dorsal/arxiv
View SchemaMassively Parallel Reductions in Multivariate Polynomial Systems: Bridging the Symbolic Preprocessing Gap on GPGPU Architectures
| Authors | Chandrasekhar Gokavarapu |
|---|---|
| Categories | |
| ArXiv ID | 2601.06765vv1 |
| URL | https://arxiv.org/abs/2601.06765 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
Gr\"obner basis computation over multivariate polynomial rings remains one of the most powerful yet computationally hostile primitives in symbolic computation. While modern algorithms (Faug\`ere-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.
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"abstract": "Gr\\\"obner basis computation over multivariate polynomial rings remains one of the most powerful yet computationally hostile primitives in symbolic computation. While modern algorithms (Faug\\`ere-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.\n 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.\n 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.",
"arxiv_id": "2601.06765",
"authors": [
"Chandrasekhar Gokavarapu"
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"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Massively Parallel Reductions in Multivariate Polynomial Systems: Bridging the Symbolic Preprocessing Gap on GPGPU Architectures",
"url": "https://arxiv.org/abs/2601.06765",
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