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Improved polynomial remainder sequences for Ore polynomials.

Maximilian Jaroschek1

  • 1Research Institute for Symbolic Computation, Johannes Kepler University, A4040 Linz, Austria.

Journal of Symbolic Computation
|November 3, 2015
PubMed
Summary
This summary is machine-generated.

This study optimizes polynomial remainder sequences for faster Euclidean algorithm computations. We introduce new methods for Ore polynomials, reducing coefficient size and improving efficiency.

Keywords:
Greatest common right divisorOre polynomialsPolynomial remainder sequencesSubresultants

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Area of Science:

  • Algebraic algorithms
  • Computational mathematics

Background:

  • Polynomial remainder sequences are crucial for the Euclidean algorithm.
  • The efficiency of these algorithms depends on the size of intermediate coefficients.
  • Existing methods like subresultant sequences offer optimal coefficients in generic cases but can be suboptimal for applied inputs.

Purpose of the Study:

  • To generalize coefficient-size reduction techniques for polynomial remainder sequences.
  • To apply these generalizations to Ore polynomials, a non-commutative setting.
  • To derive new bounds for minimal coefficient size and provide novel proofs for commutative cases.

Main Methods:

  • Generalization of two subresultant sequence improvements.
  • Application of these methods to Ore polynomials.
  • Derivation of new bounds on coefficient sizes.

Main Results:

  • Successfully generalized subresultant sequence improvements to Ore polynomials.
  • Derived a new bound for the minimal coefficient size in these sequences.
  • Provided a new perspective on extraneous factors in commutative polynomial remainder sequences.

Conclusions:

  • The generalized methods offer improved efficiency for Euclidean algorithm computations involving Ore polynomials.
  • The new bounds contribute to a better understanding of coefficient size in polynomial remainder sequences.
  • The work offers a novel viewpoint on the underlying mathematical structures in both commutative and non-commutative polynomial algebra.