solving
  13. Problem statement: symbolic preprocessing as the parallelism choke point
  14. A pipeline decomposition and where GPU parallelism actually lives
  15. An algebra--hardware interface: symbolic work as compilation to static arrays
  16. ...and 70 more sections

Key Result

Proposition 3.2

Assume $\mathcal{T}$ contains $\bigcup_{i=1}^r \operatorname{supp}(t_i g_{k_i})$. Let $A$ be as in Definition def:macaulay-batch. Then, for $v\in\mathbb{F}_p^r$,

Theorems & Definitions (29)

  • Definition 2.1: $S$-polynomial
  • Remark 2.2: From $S$-polynomials to module reduction
  • Definition 3.1: Sparse Macaulay matrix for a batch
  • Proposition 3.2: Kernel equals syzygies under dictionary closure
  • Remark 3.3: Truncation and why symbolic preprocessing matters
  • Definition 5.1: Batch specification
  • Definition 5.2: Symbolic preprocessing operator
  • Remark 5.3: Non-random sparsity and the “right” reformulation
  • Remark 8.1: Lazy reduction window
  • Remark 8.2: GPU-tuned Barrett variants
  • ...and 19 more