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<p>The computation of minimal generating sets for homogeneous polynomial ideals was resolved in the constant coefficient case by Schreyer’s syzygy theorem (1980). Still, it remained open for parametric ideals until the recent <span>MGSystem</span> algorithm (introduced by Dehghani Darmian). We present <span>Improved</span>-<span>MGSystem</span>, a significant enhancement that replaces Gröbner systems computations with parametric linear algebra techniques. Our approach reduces computational complexity through optimized algebraic operations while preserving completeness in computing minimal generator systems and enhancing scalability for parametric ideals. Our <Emphasis FontCategory="NonProportional">Maple</Emphasis> implementation demonstrates practical efficiency gains of 10–30% in computation time and 12–20% in memory usage across benchmark examples.</p>

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Efficient Computation of Minimal Generating Sets for Parametric Polynomial Ideals Using Gaussian Elimination

  • 期刊论文

  • 发表日期：

    2026年6月30日

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  • Mahdi Dehghani Darmian

摘要

The computation of minimal generating sets for homogeneous polynomial ideals was resolved in the constant coefficient case by Schreyer’s syzygy theorem (1980). Still, it remained open for parametric ideals until the recent MGSystem algorithm (introduced by Dehghani Darmian). We present Improved-MGSystem, a significant enhancement that replaces Gröbner systems computations with parametric linear algebra techniques. Our approach reduces computational complexity through optimized algebraic operations while preserving completeness in computing minimal generator systems and enhancing scalability for parametric ideals. Our Maple implementation demonstrates practical efficiency gains of 10–30% in computation time and 12–20% in memory usage across benchmark examples.