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

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.